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

### Definisi Eksponen

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 ,
*:\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:
*:\operatorname{sech} x = \frac{1}{\cosh x} = \frac {2} {e^x + e^{-x}}
  = \frac{2e^x} {e^{2x} + 1}
* Hiperbolik kosekan: untuk ,
*:\operatorname{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{align}
  \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{align}

terutama
:\begin{align}
  \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{align}

Lihat:
:\begin{align}
  \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{align}

### Pengurangan
:\begin{align}
  \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{align}

Dan juga:
:\begin{align}
  \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{align}

### Rumus setengah argumen
:\begin{align}
  \sinh\left(\frac{x}{2}\right) &= \frac{\sinh x}{\sqrt{2 (\cosh x + 1)} } &&= \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} &&= \sgn x \, \sqrt \frac{\cosh x-1}{\cosh x+1} = \frac{e^x - 1}{e^x + 1} 
\end{align}

di mana  adalah fungsi tanda.

Jika x \ne 0, maka

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

### Rumus kuadrat
:\begin{align}
\sinh^2 x &= \frac{1}{2}(\cosh 2x -1) \\
\cosh^2 x &= \frac{1}{2}(\cosh 2x +1)
\end{align}

### Pertidaksamaan

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

## Fungsi invers sebagai logaritma

:\begin{align}
  \operatorname {arsinh} (x) &= \ln \left(x + \sqrt{x^{2} + 1} \right) \\
  \operatorname {arcosh} (x) &= \ln \left(x + \sqrt{x^{2} - 1} \right) && x \geqslant 1 \\
  \operatorname {artanh} (x) &= \frac{1}{2}\ln \left( \frac{1 + x}{1 - x} \right) && | x |  1 \\
  \operatorname {arsech} (x) &= \ln \left( \frac{1}{x} + \sqrt{\frac{1}{x^2} - 1}\right) = \ln \left( \frac{1+ \sqrt{1 - x^2}}{x} \right) && 0 

## Turunan
:\begin{align}
   \frac{d}{dx}\sinh x &= \cosh x \\
   \frac{d}{dx}\cosh x &= \sinh x \\
   \frac{d}{dx}\tanh x &= 1 - \tanh^2 x = \operatorname{sech}^2 x = \frac{1}{\cosh^2 x} \\ 
   \frac{d}{dx}\coth x &= 1 - \coth^2 x = -\operatorname{csch}^2 x = -\frac{1}{\sinh^2 x} && x \neq 0 \\
   \frac{d}{dx}\operatorname{sech} x &= - \tanh x \operatorname{sech} x \\
   \frac{d}{dx}\operatorname{csch} x &= - \coth x \operatorname{csch} x && x \neq 0 \\
   \frac{d}{dx}\operatorname{arsinh} x &= \frac{1}{\sqrt{x^2+1}} \\
   \frac{d}{dx}\operatorname{arcosh} x &= \frac{1}{\sqrt{x^2 - 1}} && 1 

## Turunan detik
- Dalam pengembangan -

## Standar integral

:\begin{align}
  \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 \operatorname{sech} (ax)\,dx &= a^{-1} \arctan (\sinh (ax)) + C \\
  \int \operatorname{csch} (ax)\,dx &= a^{-1} \ln \left( \tanh \left( \frac{ax}{2} \right) \right) + C = a^{-1} \ln\left|\operatorname{csch} (ax) - \coth (ax)\right| + C
\end{align}

----

:\begin{align}
   \int {\frac{1}{\sqrt{a^2 + u^2}}\,du} & = \operatorname{arsinh} \left( \frac{u}{a} \right) + C \\
   \int {\frac{1}{\sqrt{u^2 - a^2}}\,du} &= \operatorname{arcosh} \left( \frac{u}{a} \right) + C \\
   \int {\frac{1}{a^2 - u^2}}\,du & =  a^{-1}\operatorname{artanh} \left( \frac{u}{a} \right) + C && u^2  a^2 \\
   \int {\frac{1}{u\sqrt{a^2 - u^2}}\,du} & = -a^{-1}\operatorname{arsech}\left( \frac{u}{a} \right) + C \\
   \int {\frac{1}{u\sqrt{a^2 + u^2}}\,du} & = -a^{-1}\operatorname{arcsch}\left| \frac{u}{a} \right| + C
\end{align}

## Referensi