{"id":80,"date":"2021-12-15T09:53:18","date_gmt":"2021-12-15T09:53:18","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/die-komplexe-exponentialabbildung\/"},"modified":"2021-12-15T09:53:18","modified_gmt":"2021-12-15T09:53:18","slug":"die-komplexe-exponentialabbildung","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/die-komplexe-exponentialabbildung\/","title":{"raw":"Die (komplexe) Exponentialabbildung","rendered":"Die (komplexe) Exponentialabbildung"},"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=\"z2da41a75e116\" class=\"sectionHead\"><span class=\"titlemark\">7.5 <\/span> <a id=\"x1-2030005\"><\/a>Die (komplexe) Exponentialabbildung<\/h3> <p class=\"noindent\">Wir haben in Abschnitt <a href=\"..\/..\/chapter\/die-exponentialfunktion#x1-1650003\">6.3<\/a> die reelle Exponentiabbildung gesehen und ihre wichtigsten Eigenschaften gezeigt. Insbesondere haben wir verifiziert, dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> exp<\/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-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>\u2113<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/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\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir zeigen nun, dass wir die Exponentialabbildung alternativ durch die Potenzreihe <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> exp<\/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-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-203001r7\" \/><mstyle class=\"maketag\"><mtext>(7.7)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">mit unendlichem Konvergenzradius definieren k\u00f6nnen und dadurch auf die gesamte komplexe Ebene fortsetzen k\u00f6nnen. Die Darstellung der Exponentialabbildung als Potenzreihe (<a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-203001r7\">7.7<\/a>) ist gewissermassen flexibler als die Darstellung als Grenzwert wie in Proposition <a href=\"..\/..\/chapter\/die-exponentialfunktion#x1-165002r29\">6.29<\/a>. Sie wird uns sp\u00e4ter in diesem Kapitel beispielsweise dabei helfen, das Riemann-Integral der Exponentialfunktion zu berechnen (siehe Korollar <a href=\"..\/..\/chapter\/integration-von-potenzreihen#x1-216004r86\">7.86<\/a>). <a id=\"x1-203002r202\"><\/a> <\/p> <h4 id=\"ze34038e291b8\" class=\"subsectionHead\"><span class=\"titlemark\">7.5.1 <\/span> <a id=\"x1-2040001\"><\/a>Darstellung durch die Potenzreihe<\/h4> <p class=\"noindent\">F\u00fcr ein festes <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2192<\/mo> <mn>0<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> was wegen dem Quotientenkriterium beweist, dass die Reihe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> konvergiert. Da <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> beliebig war, ist der Konvergenzradius der Potenzreihe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> in der Tat unendlich (siehe Satz&nbsp;<a href=\"..\/..\/chapter\/potenzreihen#x1-200002r56\">7.56<\/a>). <\/p><p class=\"indent\">Sei nun <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> so dass <\/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>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>N<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/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\">F\u00fcr dieses <math display=\"inline\"><mi>N<\/mi><\/math> gilt dann                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>N<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2264<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>N<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <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><mi>N<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-204001r8\" \/><mstyle class=\"maketag\"><mtext>(7.8)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> ebenso <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>N<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>\u2113<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <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>0<\/mn><\/mrow><mrow><mi>N<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u220f<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>\u2113<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">+<\/mo> <mi>\ud835\udf00<\/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 fixieren nun <math display=\"inline\"><mi>N<\/mi><\/math> und verwenden <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mi class=\"qopname\">\u220f<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>\u2113<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>N<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> woraus folgt, dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>N<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>\ud835\udf00<\/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\">Gemeinsam mit (<a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-204001r8\">7.8<\/a>) erhalten wir <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn><mi>\ud835\udf00<\/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\">Da <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war, folgt (<a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-203001r7\">7.7<\/a>). <\/p><p class=\"indent\">Die Darstellung der Exponentialfunktion als Potenzreihe liefert durch Betrachten von nur endlich vielen Termen eine Approximation von <math display=\"inline\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><\/math> durch Polynome. <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=565&amp;url=Pictures\/reihen\/exp.pdf\"><img id=\"zb85029581974\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/reihen\/exp.svg\" width=\"565\"><\/div>  <\/div> <a id=\"x1-204002r204\"><\/a> <h4 id=\"z19171183e25f\" class=\"subsectionHead\"><span class=\"titlemark\">7.5.2 <\/span> <a id=\"x1-2050002\"><\/a>Die komplexe Exponentialreihe<\/h4> <p class=\"noindent\">Wir verwenden nun die Potenzreihe aus (<a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-203001r7\">7.7<\/a>), um die Exponentialfunktion <math display=\"inline\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">:<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> auf der komplexen Zahlenebene zu definieren. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zab4ce90854fd\"> <a id=\"x1-205001r68\"><\/a> <span class=\"ecbx-1095\">Satz 7.68 <\/span>(Komplexe Exponentialabbildung)<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\">definieren wir<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/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>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/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\"><span class=\"ecti-1095\">womit eine stetige Erweiterung <\/span><math display=\"inline\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">der reellen Exponentialabbildung definiert wird. Des Weiteren gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">die<\/span> <span class=\"ecti-1095\">Additionsformel<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>w<\/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=\"x1-205002r9\" \/><mstyle class=\"maketag\"><mtext>(7.9)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und die Formel<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-205003r10\" \/><mstyle class=\"maketag\"><mtext>(7.10)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Absolutbetrag. Insbesondere gilt <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/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>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Auf Grund der Diskussion in Abschnitt <a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-2040001\">7.5.1<\/a> ist der Konvergenzradius der Reihe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> unendlich. Wiederum nach Satz&nbsp;<a href=\"..\/..\/chapter\/potenzreihen#x1-200002r56\">7.56<\/a> ist damit <math display=\"inline\"><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> eine stetige Funktion, welche wegen Abschnitt <a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-2040001\">7.5.1<\/a> die reelle Exponentialfunktion erweitert. (Wir verwenden zwar das gleiche Symbol <math display=\"inline\"><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><\/math> f\u00fcr die reelle und komplexe Exponentialfunktion, doch m\u00fcssen wir diese unterscheiden, wenn wir Eigenschaften von Funktionen wie zum Beispiel Injektivit\u00e4t besprechen wollen.) <\/p><p class=\"indent\">F\u00fcr eine positive Basis <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> und <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> setzen wir des Weiteren <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>a<\/mi><\/mrow><mrow><mi>z<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><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\">was wegen obigem mit der in Abschnitt <a href=\"..\/..\/chapter\/die-exponentialfunktion#x1-1730008\">6.3.8<\/a> eingef\u00fchrten Notation kompatibel ist. Insbesondere gilt <math display=\"inline\"><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mi>z<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z18762e98bb2b\"> <a id=\"x1-205004r69\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.69 <\/span>(Grenzwertformel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/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>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>z<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/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 alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <a id=\"x1-205005r205\"><\/a> <h4 id=\"zc6147ada6d85\" class=\"subsectionHead\"><span class=\"titlemark\">7.5.3 <\/span> <a id=\"x1-2060003\"><\/a>Die Additionsformel<\/h4> <p class=\"noindent\">Wir wollen nun die Additionsformel (<a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-205002r9\">7.9<\/a>) beweisen. In der Tat folgt f\u00fcr beliebige <math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> aus der Cauchy-Produktformel (Korollar <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-195003r37\">7.37<\/a>), dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>w<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><msup><mrow><mi>w<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>k<\/mi><\/mrow><\/msup><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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><msup><mrow><mi>w<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>k<\/mi><\/mrow><\/msup><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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow><mi>n<\/mi><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac><\/mfenced><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><msup><mrow><mi>w<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>k<\/mi><\/mrow><\/msup><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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><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> <a id=\"x1-206001r206\"><\/a> <h4 id=\"zec9aedb9515b\" class=\"subsectionHead\"><span class=\"titlemark\">7.5.4 <\/span> <a id=\"x1-2070004\"><\/a>Der Absolutbetrag der Exponentialabbildung<\/h4> <p class=\"noindent\">Es verbleibt f\u00fcr den Beweis von Satz <a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-205001r68\">7.68<\/a> die Formel (<a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-205003r10\">7.10<\/a>) f\u00fcr den Absolutbetrag zu beweisen. In der Tat gilt, da die Konjugation auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> stetig ist (wieso?), dass                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mover accent=\"false\" class=\"mml-overline\"><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/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\">Insbesondere ist nach der Additionsformel <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mover accent=\"false\" class=\"mml-overline\"><mrow><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>2<\/mn><mi class=\"qopname\">Re<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi class=\"qopname\">Re<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">womit die Formel <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> nach Wurzelziehen folgt. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z96874eaeaa17\"> <a id=\"x1-207001r70\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.70.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/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\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">die Absch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzung <\/span><span class=\"maperiod\"><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z0eea4e127c6d\"> <a id=\"x1-207002r71\"><\/a> <span class=\"ecbx-1095\">Applet 7.71 <\/span>(Komplexe Exponentialabbildung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"329px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/Gbx46jbp\/width\/688\/height\/329\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir stellen die komplexe Exponentialabbildung dar. Da der Graph dieser in <\/span><math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">liegt, k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir den Graph wohl kaum auf dem Bildschirm darstellen. Stattdessen visualisieren<\/span> <span class=\"ecti-1095\">wir die Abbildung anhand eines bewegbaren Punktes <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03c6<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">und dessen Bildpunkt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03c6<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <a id=\"x1-207003r203\"><\/a> \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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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; 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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=\"z2da41a75e116\" class=\"sectionHead\"><span class=\"titlemark\">7.5 <\/span> <a id=\"x1-2030005\"><\/a>Die (komplexe) Exponentialabbildung<\/h3> <p class=\"noindent\">Wir haben in Abschnitt <a href=\"..\/..\/chapter\/die-exponentialfunktion#x1-1650003\">6.3<\/a> die reelle Exponentiabbildung gesehen und ihre wichtigsten Eigenschaften gezeigt. Insbesondere haben wir verifiziert, dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> exp<\/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-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>\u2113<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/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\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir zeigen nun, dass wir die Exponentialabbildung alternativ durch die Potenzreihe <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> exp<\/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-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-203001r7\" \/><mstyle class=\"maketag\"><mtext>(7.7)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">mit unendlichem Konvergenzradius definieren k\u00f6nnen und dadurch auf die gesamte komplexe Ebene fortsetzen k\u00f6nnen. Die Darstellung der Exponentialabbildung als Potenzreihe (<a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-203001r7\">7.7<\/a>) ist gewissermassen flexibler als die Darstellung als Grenzwert wie in Proposition <a href=\"..\/..\/chapter\/die-exponentialfunktion#x1-165002r29\">6.29<\/a>. Sie wird uns sp\u00e4ter in diesem Kapitel beispielsweise dabei helfen, das Riemann-Integral der Exponentialfunktion zu berechnen (siehe Korollar <a href=\"..\/..\/chapter\/integration-von-potenzreihen#x1-216004r86\">7.86<\/a>). <a id=\"x1-203002r202\"><\/a> <\/p> <h4 id=\"ze34038e291b8\" class=\"subsectionHead\"><span class=\"titlemark\">7.5.1 <\/span> <a id=\"x1-2040001\"><\/a>Darstellung durch die Potenzreihe<\/h4> <p class=\"noindent\">F\u00fcr ein festes <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2192<\/mo> <mn>0<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> was wegen dem Quotientenkriterium beweist, dass die Reihe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> konvergiert. Da <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> beliebig war, ist der Konvergenzradius der Potenzreihe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> in der Tat unendlich (siehe Satz&nbsp;<a href=\"..\/..\/chapter\/potenzreihen#x1-200002r56\">7.56<\/a>). <\/p><p class=\"indent\">Sei nun <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> so dass <\/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>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>N<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/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\">F\u00fcr dieses <math display=\"inline\"><mi>N<\/mi><\/math> gilt dann                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>N<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2264<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>N<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <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><mi>N<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-204001r8\" \/><mstyle class=\"maketag\"><mtext>(7.8)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> ebenso <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>N<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>\u2113<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <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>0<\/mn><\/mrow><mrow><mi>N<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u220f<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>\u2113<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">+<\/mo> <mi>\ud835\udf00<\/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 fixieren nun <math display=\"inline\"><mi>N<\/mi><\/math> und verwenden <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mi class=\"qopname\">\u220f<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>\u2113<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>N<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> woraus folgt, dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>N<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>\ud835\udf00<\/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\">Gemeinsam mit (<a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-204001r8\">7.8<\/a>) erhalten wir <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn><mi>\ud835\udf00<\/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\">Da <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war, folgt (<a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-203001r7\">7.7<\/a>). <\/p><p class=\"indent\">Die Darstellung der Exponentialfunktion als Potenzreihe liefert durch Betrachten von nur endlich vielen Termen eine Approximation von <math display=\"inline\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><\/math> durch Polynome. <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=565&amp;url=Pictures\/reihen\/exp.pdf\"><img decoding=\"async\" id=\"zb85029581974\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/reihen\/exp.svg\" width=\"565\" \/><\/div>  <\/div> <a id=\"x1-204002r204\"><\/a> <h4 id=\"z19171183e25f\" class=\"subsectionHead\"><span class=\"titlemark\">7.5.2 <\/span> <a id=\"x1-2050002\"><\/a>Die komplexe Exponentialreihe<\/h4> <p class=\"noindent\">Wir verwenden nun die Potenzreihe aus (<a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-203001r7\">7.7<\/a>), um die Exponentialfunktion <math display=\"inline\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">:<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> auf der komplexen Zahlenebene zu definieren. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zab4ce90854fd\"> <a id=\"x1-205001r68\"><\/a> <span class=\"ecbx-1095\">Satz 7.68 <\/span>(Komplexe Exponentialabbildung)<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\">definieren wir<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/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>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/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\"><span class=\"ecti-1095\">womit eine stetige Erweiterung <\/span><math display=\"inline\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">der reellen Exponentialabbildung definiert wird. Des Weiteren gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">die<\/span> <span class=\"ecti-1095\">Additionsformel<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>w<\/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=\"x1-205002r9\" \/><mstyle class=\"maketag\"><mtext>(7.9)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und die Formel<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-205003r10\" \/><mstyle class=\"maketag\"><mtext>(7.10)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Absolutbetrag. Insbesondere gilt <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/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>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Auf Grund der Diskussion in Abschnitt <a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-2040001\">7.5.1<\/a> ist der Konvergenzradius der Reihe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> unendlich. Wiederum nach Satz&nbsp;<a href=\"..\/..\/chapter\/potenzreihen#x1-200002r56\">7.56<\/a> ist damit <math display=\"inline\"><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> eine stetige Funktion, welche wegen Abschnitt <a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-2040001\">7.5.1<\/a> die reelle Exponentialfunktion erweitert. (Wir verwenden zwar das gleiche Symbol <math display=\"inline\"><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><\/math> f\u00fcr die reelle und komplexe Exponentialfunktion, doch m\u00fcssen wir diese unterscheiden, wenn wir Eigenschaften von Funktionen wie zum Beispiel Injektivit\u00e4t besprechen wollen.) <\/p><p class=\"indent\">F\u00fcr eine positive Basis <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> und <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> setzen wir des Weiteren <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>a<\/mi><\/mrow><mrow><mi>z<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><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\">was wegen obigem mit der in Abschnitt <a href=\"..\/..\/chapter\/die-exponentialfunktion#x1-1730008\">6.3.8<\/a> eingef\u00fchrten Notation kompatibel ist. Insbesondere gilt <math display=\"inline\"><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mi>z<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z18762e98bb2b\"> <a id=\"x1-205004r69\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.69 <\/span>(Grenzwertformel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/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>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>z<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/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 alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <a id=\"x1-205005r205\"><\/a> <h4 id=\"zc6147ada6d85\" class=\"subsectionHead\"><span class=\"titlemark\">7.5.3 <\/span> <a id=\"x1-2060003\"><\/a>Die Additionsformel<\/h4> <p class=\"noindent\">Wir wollen nun die Additionsformel (<a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-205002r9\">7.9<\/a>) beweisen. In der Tat folgt f\u00fcr beliebige <math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> aus der Cauchy-Produktformel (Korollar <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-195003r37\">7.37<\/a>), dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>w<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><msup><mrow><mi>w<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>k<\/mi><\/mrow><\/msup><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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><msup><mrow><mi>w<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>k<\/mi><\/mrow><\/msup><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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow><mi>n<\/mi><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac><\/mfenced><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><msup><mrow><mi>w<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>k<\/mi><\/mrow><\/msup><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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><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> <a id=\"x1-206001r206\"><\/a> <h4 id=\"zec9aedb9515b\" class=\"subsectionHead\"><span class=\"titlemark\">7.5.4 <\/span> <a id=\"x1-2070004\"><\/a>Der Absolutbetrag der Exponentialabbildung<\/h4> <p class=\"noindent\">Es verbleibt f\u00fcr den Beweis von Satz <a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-205001r68\">7.68<\/a> die Formel (<a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-205003r10\">7.10<\/a>) f\u00fcr den Absolutbetrag zu beweisen. In der Tat gilt, da die Konjugation auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> stetig ist (wieso?), dass                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mover accent=\"false\" class=\"mml-overline\"><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/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\">Insbesondere ist nach der Additionsformel <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mover accent=\"false\" class=\"mml-overline\"><mrow><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>2<\/mn><mi class=\"qopname\">Re<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi class=\"qopname\">Re<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">womit die Formel <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> nach Wurzelziehen folgt. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z96874eaeaa17\"> <a id=\"x1-207001r70\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.70.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/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\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">die Absch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzung <\/span><span class=\"maperiod\"><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z0eea4e127c6d\"> <a id=\"x1-207002r71\"><\/a> <span class=\"ecbx-1095\">Applet 7.71 <\/span>(Komplexe Exponentialabbildung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"329px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/Gbx46jbp\/width\/688\/height\/329\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir stellen die komplexe Exponentialabbildung dar. Da der Graph dieser in <\/span><math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">liegt, k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir den Graph wohl kaum auf dem Bildschirm darstellen. Stattdessen visualisieren<\/span> <span class=\"ecti-1095\">wir die Abbildung anhand eines bewegbaren Punktes <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03c6<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">und dessen Bildpunkt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03c6<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <a id=\"x1-207003r203\"><\/a> \n","protected":false},"author":1089,"menu_order":5,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-80","chapter","type-chapter","status-publish","hentry"],"part":75,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/80","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/users\/1089"}],"version-history":[{"count":0,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/80\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/75"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/80\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=80"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=80"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=80"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=80"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}