{"id":95,"date":"2021-12-15T09:53:27","date_gmt":"2021-12-15T09:53:27","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/das-uneigentliche-integral\/"},"modified":"2021-12-15T09:53:27","modified_gmt":"2021-12-15T09:53:27","slug":"das-uneigentliche-integral","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/das-uneigentliche-integral\/","title":{"raw":"Das uneigentliche Integral","rendered":"Das uneigentliche Integral"},"content":{"raw":"\n<style>.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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\n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z0842d6de6d17\" class=\"sectionHead\"><span class=\"titlemark\">9.3 <\/span> <a id=\"x1-2720003\"><\/a>Das uneigentliche Integral<\/h3> <p class=\"noindent\">Wir wollen nun den Begriff des Riemann-Integrals auf mehrere Arten erweitern. <a id=\"x1-272001r271\"><\/a> <\/p> <h4 id=\"z250aeed47237\" class=\"subsectionHead\"><span class=\"titlemark\">9.3.1 <\/span> <a id=\"x1-2730001\"><\/a>Uneigentliche Integrationsgrenzen<\/h4> <p class=\"noindent\">F\u00fcr <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> und eine komplexwertige Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> mit <math display=\"inline\"><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>a<\/mi><\/math> definieren wir das <span class=\"ecbx-1095\">uneigentliche Integral<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">falls der Grenzwert existiert. Weiter sagen wir, dass das uneigentliche Integral <span class=\"ecbx-1095\">konvergiert<\/span>, falls der obige Grenzwert in <math display=\"inline\"><mi>\u2102<\/mi><\/math> existiert. Ansonsten nennen wir das uneigentliche Integral <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecbx-1095\">divergent<\/span>. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z89b32aaffd5e\"> <a id=\"x1-273001r29\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.29.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Es gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi class=\"qopname\">arctan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z2de06ddd4a0e\"> <a id=\"x1-273002r30\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.30.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Es gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b1<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\u03b1<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac> <\/mtd><mtd class=\"array\" columnalign=\"center\"> <mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u221e<\/mi><\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Insbesondere ist das obige uneigentliche Integral genau dann konvergent, wenn<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">In der Tat ist<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b1<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><msup><mrow><mi>b<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>\u03b1<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mn>1<\/mn> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><msup><mrow><mi>b<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u221e<\/mi><\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>0<\/mn> <\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zffa448e2de49\"> <a id=\"x1-273003r31\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.31.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b1<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Uneigentliche Integrale der Form <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> sind \u00e4hnlich definiert. Ebenso definieren wir f\u00fcr eine komplexwertige Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> mit <math display=\"inline\"><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> das uneigentliche Integral                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>a<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">falls beide Grenzwerte existieren. Wir m\u00f6chten dazu anmerken, dass man sich bewusst dazu entscheidet, die Bewegungen gegen <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/math> respektive <math display=\"inline\"> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u221e<\/mi><\/math> komplett getrennt zu behandeln. Alles andere w\u00fcrde zu komischen Ph\u00e4nomenen f\u00fchren, wie folgendes Beispiel zeigt. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z9884555ee5dd\"> <a id=\"x1-273004r32\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.32.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Das uneigentliche Integral <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">existiert nicht, da <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msubsup><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">sowie <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">nicht existieren. Wir bemerken aber, dass der Grenzwert <\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>c<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msubsup><mrow><mi class=\"qopname\">\u222b  <\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>c<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>c<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mn>0<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">existieren w<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rde aber<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>c<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msubsup><mrow><mi class=\"qopname\">\u222b  <\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>c<\/mi><\/mrow><mrow><mi>c<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>c<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>c<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>c<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re.<\/span> <\/p> <\/div> <p class=\"indent\">Wie wir nun besprechen wollen, haben uneigentlichen Integrale oft sehr enge Beziehungen zu Reihen. Genau wie bei Folgen und Reihen (siehe Satz <a href=\"..\/..\/chapter\/reelle-folgen#x1-158001r5\">6.5<\/a> und Proposition <a href=\"..\/..\/chapter\/reihen#x1-188001r11\">7.11<\/a>) ist es bei uneigentlichen Integralen nicht-negativer Funktionen einfacher \u00fcber Konvergenz zu entscheiden. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"zd7d8cccbe1ff\"> <a id=\"x1-273005r33\"><\/a> <span class=\"ecbx-1095\">Lemma 9.33.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine nicht-negative<\/span> <span class=\"ecti-1095\">Funktion mit <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>a<\/mi><\/math><span class=\"ecti-1095\">. Entweder konvergiert das<\/span> <span class=\"ecti-1095\">uneigentliche Integral <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">oder es divergiert gegen Unendlich. In beiden F<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">llen gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/munderover><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Die Funktion <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> ist monoton wachsend. Wenn das Supremum <math display=\"inline\"><mi>S<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mi class=\"qopname\">\u222b  <\/mi><mo>  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/munderover><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> Unendlich ist, dann divergiert das uneigentliche Integral auf Grund der Monotonie gegen Unendlich. Wenn <math display=\"inline\"><mi>S<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/math> ist, dann gibt es zu <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>a<\/mi><\/math> mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>S<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>B<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>S<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Insbesondere gilt f\u00fcr <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>B<\/mi><\/math> auf Grund der Monotonie und der Definition von&nbsp;<math display=\"inline\"><mi>S<\/mi><\/math> dieselbe Ungleichung auch f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dies beweist die Konvergenz des uneigentlichen Integrals. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z90c74d86a6e0\"> <a id=\"x1-273006r34\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.34 <\/span>(Gaussche Glockenkurve)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir wollen das uneigentliche Integral<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">besprechen, wobei die Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><\/math> <span class=\"ecti-1095\">die <\/span>Gaussche Glockenkurve <span class=\"ecti-1095\">genannt wird. Auf Grund von Lemma <\/span><a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-273005r33\"><span class=\"ecti-1095\">9.33<\/span><\/a> <span class=\"ecti-1095\">reicht es aus eine<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">Majorantenfunktion<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">zu finden, die ein konvergentes uneigentliches Integral definiert. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gilt zum<\/span> <span class=\"ecti-1095\">Beispiel <\/span><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>x<\/mi><\/math> <span class=\"ecti-1095\">und daher <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">woraus<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">folgt. Dies zeigt die Konvergenz des zweiten uneigentlichen Integrals, auf Grund der Symmetrie der Funktion ist daher<\/span> <span class=\"ecti-1095\">auch <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">konvergent. Wir<\/span> <span class=\"ecti-1095\">werden den Wert <\/span><math display=\"inline\"><mi>I<\/mi><\/math> <span class=\"ecti-1095\">dieses Integral erst im zweiten Semester berechnen k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen. Doch wollen wir noch erw<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hnen, dass<\/span> <span class=\"ecti-1095\">die streng monoton wachsende Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03a6<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>I<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">die Verteilungsfunktion der Normalverteilung mit Erwartungswert<\/span> <math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">Standardabweichung <\/span><math display=\"inline\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>2<\/mn><\/mrow><\/msqrt><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">genannt wird. Diese Funktion l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich ebenso wie die Funktionen aus Abschnitt <\/span><a href=\"..\/..\/chapter\/integrationsmethoden#x1-2710008\"><span class=\"ecti-1095\">9.2.8<\/span><\/a> <span class=\"ecti-1095\">nicht durch<\/span> <span class=\"ecti-1095\">die sonst <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blichen Funktionen ausdr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">cken und ist in der Wahrscheinlichkeitsrechnung, der Statstik<\/span> <span class=\"ecti-1095\">und in vielen Anwendungen von fundamentaler Bedeutung.<\/span> <\/p> <\/div> <p class=\"indent\">Der folgende Satz charakterisiert nun Konvergenz uneigentlicher Integrale wie in obigem Lemma durch Konvergenz von Reihen (auf hinreichende und notwendige Weise). <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z6364452cc400\"> <a id=\"x1-273007r35\"><\/a> <span class=\"ecbx-1095\">Satz 9.35 <\/span>(Integraltest f\u00fcr Reihen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine monoton fallende Funktion. Dann gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>2<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Insbesondere konvergiert die Reihe <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">genau dann, wenn das uneigentliche Integral<\/span> <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">konvergiert. Dies gilt analog<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r Integrale der Form<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>N<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Wir bemerken, dass auf Grund der Monotonieannahme an <math display=\"inline\"><mi>f<\/mi><\/math> in obigem Satz die Eigenschaft <math display=\"inline\"><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> erf\u00fcllt ist nach Satz <a href=\"..\/..\/chapter\/integrierbarkeit-monotoner-funktionen#x1-121001r31\">4.31<\/a>. <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">F\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und einen beliebigen Zwischenpunkt <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> gilt nach Monotonie von <math display=\"inline\"><mi>f<\/mi><\/math> die Ungleichung <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und somit                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was auch in folgendem Bild ersichtlich ist. <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=517&amp;url=Pictures\/fundsatz\/integraltest.pdf\"><img id=\"z79d40f8f78a0\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/fundsatz\/integraltest.svg\" width=\"517\"><\/div>  <\/div> <p class=\"noindent\">Nach Summation von <math display=\"inline\"><mn>1<\/mn><\/math> bis <math display=\"inline\"><mi>n<\/mi><\/math> erh\u00e4lt man mit Intervalladditivit\u00e4t des Riemann-Integrals <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/munderover><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u2113<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Falls das uneigentliche Integral <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> existiert, dann folgt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/munderover><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u2113<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Daher ist die monoton wachsende Folge <math display=\"inline\"><msub><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u2113<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> nach oben beschr\u00e4nkt und konvergiert somit nach Satz&nbsp;<a href=\"..\/..\/chapter\/reelle-folgen#x1-158001r5\">6.5<\/a>. Insbesondere gilt auch <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>2<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Falls <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> konvergiert, dann ist f\u00fcr <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> und <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">\u230a<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">\u230b<\/mo><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Nach Lemma <a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-273005r33\">9.33<\/a> ist somit das uneigentliche Integral <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> konvergent und durch die Zahl <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> beschr\u00e4nkt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z74f0ac672572\"> <a id=\"x1-273008r36\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.36 <\/span>(Divergenzrate der harmonischen Reihe)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Verwenden Sie obigen Satz, um den <\/span><math display=\"inline\"><mi>p<\/mi><\/math><span class=\"ecti-1095\">-Test<\/span> <span class=\"ecti-1095\">in Beispiel <\/span><a href=\"..\/..\/chapter\/reihen#x1-188007r17\"><span class=\"ecti-1095\">7.17<\/span><\/a> <span class=\"ecti-1095\">zu erhalten. Imitieren Sie des Weiteren die Methodik im obigen Beweis von Satz<\/span> <a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-273007r35\"><span class=\"ecti-1095\">9.35<\/span><\/a><span class=\"ecti-1095\">, um die Divergenzrate<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>N<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>N<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die harmonische Reihe zu beweisen.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z373f96ed5f4e\"> <a id=\"x1-273009r37\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.37 <\/span>(Ein oszillierendes Integral)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Entscheiden                          Sie                          f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r                          welche<\/span> <math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">das                                          uneigentliche                                          Integral<\/span> <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>x<\/mi><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">konvergiert.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">verwenden Sie am besten die Substitution <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r das Integral<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>x<\/mi><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und ziehen je nach Fall auf geeignete Weise das Leibniz-Kriterium hinzu.<\/span><\/p><\/details>  <\/div> <a id=\"x1-273010r273\"><\/a> <h4 id=\"z26f8abc3f19c\" class=\"subsectionHead\"><span class=\"titlemark\">9.3.2 <\/span> <a id=\"x1-2740002\"><\/a>Das Integral \u00fcber unbeschr\u00e4nkte Funktionen<\/h4> <p class=\"noindent\">F\u00fcr <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>B<\/mi><\/math> in <math display=\"inline\"><mi>\u211d<\/mi><\/math> und eine Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> mit <math display=\"inline\"><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> definieren wir das <span class=\"ecbx-1095\">uneigentliche Integral<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>B<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>b<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mi>B<\/mi><\/mrow><\/munder><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">falls der Grenzwert existiert. <\/p><p class=\"indent\">Wie folgende \u00dcbung zeigt, steht diese Notation nicht im Widerspruch zum Riemann-Integral. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zc798c3f7fb1c\"> <a id=\"x1-274001r38\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.38 <\/span>(Kompatibilit\u00e4t)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">wie                                           oben.                                            Angenommen<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt.       Zeigen       Sie,       dass       das       uneigentliche       Integral<\/span> <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>B<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">existiert                 und                 gleich                 dem                 Riemann-Integral<\/span> <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>B<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">ist,                                                 wobei                                                 man<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">auf                beliebige                Weise                auf                den                Punkt<\/span> <math display=\"inline\"><mi>B<\/mi><\/math> <span class=\"ecti-1095\">erweitert.<\/span> <\/p> <\/div> <p class=\"indent\">F\u00fcr <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> in <math display=\"inline\"><mi>\u211d<\/mi><\/math> und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> eine Funktion mit <math display=\"inline\"><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> definieren wir analog das uneigentliche Integral                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>A<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>a<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mi>A<\/mi><\/mrow><\/munder><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zec193d64d3fa\"> <a id=\"x1-274002r39\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.39.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir berechnen <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">mittels<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>a<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>a<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>x<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>a<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">nach Beispiel <\/span><a href=\"..\/..\/chapter\/integrationsmethoden#x1-264002r15\"><span class=\"ecti-1095\">9.15<\/span><\/a> <span class=\"ecti-1095\">und Beispiel <\/span><a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-179001r44\"><span class=\"ecti-1095\">6.44<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <p class=\"indent\">Weitere uneigentliche Integrale f\u00fchren wir mittels Intervalladditivit\u00e4t auf obige uneigentliche Integrale zur\u00fcck. Wir \u00fcberlassen es Interessierten, sich hier einige M\u00f6glichkeiten auszudenken, und f\u00fchren stattdessen ein Beispiel vor. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zb5db8d71830c\"> <a id=\"x1-274003r40\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.40.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten das Integral <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dieses ist uneigentlich, da <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">auf jeder Umgebung von <\/span><math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">unbeschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist. Es gilt daher auf Grund der Definition in diesem Fall, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei beide Integrale rechts uneigentlich sind und das uneigentliche Integral<\/span> <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">per<\/span> <span class=\"ecti-1095\">Definition genau dann existiert, wenn die beiden Integrale rechts existieren. Des Weiteren<\/span> <span class=\"ecti-1095\">gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mn>0<\/mn><\/mrow><\/munder><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mn>0<\/mn><\/mrow><\/munder><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mn>0<\/mn><\/mrow><\/munder><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>a<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>a<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wodurch <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">nicht existiert (und wir<\/span> <span class=\"ecti-1095\">diesem auch nicht das Symbol <\/span><math display=\"inline\"><mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">oder <\/span><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">zuweisen).<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z364f41cb5548\"> <a id=\"x1-274004r41\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.41 <\/span>(Bogenl\u00e4nge des Kreises)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir wollen nochmals die Bogenl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge des Kreises berechnen. Doch verwenden wir diesmal die<\/span> <span class=\"ecti-1095\">Gleichung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/math> <span class=\"ecti-1095\">als Definition des oberen Halbkreises. Die Bogenl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge des Kreises ist demnach gegeben durch das<\/span> <span class=\"ecti-1095\">Integral<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>2<\/mn><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mn>4<\/mn><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac> <mrow> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mn>4<\/mn><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mn>1<\/mn><\/mrow><\/munder><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mn>4<\/mn><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mn>1<\/mn><\/mrow><\/munder><mi class=\"qopname\"> arcsin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mn>4<\/mn><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>\u03c0<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zf995c7500dbf\"> <a id=\"x1-274005r42\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.42.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mi>x<\/mi><\/mrow><\/msqrt><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msubsup><mi class=\"qopname\"> tan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z561aa2868a18\"> <a id=\"x1-274006r43\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.43 <\/span>(Absolute Konvergenz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">eine komplexwertige Funktion mit <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>a<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Wir nennen das uneigentliche Integral <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecbi-1095\">absolut konvergent<\/span><span class=\"ecti-1095\">, falls <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">konvergent ist. Zeigen Sie, dass absolute Konvergenz des uneigentlichen Integrals <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">auch die Konvergenz dieses Integrals impliziert.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z61b58e9b2904\"> <span class=\"ecti-1095\">Bemerkung.<\/span><\/h4> <p class=\"indent\">Zusammenfassend  haben  wir  bei  den  Definitionen  in  diesem  Abschnitt  bei  jedem Problempunkt  eines  m\u00f6glichen  Riemann-Integrals  einen  Grenzwert  verwendet,  um  den Integralbegriff  zu  erweiteren.  Dies  wirft  nochmals  die  Frage  auf,  ob  es  nicht  vielleicht einen Integralbegriff  gibt,  der  diese  und  auch  andere  bereits  erw\u00e4hnte  Probleme  des Riemann-Integrals  auf  nat\u00fcrliche  Art  und  Weise  l\u00f6st.  Diese  Frage  wird  im  zweiten Studienjahr des Mathematikstudiums mit der Theorie des Lebesgue-Integrals in der Vorlesung \u201eMass und Integral\u201c positiv beantwortet. <\/p> <\/div> <a id=\"x1-274007r274\"><\/a> <h4 id=\"z18b7aff68f95\" class=\"subsectionHead\"><span class=\"titlemark\">9.3.3 <\/span> <a id=\"x1-2750003\"><\/a>Die Gamma-Funktion<\/h4> <p class=\"noindent\">Die <span class=\"ecbx-1095\">Gamma-Funktion <\/span><math display=\"inline\"><mi>\u0393<\/mi><\/math> ist bei <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> durch das konvergente uneigentliche Integral                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-275001r6\" \/><mstyle class=\"maketag\"><mtext>(9.6)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">definiert. F\u00fcr&nbsp;<math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> ist dies aus zwei Gr\u00fcnden ein uneigentliches Integral und wir m\u00fcssen die Integrationsgrenzen&nbsp;<math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> und&nbsp;<math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi><\/math> getrennt untersuchen. F\u00fcr <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>a<\/mi><\/math> gilt jedoch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><\/mrow><\/mfrac><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <\/mrow><mrow> <mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><\/mrow><\/mfrac><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir setzen&nbsp;<math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> und erhalten <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>a<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><\/mrow><\/mfrac><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <\/mrow><mrow> <mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><\/mrow><\/mfrac><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><\/mrow><\/mfrac><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei das Integral rechts (f\u00fcr alle&nbsp;<math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math>) ein eigentliches Riemann-Integral darstellt. F\u00fcr&nbsp;<math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> erhalten wir <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><\/mrow><\/mfrac><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <\/mrow><mrow> <mn>1<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><\/mrow><\/mfrac><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><\/mrow><\/mfrac><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Um die Konvergenz von&nbsp;<math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> zu zeigen, wollen wir den Integraltest f\u00fcr Reihen in Satz&nbsp;<a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-273007r35\">9.35<\/a> verwenden. Die erste Vorraussetzung des Integraltests ist erf\u00fcllt, da die Funktion&nbsp;<math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/math> nicht-negativ ist. Die zweite Vorraussetzung ist, dass <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> monoton abnehmend sein soll. Wir berechnen daher die Ableitung und sehen, dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Da&nbsp;<math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> f\u00fcr alle&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>s<\/mi><\/math><\/span><span class=\"period\">,<\/span> sehen wir, dass diese Vorraussetzung zumindest f\u00fcr&nbsp;<math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">\u230a<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">\u230b<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> erf\u00fcllt ist. Es folgt daher, dass das Integral&nbsp;<math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>N<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> genau dann konvergiert wenn                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>N<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">konvergiert, was aber nach dem Quotientenkriterium in Korollar&nbsp;<a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193004r32\">7.32<\/a> (oder dem Wurzelkriterium in Korollar&nbsp;<a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193002r30\">7.30<\/a>) f\u00fcr Reihen in der Tat zutrifft. <\/p><p class=\"indent\">Addieren wir die beiden Integrale wieder und verwenden wir die Definition in (<a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-275001r6\">9.6<\/a>) so erhalten wir, dass&nbsp;<math display=\"inline\"><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle&nbsp;<math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> wohldefiniert ist und <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-275002r7\" \/><mstyle class=\"maketag\"><mtext>(9.7)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">erf\u00fcllt. <\/p><p class=\"indent\">Oft wird die Gamma-Funktion als eine Erweiterung der Fakult\u00e4tsfunktion auf <math display=\"inline\"><mi>\u2115<\/mi><\/math> bezeichnet. In der Tat gilt f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was wir hier noch beweisen m\u00f6chten. F\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u0393<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <\/mrow><mrow> <mn>0<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und somit folgt nach (<a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-275002r7\">9.7<\/a>) und Induktion <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir werden in der Fortsetzung dieser Vorlesung weitere Eigenschaften der Gamma-Funktion nachweisen k\u00f6nnen. Beispielsweise stellt sich heraus, dass die Gamma-Funktion glatt ist. Wir k\u00f6nnen dies hier aber nicht zeigen, da <math display=\"inline\"><mi>\u0393<\/mi><\/math> in (<a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-275001r6\">9.6<\/a>) durch ein sogenanntes Parameterintegral definiert ist. Auch k\u00f6nnen wir den Wert                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mi>x<\/mi><\/mrow><\/msqrt><\/mrow><\/mfrac><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mi>x<\/mi><\/mrow><\/msqrt><\/mrow><\/mfrac><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mn>2<\/mn><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msqrt><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msqrt><\/mrow><mrow><msqrt><mrow><mi>b<\/mi><\/mrow><\/msqrt><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">mit den uns bis jetzt bekannten Integrationsmethoden nicht berechnen, werden jedoch sp\u00e4ter mittels einem zweidimensionalen Integral sehen, dass dieser <math display=\"inline\"><msqrt><mrow><mi>\u03c0<\/mi><\/mrow><\/msqrt><\/math> ist. <\/p><p class=\"indent\">Die Gamma-Funktion enth\u00fcllt ihre wahre Sch\u00f6nheit erst wenn man komplexe Parameter&nbsp;<math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">}<\/mo><\/math> erlaubt. Wir laden Interessierte ein, diese Funktion in folgender \u00dcbung zu konstruieren. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z330a57a7da98\"> <a id=\"x1-275003r44\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.44 <\/span>(Challenge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>\u211c<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">definiert man<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>z<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-275004r8\" \/><mstyle class=\"maketag\"><mtext>(9.8)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">(a) Zeigen Sie, dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>z<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u211c<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">konvergiert.<\/span> <\/p><p class=\"noindent\"><span class=\"ecti-1095\">(b) Zeigen Sie, dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>z<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u211c<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">konvergiert.<\/span> <\/p><p class=\"noindent\"><span class=\"ecti-1095\">(c) Zeigen Sie<\/span><span class=\"ecti-1095\">&nbsp;<\/span>(<a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-275002r7\">9.7<\/a>) <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211c<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"noindent\"><span class=\"ecti-1095\">(d) Verwenden Sie<\/span><span class=\"ecti-1095\">&nbsp;<\/span>(<a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-275002r7\">9.7<\/a>) <span class=\"ecti-1095\">um rekursiv<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211c<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">oder<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u211c<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/math><span class=\"ecti-1095\">, <\/span><span class=\"ecti-1095\">\u2026, zu definieren,<\/span> <span class=\"ecti-1095\">so dass anschliessend<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">definiert ist und<\/span><span class=\"ecti-1095\">&nbsp;<\/span>(<a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-275002r7\">9.7<\/a>) <span class=\"ecti-1095\">auf dem ganzen Definitionsbereich erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Hinweis: Sie k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen in (a) und (b) <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-274006r43\"><span class=\"ecti-1095\">9.43<\/span><\/a> <span class=\"ecti-1095\">verwenden.<\/span> <\/p> <\/div> <p class=\"indent\">Hilbert (1862\u20131943) verwendete in seinem Artikel <span class=\"cite\">[<a href=\"#XHilberteundpi\">Hil93<\/a>]<\/span> von 1893 uneigentliche Integrale im Stile der Gamma-Funktion, um zu beweisen, dass <math display=\"inline\"><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/math> (wie erstmals von Hermite in 1873 bewiesen) und <math display=\"inline\"><mi>\u03c0<\/mi><\/math> (wie erstmals von Lindemann 1882 bewiesen) transzendent sind. Wir bemerken dabei, dass sich die blosse Irrationalit\u00e4t dieser Zahlen deutlich einfacher beweisen l\u00e4sst \u2013 f\u00fcr <math display=\"inline\"><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/math> gibt es hierzu eine \u00dcbung in Abschnitt <a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-2240002\">7.9.2<\/a> und f\u00fcr <math display=\"inline\"><mi>\u03c0<\/mi><\/math> eine \u00dcbung in Abschnitt <a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-2930002\">9.8.2<\/a>. Transzendenzbeweise sind jedoch im Allgemeinen deutlich schwieriger. Wie schwierige derartige Aussagen tats\u00e4chlich sind, illustriert vielleicht die Tatsache, dass immer noch nicht bekannt ist, ob <math display=\"inline\"><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><mo class=\"MathClass-bin\">+<\/mo><mi>\u03c0<\/mi><\/math> eine transzendente Zahl ist oder nicht. Hilbert\u2019s Beweis der Transzendenz von <math display=\"inline\"><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/math> und <math display=\"inline\"><mi>\u03c0<\/mi><\/math> ist mit den uns bisher bekannten Hilfsmitteln allerdings gut lesbar, weswegen wir Ihnen einen Blick auf diese Lekt\u00fcre und die damit verbundene Zeitreise empfehlen m\u00f6chten.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               <a id=\"x1-275005r272\"><\/a> <\/p> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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}\n.hline hr, .cline hr{border:none;border-top:1px solid black;}\n.equation-star td{text-align:center; vertical-align:middle; }\ntable.equation-star { width:100%; border-bottom-color: rgb(255,255,255); }\n#content table.equation-star, #content table.equation-star tbody tr td { border: 0px none rgb(255,255,255); }\nmtd.align-odd{margin-left:2em; text-align:right;}\nmtd.align-even{margin-right:2em; text-align:left;}\n.boxed{border: 1px solid black; padding-left:2px; padding-right:2px;}\n.rotatebox{display: inline-block;}\n.item-head{float:left;width:2em;clear:left;}\n.item-content{margin-left:2em;}\n .foreignobject {line-height:100%; font-size:120%; font-family:STIXgeneral,Times,Symbol,cmr10,CMSY10,CMEX10;padding:0; margin:0; text-align:center; }\nmath {vertical-align:baseline; line-height:100%; font-size:100%; font-family:STIXGeneral,Times,Symbol, cmr10,cmsy10,cmex10,cmmi10; font-style: normal; margin:0; padding:0; }\n\n.entry-title{display: none}\n\ndiv.newtheorem { margin-bottom: 2em; 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width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style scoped=\"scoped\">\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z0842d6de6d17\" class=\"sectionHead\"><span class=\"titlemark\">9.3 <\/span> <a id=\"x1-2720003\"><\/a>Das uneigentliche Integral<\/h3> <p class=\"noindent\">Wir wollen nun den Begriff des Riemann-Integrals auf mehrere Arten erweitern. <a id=\"x1-272001r271\"><\/a> <\/p> <h4 id=\"z250aeed47237\" class=\"subsectionHead\"><span class=\"titlemark\">9.3.1 <\/span> <a id=\"x1-2730001\"><\/a>Uneigentliche Integrationsgrenzen<\/h4> <p class=\"noindent\">F\u00fcr <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> und eine komplexwertige Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> mit <math display=\"inline\"><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>a<\/mi><\/math> definieren wir das <span class=\"ecbx-1095\">uneigentliche Integral<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">falls der Grenzwert existiert. Weiter sagen wir, dass das uneigentliche Integral <span class=\"ecbx-1095\">konvergiert<\/span>, falls der obige Grenzwert in <math display=\"inline\"><mi>\u2102<\/mi><\/math> existiert. Ansonsten nennen wir das uneigentliche Integral <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecbx-1095\">divergent<\/span>. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z89b32aaffd5e\"> <a id=\"x1-273001r29\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.29.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Es gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi class=\"qopname\">arctan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z2de06ddd4a0e\"> <a id=\"x1-273002r30\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.30.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Es gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b1<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\u03b1<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac> <\/mtd><mtd class=\"array\" columnalign=\"center\"> <mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u221e<\/mi><\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Insbesondere ist das obige uneigentliche Integral genau dann konvergent, wenn<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">In der Tat ist<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b1<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><msup><mrow><mi>b<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>\u03b1<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mn>1<\/mn> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><msup><mrow><mi>b<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u221e<\/mi><\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>0<\/mn> <\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zffa448e2de49\"> <a id=\"x1-273003r31\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.31.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b1<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Uneigentliche Integrale der Form <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> sind \u00e4hnlich definiert. Ebenso definieren wir f\u00fcr eine komplexwertige Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> mit <math display=\"inline\"><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> das uneigentliche Integral                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>a<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">falls beide Grenzwerte existieren. Wir m\u00f6chten dazu anmerken, dass man sich bewusst dazu entscheidet, die Bewegungen gegen <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/math> respektive <math display=\"inline\"> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u221e<\/mi><\/math> komplett getrennt zu behandeln. Alles andere w\u00fcrde zu komischen Ph\u00e4nomenen f\u00fchren, wie folgendes Beispiel zeigt. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z9884555ee5dd\"> <a id=\"x1-273004r32\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.32.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Das uneigentliche Integral <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">existiert nicht, da <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msubsup><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">sowie <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">nicht existieren. Wir bemerken aber, dass der Grenzwert <\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>c<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msubsup><mrow><mi class=\"qopname\">\u222b  <\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>c<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>c<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mn>0<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">existieren w<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rde aber<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>c<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msubsup><mrow><mi class=\"qopname\">\u222b  <\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>c<\/mi><\/mrow><mrow><mi>c<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>c<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>c<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>c<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re.<\/span> <\/p> <\/div> <p class=\"indent\">Wie wir nun besprechen wollen, haben uneigentlichen Integrale oft sehr enge Beziehungen zu Reihen. Genau wie bei Folgen und Reihen (siehe Satz <a href=\"..\/..\/chapter\/reelle-folgen#x1-158001r5\">6.5<\/a> und Proposition <a href=\"..\/..\/chapter\/reihen#x1-188001r11\">7.11<\/a>) ist es bei uneigentlichen Integralen nicht-negativer Funktionen einfacher \u00fcber Konvergenz zu entscheiden. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd7d8cccbe1ff\"> <a id=\"x1-273005r33\"><\/a> <span class=\"ecbx-1095\">Lemma 9.33.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine nicht-negative<\/span> <span class=\"ecti-1095\">Funktion mit <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>a<\/mi><\/math><span class=\"ecti-1095\">. Entweder konvergiert das<\/span> <span class=\"ecti-1095\">uneigentliche Integral <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">oder es divergiert gegen Unendlich. In beiden F<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">llen gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/munderover><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Die Funktion <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> ist monoton wachsend. Wenn das Supremum <math display=\"inline\"><mi>S<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mi class=\"qopname\">\u222b  <\/mi><mo>  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/munderover><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> Unendlich ist, dann divergiert das uneigentliche Integral auf Grund der Monotonie gegen Unendlich. Wenn <math display=\"inline\"><mi>S<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/math> ist, dann gibt es zu <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>a<\/mi><\/math> mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>S<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>B<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>S<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Insbesondere gilt f\u00fcr <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>B<\/mi><\/math> auf Grund der Monotonie und der Definition von&nbsp;<math display=\"inline\"><mi>S<\/mi><\/math> dieselbe Ungleichung auch f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dies beweist die Konvergenz des uneigentlichen Integrals. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z90c74d86a6e0\"> <a id=\"x1-273006r34\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.34 <\/span>(Gaussche Glockenkurve)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir wollen das uneigentliche Integral<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">besprechen, wobei die Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><\/math> <span class=\"ecti-1095\">die <\/span>Gaussche Glockenkurve <span class=\"ecti-1095\">genannt wird. Auf Grund von Lemma <\/span><a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-273005r33\"><span class=\"ecti-1095\">9.33<\/span><\/a> <span class=\"ecti-1095\">reicht es aus eine<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">Majorantenfunktion<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">zu finden, die ein konvergentes uneigentliches Integral definiert. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gilt zum<\/span> <span class=\"ecti-1095\">Beispiel <\/span><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>x<\/mi><\/math> <span class=\"ecti-1095\">und daher <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">woraus<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">folgt. Dies zeigt die Konvergenz des zweiten uneigentlichen Integrals, auf Grund der Symmetrie der Funktion ist daher<\/span> <span class=\"ecti-1095\">auch <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">konvergent. Wir<\/span> <span class=\"ecti-1095\">werden den Wert <\/span><math display=\"inline\"><mi>I<\/mi><\/math> <span class=\"ecti-1095\">dieses Integral erst im zweiten Semester berechnen k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen. Doch wollen wir noch erw<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hnen, dass<\/span> <span class=\"ecti-1095\">die streng monoton wachsende Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03a6<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>I<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">die Verteilungsfunktion der Normalverteilung mit Erwartungswert<\/span> <math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">Standardabweichung <\/span><math display=\"inline\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>2<\/mn><\/mrow><\/msqrt><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">genannt wird. Diese Funktion l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich ebenso wie die Funktionen aus Abschnitt <\/span><a href=\"..\/..\/chapter\/integrationsmethoden#x1-2710008\"><span class=\"ecti-1095\">9.2.8<\/span><\/a> <span class=\"ecti-1095\">nicht durch<\/span> <span class=\"ecti-1095\">die sonst <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blichen Funktionen ausdr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">cken und ist in der Wahrscheinlichkeitsrechnung, der Statstik<\/span> <span class=\"ecti-1095\">und in vielen Anwendungen von fundamentaler Bedeutung.<\/span> <\/p> <\/div> <p class=\"indent\">Der folgende Satz charakterisiert nun Konvergenz uneigentlicher Integrale wie in obigem Lemma durch Konvergenz von Reihen (auf hinreichende und notwendige Weise). <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z6364452cc400\"> <a id=\"x1-273007r35\"><\/a> <span class=\"ecbx-1095\">Satz 9.35 <\/span>(Integraltest f\u00fcr Reihen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine monoton fallende Funktion. Dann gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>2<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Insbesondere konvergiert die Reihe <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">genau dann, wenn das uneigentliche Integral<\/span> <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">konvergiert. Dies gilt analog<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r Integrale der Form<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>N<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Wir bemerken, dass auf Grund der Monotonieannahme an <math display=\"inline\"><mi>f<\/mi><\/math> in obigem Satz die Eigenschaft <math display=\"inline\"><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> erf\u00fcllt ist nach Satz <a href=\"..\/..\/chapter\/integrierbarkeit-monotoner-funktionen#x1-121001r31\">4.31<\/a>. <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">F\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und einen beliebigen Zwischenpunkt <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> gilt nach Monotonie von <math display=\"inline\"><mi>f<\/mi><\/math> die Ungleichung <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und somit                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was auch in folgendem Bild ersichtlich ist. <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=517&amp;url=Pictures\/fundsatz\/integraltest.pdf\"><img decoding=\"async\" id=\"z79d40f8f78a0\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/fundsatz\/integraltest.svg\" width=\"517\" \/><\/div>  <\/div> <p class=\"noindent\">Nach Summation von <math display=\"inline\"><mn>1<\/mn><\/math> bis <math display=\"inline\"><mi>n<\/mi><\/math> erh\u00e4lt man mit Intervalladditivit\u00e4t des Riemann-Integrals <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/munderover><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u2113<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Falls das uneigentliche Integral <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> existiert, dann folgt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/munderover><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u2113<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Daher ist die monoton wachsende Folge <math display=\"inline\"><msub><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u2113<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> nach oben beschr\u00e4nkt und konvergiert somit nach Satz&nbsp;<a href=\"..\/..\/chapter\/reelle-folgen#x1-158001r5\">6.5<\/a>. Insbesondere gilt auch <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>2<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Falls <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> konvergiert, dann ist f\u00fcr <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> und <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">\u230a<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">\u230b<\/mo><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Nach Lemma <a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-273005r33\">9.33<\/a> ist somit das uneigentliche Integral <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> konvergent und durch die Zahl <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> beschr\u00e4nkt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z74f0ac672572\"> <a id=\"x1-273008r36\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.36 <\/span>(Divergenzrate der harmonischen Reihe)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Verwenden Sie obigen Satz, um den <\/span><math display=\"inline\"><mi>p<\/mi><\/math><span class=\"ecti-1095\">-Test<\/span> <span class=\"ecti-1095\">in Beispiel <\/span><a href=\"..\/..\/chapter\/reihen#x1-188007r17\"><span class=\"ecti-1095\">7.17<\/span><\/a> <span class=\"ecti-1095\">zu erhalten. Imitieren Sie des Weiteren die Methodik im obigen Beweis von Satz<\/span> <a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-273007r35\"><span class=\"ecti-1095\">9.35<\/span><\/a><span class=\"ecti-1095\">, um die Divergenzrate<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>N<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>N<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die harmonische Reihe zu beweisen.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z373f96ed5f4e\"> <a id=\"x1-273009r37\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.37 <\/span>(Ein oszillierendes Integral)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Entscheiden                          Sie                          f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r                          welche<\/span> <math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">das                                          uneigentliche                                          Integral<\/span> <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>x<\/mi><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">konvergiert.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">verwenden Sie am besten die Substitution <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r das Integral<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>x<\/mi><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und ziehen je nach Fall auf geeignete Weise das Leibniz-Kriterium hinzu.<\/span><\/p><\/details>  <\/div> <a id=\"x1-273010r273\"><\/a> <h4 id=\"z26f8abc3f19c\" class=\"subsectionHead\"><span class=\"titlemark\">9.3.2 <\/span> <a id=\"x1-2740002\"><\/a>Das Integral \u00fcber unbeschr\u00e4nkte Funktionen<\/h4> <p class=\"noindent\">F\u00fcr <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>B<\/mi><\/math> in <math display=\"inline\"><mi>\u211d<\/mi><\/math> und eine Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> mit <math display=\"inline\"><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> definieren wir das <span class=\"ecbx-1095\">uneigentliche Integral<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>B<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>b<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mi>B<\/mi><\/mrow><\/munder><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">falls der Grenzwert existiert. <\/p><p class=\"indent\">Wie folgende \u00dcbung zeigt, steht diese Notation nicht im Widerspruch zum Riemann-Integral. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zc798c3f7fb1c\"> <a id=\"x1-274001r38\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.38 <\/span>(Kompatibilit\u00e4t)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">wie                                           oben.                                            Angenommen<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt.       Zeigen       Sie,       dass       das       uneigentliche       Integral<\/span> <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>B<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">existiert                 und                 gleich                 dem                 Riemann-Integral<\/span> <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>B<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">ist,                                                 wobei                                                 man<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">auf                beliebige                Weise                auf                den                Punkt<\/span> <math display=\"inline\"><mi>B<\/mi><\/math> <span class=\"ecti-1095\">erweitert.<\/span> <\/p> <\/div> <p class=\"indent\">F\u00fcr <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> in <math display=\"inline\"><mi>\u211d<\/mi><\/math> und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> eine Funktion mit <math display=\"inline\"><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> definieren wir analog das uneigentliche Integral                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>A<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>a<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mi>A<\/mi><\/mrow><\/munder><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zec193d64d3fa\"> <a id=\"x1-274002r39\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.39.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir berechnen <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">mittels<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>a<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>a<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>x<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>a<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">nach Beispiel <\/span><a href=\"..\/..\/chapter\/integrationsmethoden#x1-264002r15\"><span class=\"ecti-1095\">9.15<\/span><\/a> <span class=\"ecti-1095\">und Beispiel <\/span><a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-179001r44\"><span class=\"ecti-1095\">6.44<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <p class=\"indent\">Weitere uneigentliche Integrale f\u00fchren wir mittels Intervalladditivit\u00e4t auf obige uneigentliche Integrale zur\u00fcck. Wir \u00fcberlassen es Interessierten, sich hier einige M\u00f6glichkeiten auszudenken, und f\u00fchren stattdessen ein Beispiel vor. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zb5db8d71830c\"> <a id=\"x1-274003r40\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.40.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten das Integral <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dieses ist uneigentlich, da <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">auf jeder Umgebung von <\/span><math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">unbeschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist. Es gilt daher auf Grund der Definition in diesem Fall, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei beide Integrale rechts uneigentlich sind und das uneigentliche Integral<\/span> <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">per<\/span> <span class=\"ecti-1095\">Definition genau dann existiert, wenn die beiden Integrale rechts existieren. Des Weiteren<\/span> <span class=\"ecti-1095\">gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mn>0<\/mn><\/mrow><\/munder><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mn>0<\/mn><\/mrow><\/munder><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mn>0<\/mn><\/mrow><\/munder><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>a<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>a<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wodurch <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">nicht existiert (und wir<\/span> <span class=\"ecti-1095\">diesem auch nicht das Symbol <\/span><math display=\"inline\"><mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">oder <\/span><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">zuweisen).<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z364f41cb5548\"> <a id=\"x1-274004r41\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.41 <\/span>(Bogenl\u00e4nge des Kreises)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir wollen nochmals die Bogenl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge des Kreises berechnen. Doch verwenden wir diesmal die<\/span> <span class=\"ecti-1095\">Gleichung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/math> <span class=\"ecti-1095\">als Definition des oberen Halbkreises. Die Bogenl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge des Kreises ist demnach gegeben durch das<\/span> <span class=\"ecti-1095\">Integral<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>2<\/mn><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mn>4<\/mn><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac> <mrow> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mn>4<\/mn><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mn>1<\/mn><\/mrow><\/munder><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mn>4<\/mn><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mn>1<\/mn><\/mrow><\/munder><mi class=\"qopname\"> arcsin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mn>4<\/mn><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>\u03c0<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf995c7500dbf\"> <a id=\"x1-274005r42\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.42.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mi>x<\/mi><\/mrow><\/msqrt><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msubsup><mi class=\"qopname\"> tan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z561aa2868a18\"> <a id=\"x1-274006r43\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.43 <\/span>(Absolute Konvergenz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">eine komplexwertige Funktion mit <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>a<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Wir nennen das uneigentliche Integral <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecbi-1095\">absolut konvergent<\/span><span class=\"ecti-1095\">, falls <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">konvergent ist. Zeigen Sie, dass absolute Konvergenz des uneigentlichen Integrals <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">auch die Konvergenz dieses Integrals impliziert.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z61b58e9b2904\"> <span class=\"ecti-1095\">Bemerkung.<\/span><\/h4> <p class=\"indent\">Zusammenfassend  haben  wir  bei  den  Definitionen  in  diesem  Abschnitt  bei  jedem Problempunkt  eines  m\u00f6glichen  Riemann-Integrals  einen  Grenzwert  verwendet,  um  den Integralbegriff  zu  erweiteren.  Dies  wirft  nochmals  die  Frage  auf,  ob  es  nicht  vielleicht einen Integralbegriff  gibt,  der  diese  und  auch  andere  bereits  erw\u00e4hnte  Probleme  des Riemann-Integrals  auf  nat\u00fcrliche  Art  und  Weise  l\u00f6st.  Diese  Frage  wird  im  zweiten Studienjahr des Mathematikstudiums mit der Theorie des Lebesgue-Integrals in der Vorlesung \u201eMass und Integral\u201c positiv beantwortet. <\/p> <\/div> <a id=\"x1-274007r274\"><\/a> <h4 id=\"z18b7aff68f95\" class=\"subsectionHead\"><span class=\"titlemark\">9.3.3 <\/span> <a id=\"x1-2750003\"><\/a>Die Gamma-Funktion<\/h4> <p class=\"noindent\">Die <span class=\"ecbx-1095\">Gamma-Funktion <\/span><math display=\"inline\"><mi>\u0393<\/mi><\/math> ist bei <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> durch das konvergente uneigentliche Integral                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-275001r6\" \/><mstyle class=\"maketag\"><mtext>(9.6)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">definiert. F\u00fcr&nbsp;<math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> ist dies aus zwei Gr\u00fcnden ein uneigentliches Integral und wir m\u00fcssen die Integrationsgrenzen&nbsp;<math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> und&nbsp;<math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi><\/math> getrennt untersuchen. F\u00fcr <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>a<\/mi><\/math> gilt jedoch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><\/mrow><\/mfrac><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <\/mrow><mrow> <mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><\/mrow><\/mfrac><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir setzen&nbsp;<math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> und erhalten <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>a<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><\/mrow><\/mfrac><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <\/mrow><mrow> <mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><\/mrow><\/mfrac><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><\/mrow><\/mfrac><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei das Integral rechts (f\u00fcr alle&nbsp;<math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math>) ein eigentliches Riemann-Integral darstellt. F\u00fcr&nbsp;<math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> erhalten wir <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><\/mrow><\/mfrac><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <\/mrow><mrow> <mn>1<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><\/mrow><\/mfrac><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><\/mrow><\/mfrac><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Um die Konvergenz von&nbsp;<math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> zu zeigen, wollen wir den Integraltest f\u00fcr Reihen in Satz&nbsp;<a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-273007r35\">9.35<\/a> verwenden. Die erste Vorraussetzung des Integraltests ist erf\u00fcllt, da die Funktion&nbsp;<math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/math> nicht-negativ ist. Die zweite Vorraussetzung ist, dass <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> monoton abnehmend sein soll. Wir berechnen daher die Ableitung und sehen, dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Da&nbsp;<math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> f\u00fcr alle&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>s<\/mi><\/math><\/span><span class=\"period\">,<\/span> sehen wir, dass diese Vorraussetzung zumindest f\u00fcr&nbsp;<math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">\u230a<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">\u230b<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> erf\u00fcllt ist. Es folgt daher, dass das Integral&nbsp;<math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>N<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> genau dann konvergiert wenn                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>N<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">konvergiert, was aber nach dem Quotientenkriterium in Korollar&nbsp;<a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193004r32\">7.32<\/a> (oder dem Wurzelkriterium in Korollar&nbsp;<a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193002r30\">7.30<\/a>) f\u00fcr Reihen in der Tat zutrifft. <\/p><p class=\"indent\">Addieren wir die beiden Integrale wieder und verwenden wir die Definition in (<a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-275001r6\">9.6<\/a>) so erhalten wir, dass&nbsp;<math display=\"inline\"><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle&nbsp;<math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> wohldefiniert ist und <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-275002r7\" \/><mstyle class=\"maketag\"><mtext>(9.7)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">erf\u00fcllt. <\/p><p class=\"indent\">Oft wird die Gamma-Funktion als eine Erweiterung der Fakult\u00e4tsfunktion auf <math display=\"inline\"><mi>\u2115<\/mi><\/math> bezeichnet. In der Tat gilt f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was wir hier noch beweisen m\u00f6chten. F\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u0393<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <\/mrow><mrow> <mn>0<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und somit folgt nach (<a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-275002r7\">9.7<\/a>) und Induktion <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir werden in der Fortsetzung dieser Vorlesung weitere Eigenschaften der Gamma-Funktion nachweisen k\u00f6nnen. Beispielsweise stellt sich heraus, dass die Gamma-Funktion glatt ist. Wir k\u00f6nnen dies hier aber nicht zeigen, da <math display=\"inline\"><mi>\u0393<\/mi><\/math> in (<a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-275001r6\">9.6<\/a>) durch ein sogenanntes Parameterintegral definiert ist. Auch k\u00f6nnen wir den Wert                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mi>x<\/mi><\/mrow><\/msqrt><\/mrow><\/mfrac><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mi>x<\/mi><\/mrow><\/msqrt><\/mrow><\/mfrac><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>b<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mn>2<\/mn><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msqrt><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msqrt><\/mrow><mrow><msqrt><mrow><mi>b<\/mi><\/mrow><\/msqrt><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">mit den uns bis jetzt bekannten Integrationsmethoden nicht berechnen, werden jedoch sp\u00e4ter mittels einem zweidimensionalen Integral sehen, dass dieser <math display=\"inline\"><msqrt><mrow><mi>\u03c0<\/mi><\/mrow><\/msqrt><\/math> ist. <\/p><p class=\"indent\">Die Gamma-Funktion enth\u00fcllt ihre wahre Sch\u00f6nheit erst wenn man komplexe Parameter&nbsp;<math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">}<\/mo><\/math> erlaubt. Wir laden Interessierte ein, diese Funktion in folgender \u00dcbung zu konstruieren. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z330a57a7da98\"> <a id=\"x1-275003r44\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.44 <\/span>(Challenge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>\u211c<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">definiert man<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>z<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-275004r8\" \/><mstyle class=\"maketag\"><mtext>(9.8)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">(a) Zeigen Sie, dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>z<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u211c<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">konvergiert.<\/span> <\/p><p class=\"noindent\"><span class=\"ecti-1095\">(b) Zeigen Sie, dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>z<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u211c<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">konvergiert.<\/span> <\/p><p class=\"noindent\"><span class=\"ecti-1095\">(c) Zeigen Sie<\/span><span class=\"ecti-1095\">&nbsp;<\/span>(<a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-275002r7\">9.7<\/a>) <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211c<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"noindent\"><span class=\"ecti-1095\">(d) Verwenden Sie<\/span><span class=\"ecti-1095\">&nbsp;<\/span>(<a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-275002r7\">9.7<\/a>) <span class=\"ecti-1095\">um rekursiv<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211c<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">oder<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u211c<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/math><span class=\"ecti-1095\">, <\/span><span class=\"ecti-1095\">\u2026, zu definieren,<\/span> <span class=\"ecti-1095\">so dass anschliessend<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u0393<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">definiert ist und<\/span><span class=\"ecti-1095\">&nbsp;<\/span>(<a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-275002r7\">9.7<\/a>) <span class=\"ecti-1095\">auf dem ganzen Definitionsbereich erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Hinweis: Sie k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen in (a) und (b) <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-274006r43\"><span class=\"ecti-1095\">9.43<\/span><\/a> <span class=\"ecti-1095\">verwenden.<\/span> <\/p> <\/div> <p class=\"indent\">Hilbert (1862\u20131943) verwendete in seinem Artikel <span class=\"cite\">[<a href=\"#XHilberteundpi\">Hil93<\/a>]<\/span> von 1893 uneigentliche Integrale im Stile der Gamma-Funktion, um zu beweisen, dass <math display=\"inline\"><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/math> (wie erstmals von Hermite in 1873 bewiesen) und <math display=\"inline\"><mi>\u03c0<\/mi><\/math> (wie erstmals von Lindemann 1882 bewiesen) transzendent sind. Wir bemerken dabei, dass sich die blosse Irrationalit\u00e4t dieser Zahlen deutlich einfacher beweisen l\u00e4sst \u2013 f\u00fcr <math display=\"inline\"><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/math> gibt es hierzu eine \u00dcbung in Abschnitt <a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-2240002\">7.9.2<\/a> und f\u00fcr <math display=\"inline\"><mi>\u03c0<\/mi><\/math> eine \u00dcbung in Abschnitt <a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-2930002\">9.8.2<\/a>. Transzendenzbeweise sind jedoch im Allgemeinen deutlich schwieriger. Wie schwierige derartige Aussagen tats\u00e4chlich sind, illustriert vielleicht die Tatsache, dass immer noch nicht bekannt ist, ob <math display=\"inline\"><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><mo class=\"MathClass-bin\">+<\/mo><mi>\u03c0<\/mi><\/math> eine transzendente Zahl ist oder nicht. 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