{"id":89,"date":"2021-12-15T09:53:24","date_gmt":"2021-12-15T09:53:24","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/hyperbolische-funktionen\/"},"modified":"2021-12-15T09:53:24","modified_gmt":"2021-12-15T09:53:24","slug":"hyperbolische-funktionen","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/hyperbolische-funktionen\/","title":{"raw":"Hyperbolische Funktionen","rendered":"Hyperbolische Funktionen"},"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=\"zb9134dbcb371\" class=\"sectionHead\"><span class=\"titlemark\">8.4 <\/span> <a id=\"x1-2460004\"><\/a>Hyperbolische Funktionen<\/h3> <p class=\"noindent\">Wir m\u00f6chten in diesem kurzen Abschnitt die zu Abschnitt <a href=\"..\/..\/chapter\/trigonometrische-funktionen#x1-2380003\">8.3<\/a> analoge Diskussion f\u00fcr die in Abschnitt <a href=\"..\/..\/chapter\/integration-von-potenzreihen#x1-2170001\">7.7.1<\/a> eingef\u00fchrten hyperbolischen Funktionen durchf\u00fchren. Wir erinnern daran, dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> sinh<\/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> <mfrac><mrow><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi class=\"qopname\">cosh<\/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> <mfrac><mrow><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi class=\"qopname\">tanh<\/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> <mfrac><mrow><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><\/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> <a id=\"x1-246001r245\"><\/a> <\/p> <h4 id=\"zb7a3732c5830\" class=\"subsectionHead\"><span class=\"titlemark\">8.4.1 <\/span> <a id=\"x1-2470001\"><\/a>Der Areasinus Hyperbolicus<\/h4> <p class=\"noindent\">Nach \u00dcbung <a href=\"..\/..\/chapter\/die-ableitung#x1-228005r4\">8.4<\/a> gilt <math display=\"inline\"><msup><mrow><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> 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> Somit ist also nach Korollar&nbsp;<a href=\"..\/..\/chapter\/zentrale-saetze-der-differentialrechnung#x1-234003r35\">8.35<\/a> der Sinus Hyperbolicus streng monoton wachsend. Da <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi class=\"qopname\">sinh<\/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> <mi>\u221e<\/mi><\/math> und <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi class=\"qopname\">sinh<\/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> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/math> gilt, erhalten wir nach dem Zwischenwertsatz <a href=\"..\/..\/chapter\/der-zwischenwertsatz#x1-96001r58\">3.58<\/a>, dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">streng monoton wachsend und bijektiv ist. Die Umkehrabbildung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> arsinh<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">nennen wir den <span class=\"ecbx-1095\">Areasinus Hyperbolicus<\/span>. Nach dem Satz zur Differenzierbarkeit der inversen Funktion ist <math display=\"inline\"><mi class=\"qopname\">arsinh<\/mi><mo>  <\/mo><\/math> differenzierbar und es gilt f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> und <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi class=\"qopname\">arsinh<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo>  <\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msqrt><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Der <span class=\"ecbx-1095\">Areasinus Hyperbolicus <\/span>besitzt im Gegensatz zu den Umkehrfunktionen <math display=\"inline\"><mi class=\"qopname\">arcsin<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> arccos<\/mi><mo>  <\/mo><\/math> und <math display=\"inline\"><mi class=\"qopname\">arctan<\/mi><mo>  <\/mo><\/math> eine geschlossene Form. In der Tat gilt                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> arsinh<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> wobei man beachten sollte, dass der Ausdruck rechts f\u00fcr alle <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> Sinn ergibt. Kurzes Nachrechnen ergibt f\u00fcr <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> und <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mo class=\"MathClass-close\">)<\/mo><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mi>s<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>s<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow> <mrow><msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wie gew\u00fcnscht. <a id=\"x1-247001r247\"><\/a> <\/p> <h4 id=\"zb712c43a6b8e\" class=\"subsectionHead\"><span class=\"titlemark\">8.4.2 <\/span> <a id=\"x1-2480002\"><\/a>Der Areakosinus Hyperbolicus<\/h4> <p class=\"noindent\">Der Kosinus Hyperbolicus erf\u00fcllt <math display=\"inline\"><msup><mrow><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><msup><mrow><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo>  <\/mrow><mrow><mi class=\"qopname\">\u2033<\/mi><mo>  <\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/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\">&gt;<\/mo> <mn>0<\/mn><\/math> f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> nach \u00dcbung <a href=\"..\/..\/chapter\/die-ableitung#x1-228005r4\">8.4<\/a>. Insbesondere ist der Kosinus Hyperbolicus streng konvex nach Korollar <a href=\"..\/..\/chapter\/zentrale-saetze-der-differentialrechnung#x1-235008r42\">8.42<\/a> und hat ein globales Minimum bei <math display=\"inline\"><mn>0<\/mn><\/math> nach Korollar <a href=\"..\/..\/chapter\/zentrale-saetze-der-differentialrechnung#x1-234005r37\">8.37<\/a> (wieso?). F\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> gilt <math display=\"inline\"><msup><mrow><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo>  <\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und somit ist <math display=\"inline\"><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><\/math> auf <math display=\"inline\"><msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn> <\/mrow> <\/msub> <\/math> streng monoton wachsend. Da <math display=\"inline\"><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> und <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi class=\"qopname\">cosh<\/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> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> folgt, dass                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">streng monoton wachsend und bijektiv ist. Die Umkehrabbildung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> arcosh<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wird der <span class=\"ecbx-1095\">Areakosinus Hyperbolicus <\/span>genannt, ist auf&nbsp;<math display=\"inline\"><msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <\/math> differenzierbar und erf\u00fcllt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi class=\"qopname\"> arcosh<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/msqrt><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> und <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> mit&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Des Weiteren gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> arcosh<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msqrt><mrow><msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/msqrt><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> Der Nachweis der obigen Eigenschaften des Areakosinus Hyperbolicus und der noch folgenden Eigenschaften \u00fcberlassen wir Interessierten. <a id=\"x1-248001r248\"><\/a> <\/p> <h4 id=\"z6311830cc99e\" class=\"subsectionHead\"><span class=\"titlemark\">8.4.3 <\/span> <a id=\"x1-2490003\"><\/a>Der Areatangens Hyperbolicus<\/h4> <p class=\"noindent\">Der <span class=\"ecbx-1095\">Areatangens Hyperbolicus <\/span>ist die Umkehrfunktion <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> artanh<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">der streng monoton wachsenden Bijektion                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> tanh<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><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\">Des Weiteren ist nach dem Satz zur inversen Funktion (Satz <a href=\"..\/..\/chapter\/die-ableitung#x1-228020r14\">8.14<\/a>) <math display=\"inline\"><mi class=\"qopname\">artanh<\/mi><mo>  <\/mo><\/math> differenzierbar und es gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi class=\"qopname\"> artanh<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/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>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zb1a002b5d11c\"> <a id=\"x1-249001r56\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 8.56.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Verifizieren Sie die oben aufgestellten Behauptungen.<\/span> <\/p> <\/div> <a id=\"x1-249002r246\"><\/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: 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1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"zb9134dbcb371\" class=\"sectionHead\"><span class=\"titlemark\">8.4 <\/span> <a id=\"x1-2460004\"><\/a>Hyperbolische Funktionen<\/h3> <p class=\"noindent\">Wir m\u00f6chten in diesem kurzen Abschnitt die zu Abschnitt <a href=\"..\/..\/chapter\/trigonometrische-funktionen#x1-2380003\">8.3<\/a> analoge Diskussion f\u00fcr die in Abschnitt <a href=\"..\/..\/chapter\/integration-von-potenzreihen#x1-2170001\">7.7.1<\/a> eingef\u00fchrten hyperbolischen Funktionen durchf\u00fchren. Wir erinnern daran, dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> sinh<\/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> <mfrac><mrow><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi class=\"qopname\">cosh<\/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> <mfrac><mrow><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi class=\"qopname\">tanh<\/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> <mfrac><mrow><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><\/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> <a id=\"x1-246001r245\"><\/a> <\/p> <h4 id=\"zb7a3732c5830\" class=\"subsectionHead\"><span class=\"titlemark\">8.4.1 <\/span> <a id=\"x1-2470001\"><\/a>Der Areasinus Hyperbolicus<\/h4> <p class=\"noindent\">Nach \u00dcbung <a href=\"..\/..\/chapter\/die-ableitung#x1-228005r4\">8.4<\/a> gilt <math display=\"inline\"><msup><mrow><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> 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> Somit ist also nach Korollar&nbsp;<a href=\"..\/..\/chapter\/zentrale-saetze-der-differentialrechnung#x1-234003r35\">8.35<\/a> der Sinus Hyperbolicus streng monoton wachsend. Da <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi class=\"qopname\">sinh<\/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> <mi>\u221e<\/mi><\/math> und <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi class=\"qopname\">sinh<\/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> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/math> gilt, erhalten wir nach dem Zwischenwertsatz <a href=\"..\/..\/chapter\/der-zwischenwertsatz#x1-96001r58\">3.58<\/a>, dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">streng monoton wachsend und bijektiv ist. Die Umkehrabbildung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> arsinh<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">nennen wir den <span class=\"ecbx-1095\">Areasinus Hyperbolicus<\/span>. Nach dem Satz zur Differenzierbarkeit der inversen Funktion ist <math display=\"inline\"><mi class=\"qopname\">arsinh<\/mi><mo>  <\/mo><\/math> differenzierbar und es gilt f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> und <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi class=\"qopname\">arsinh<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo>  <\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msqrt><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Der <span class=\"ecbx-1095\">Areasinus Hyperbolicus <\/span>besitzt im Gegensatz zu den Umkehrfunktionen <math display=\"inline\"><mi class=\"qopname\">arcsin<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> arccos<\/mi><mo>  <\/mo><\/math> und <math display=\"inline\"><mi class=\"qopname\">arctan<\/mi><mo>  <\/mo><\/math> eine geschlossene Form. In der Tat gilt                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> arsinh<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> wobei man beachten sollte, dass der Ausdruck rechts f\u00fcr alle <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> Sinn ergibt. Kurzes Nachrechnen ergibt f\u00fcr <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> und <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mo class=\"MathClass-close\">)<\/mo><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mi>s<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>s<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow> <mrow><msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wie gew\u00fcnscht. <a id=\"x1-247001r247\"><\/a> <\/p> <h4 id=\"zb712c43a6b8e\" class=\"subsectionHead\"><span class=\"titlemark\">8.4.2 <\/span> <a id=\"x1-2480002\"><\/a>Der Areakosinus Hyperbolicus<\/h4> <p class=\"noindent\">Der Kosinus Hyperbolicus erf\u00fcllt <math display=\"inline\"><msup><mrow><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><msup><mrow><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo>  <\/mrow><mrow><mi class=\"qopname\">\u2033<\/mi><mo>  <\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/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\">&gt;<\/mo> <mn>0<\/mn><\/math> f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> nach \u00dcbung <a href=\"..\/..\/chapter\/die-ableitung#x1-228005r4\">8.4<\/a>. Insbesondere ist der Kosinus Hyperbolicus streng konvex nach Korollar <a href=\"..\/..\/chapter\/zentrale-saetze-der-differentialrechnung#x1-235008r42\">8.42<\/a> und hat ein globales Minimum bei <math display=\"inline\"><mn>0<\/mn><\/math> nach Korollar <a href=\"..\/..\/chapter\/zentrale-saetze-der-differentialrechnung#x1-234005r37\">8.37<\/a> (wieso?). F\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> gilt <math display=\"inline\"><msup><mrow><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo>  <\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und somit ist <math display=\"inline\"><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><\/math> auf <math display=\"inline\"><msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn> <\/mrow> <\/msub> <\/math> streng monoton wachsend. Da <math display=\"inline\"><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> und <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi class=\"qopname\">cosh<\/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> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> folgt, dass                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">streng monoton wachsend und bijektiv ist. Die Umkehrabbildung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> arcosh<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wird der <span class=\"ecbx-1095\">Areakosinus Hyperbolicus <\/span>genannt, ist auf&nbsp;<math display=\"inline\"><msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <\/math> differenzierbar und erf\u00fcllt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi class=\"qopname\"> arcosh<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/msqrt><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> und <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> mit&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Des Weiteren gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> arcosh<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msqrt><mrow><msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/msqrt><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> Der Nachweis der obigen Eigenschaften des Areakosinus Hyperbolicus und der noch folgenden Eigenschaften \u00fcberlassen wir Interessierten. <a id=\"x1-248001r248\"><\/a> <\/p> <h4 id=\"z6311830cc99e\" class=\"subsectionHead\"><span class=\"titlemark\">8.4.3 <\/span> <a id=\"x1-2490003\"><\/a>Der Areatangens Hyperbolicus<\/h4> <p class=\"noindent\">Der <span class=\"ecbx-1095\">Areatangens Hyperbolicus <\/span>ist die Umkehrfunktion <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> artanh<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">der streng monoton wachsenden Bijektion                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> tanh<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><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\">Des Weiteren ist nach dem Satz zur inversen Funktion (Satz <a href=\"..\/..\/chapter\/die-ableitung#x1-228020r14\">8.14<\/a>) <math display=\"inline\"><mi class=\"qopname\">artanh<\/mi><mo>  <\/mo><\/math> differenzierbar und es gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi class=\"qopname\"> artanh<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/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>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zb1a002b5d11c\"> <a id=\"x1-249001r56\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 8.56.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Verifizieren Sie die oben aufgestellten Behauptungen.<\/span> <\/p> <\/div> <a id=\"x1-249002r246\"><\/a> 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