Fungsi hiperbolik

Fungsi hiperbolik

Fungsi hiperbolik adalah salah satu hasil kombinasi dari fungsi-fungsi eksponen. Fungsi hiperbolik memiliki rumus. Selain itu memiliki invers serta turunan dan anti turunan fungsi hiperbolik dan inversnya.

Definisi

sinh, cosh dan tanh

csch, sech dan coth

Definisi Eksponen

Templat:Math adalah separuh selisih Templat:Math dan Templat:Math

Templat:Math adalah rerata Templat:Math dan Templat:Math

Dalam istilah dari fungsi eksponensial:

• Hiperbolik sinus:

  $\sinh x=\frac{e^x-e^{-x}}{2}=\frac{e^{2x}-1}{2e^x}=\frac{1-e^{-2x}}{2e^{-x}}.$

• Hiperbolik kosinus:

  $\cosh x=\frac{e^x+e^{-x}}{2}=\frac{e^{2x}+1}{2e^x}=\frac{1+e^{-2x}}{2e^{-x}}.$

• Hiperbolik tangen:

  $\tanh x=\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}=\frac{e^{2x}-1}{e^{2x}+1}$

• Hiperbolik kotangen: untuk Templat:Math,

  $\coth x=\frac{\cosh x}{\sinh x}=\frac{e^x+e^{-x}}{e^x-e^{-x}}=\frac{e^{2x}+1}{e^{2x}-1}$

• Hiperbolik sekan:

  $\mathrm{sech}x=\frac{1}{\cosh x}=\frac{2}{e^x+e^{-x}}=\frac{2e^x}{e^{2x}+1}$

• Hiperbolik kosekan: untuk Templat:Math,

  $\mathrm{csch}x=\frac{1}{\sinh x}=\frac{2}{e^x-e^{-x}}=\frac{2e^x}{e^{2x}-1}$

Definisi persamaan diferensial

- Dalam pengembangan -

Definisi kompleks trigonometri

-Dalam pengembangan -

Sifat karakteristik

- Dalam pengembangan -

Penambahan

$\begin{array}{cc}\sinh (x+y)& =\sinh x\cosh y+\cosh x\sinh y\\ \cosh (x+y)& =\cosh x\cosh y+\sinh x\sinh y\\ [6px]\tanh (x+y)& =\frac{\tanh x+\tanh y}{1+\tanh x\tanh y}\\ \end{array}$

terutama

$\begin{array}{cc}\cosh (2x)& ={\sinh }^{2}x+{\cosh }^{2}x=2{\sinh }^{2}x+1=2{\cosh }^{2}x-1\\ \sinh (2x)& =2\sinh x\cosh x\\ \tanh (2x)& =\frac{2\tanh x}{1+{\tanh }^{2}x}\\ \end{array}$

Lihat:

$\begin{array}{cc}\sinh x+\sinh y& =2\sinh \left(\frac{x+y}{2}\right)\cosh \left(\frac{x-y}{2}\right)\\ \cosh x+\cosh y& =2\cosh \left(\frac{x+y}{2}\right)\cosh \left(\frac{x-y}{2}\right)\\ \end{array}$

Pengurangan

$\begin{array}{cc}\sinh (x-y)& =\sinh x\cosh y-\cosh x\sinh y\\ \cosh (x-y)& =\cosh x\cosh y-\sinh x\sinh y\\ \tanh (x-y)& =\frac{\tanh x-\tanh y}{1-\tanh x\tanh y}\\ \end{array}$

Dan juga:

$\begin{array}{cc}\sinh x-\sinh y& =2\cosh \left(\frac{x+y}{2}\right)\sinh \left(\frac{x-y}{2}\right)\\ \cosh x-\cosh y& =2\sinh \left(\frac{x+y}{2}\right)\sinh \left(\frac{x-y}{2}\right)\\ \end{array}$

Rumus setengah argumen

$\begin{array}{cccc}\sinh \left(\frac{x}{2}\right)& =\frac{\sinh x}{\sqrt{2(\cosh x+1)}}& & =\mathrm{sgn}x\sqrt{\frac{\cosh x-1}{2}}\\ [6px]\cosh \left(\frac{x}{2}\right)& =\sqrt{\frac{\cosh x+1}{2}}\\ [6px]\tanh \left(\frac{x}{2}\right)& =\frac{\sinh x}{\cosh x+1}& & =\mathrm{sgn}x\sqrt{\frac{\cosh x-1}{\cosh x+1}}=\frac{e^x-1}{e^x+1}\end{array}$

di mana Templat:Math adalah fungsi tanda.

Jika $x\ne 0$, maka

$\tanh \left(\frac{x}{2}\right)=\frac{\cosh x-1}{\sinh x}=\coth x-\mathrm{csch}x$

Rumus kuadrat

$\begin{array}{cc}{\sinh }^{2}x& =\frac{1}{2}(\cosh 2x-1)\\ {\cosh }^{2}x& =\frac{1}{2}(\cosh 2x+1)\end{array}$

Pertidaksamaan

Pertidaksamaan berikut sangat berguna dalam statistik, yaitu $\cosh (t)\le e^{t^2/2}$

Fungsi invers sebagai logaritma

$\begin{array}{cc}\mathrm{arsinh}(x)& =\ln (x+\sqrt{x^2+1})\\ \mathrm{arcosh}(x)& =\ln (x+\sqrt{x^2-1})& & x⩾1\\ \mathrm{artanh}(x)& =\frac{1}{2}\ln \left(\frac{1+x}{1-x}\right)& & |x|<1\\ \mathrm{arcoth}(x)& =\frac{1}{2}\ln \left(\frac{x+1}{x-1}\right)& & |x|>1\\ \mathrm{arsech}(x)& =\ln (\frac{1}{x}+\sqrt{\frac{1}{x^2}-1})=\ln \left(\frac{1+\sqrt{1-x^2}}{x}\right)& & 0<x⩽1\\ \mathrm{arcsch}(x)& =\ln (\frac{1}{x}+\sqrt{\frac{1}{x^2}+1})=\ln \left(\frac{1+\sqrt{1+x^2}}{x}\right)& & x\ne 0\end{array}$

Turunan

$\begin{array}{cc}\frac{d}{dx}\sinh x& =\cosh x\\ \frac{d}{dx}\cosh x& =\sinh x\\ \frac{d}{dx}\tanh x& =1-{\tanh }^{2}x={\mathrm{sech}}^{2}x=\frac{1}{{\cosh }^{2}x}\\ \frac{d}{dx}\coth x& =1-{\coth }^{2}x=-{\mathrm{csch}}^{2}x=-\frac{1}{{\sinh }^{2}x}& & x\ne 0\\ \frac{d}{dx}\mathrm{sech}x& =-\tanh x\mathrm{sech}x\\ \frac{d}{dx}\mathrm{csch}x& =-\coth x\mathrm{csch}x& & x\ne 0\\ \frac{d}{dx}\mathrm{arsinh}x& =\frac{1}{\sqrt{x^2+1}}\\ \frac{d}{dx}\mathrm{arcosh}x& =\frac{1}{\sqrt{x^2-1}}& & 1<x\\ \frac{d}{dx}\mathrm{artanh}x& =\frac{1}{1-x^2}& & |x|<1\\ \frac{d}{dx}\mathrm{arcoth}x& =\frac{1}{1-x^2}& & 1<|x|\\ \frac{d}{dx}\mathrm{arsech}x& =-\frac{1}{x\sqrt{1-x^2}}& & 0<x<1\\ \frac{d}{dx}\mathrm{arcsch}x& =-\frac{1}{|x|\sqrt{1+x^2}}& & x\ne 0\end{array}$

Turunan detik

- Dalam pengembangan -

Standar integral

$\begin{array}{cc}\int \sinh (ax)dx& =a^{-1}\cosh (ax)+C\\ \int \cosh (ax)dx& =a^{-1}\sinh (ax)+C\\ \int \tanh (ax)dx& =a^{-1}\ln (\cosh (ax))+C\\ \int \coth (ax)dx& =a^{-1}\ln (\sinh (ax))+C\\ \int \mathrm{sech}(ax)dx& =a^{-1}\arctan (\sinh (ax))+C\\ \int \mathrm{csch}(ax)dx& =a^{-1}\ln (\tanh \left(\frac{ax}{2}\right))+C=a^{-1}\ln |\mathrm{csch}(ax)-\coth (ax)|+C\end{array}$

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$\begin{array}{cc}\int \frac{1}{\sqrt{a^2+u^2}}du& =\mathrm{arsinh}\left(\frac{u}{a}\right)+C\\ \int \frac{1}{\sqrt{u^2-a^2}}du& =\mathrm{arcosh}\left(\frac{u}{a}\right)+C\\ \int \frac{1}{a^2-u^2}du& =a^{-1}\mathrm{artanh}\left(\frac{u}{a}\right)+C& & u^2<a^2\\ \int \frac{1}{a^2-u^2}du& =a^{-1}\mathrm{arcoth}\left(\frac{u}{a}\right)+C& & u^2>a^2\\ \int \frac{1}{u\sqrt{a^2-u^2}}du& =-a^{-1}\mathrm{arsech}\left(\frac{u}{a}\right)+C\\ \int \frac{1}{u\sqrt{a^2+u^2}}du& =-a^{-1}\mathrm{arcsch}\left|\frac{u}{a}\right|+C\end{array}$