# 反双曲函数

## 定义

### 反双曲正弦函数

$\operatorname{arsinh}\, x = \ln(x + \sqrt{x^2 + 1})$，其中$x \in \mathbb{R}$

#### 證明

$\displaystyle y=\sinh ^{-1}x$，那么$\displaystyle \sinh y=x$

$\displaystyle 2x=e^{y}-e^{-y}$，因此有二次方程$\displaystyle e^{2y}-2xe^{y}-1=0$

$y=\ln \left( x+\sqrt{x^{2}+1} \right)$

### 反双曲余弦函数

$\operatorname{arcosh}\,x=\ln (x+\sqrt{x^{2}-1})$，其中$1 \le x < +\infty$

#### 證明

$1 \le x < +\infty$，令$y=\cosh ^{-1}x$，于是$\cosh y=x$

$\displaystyle 2x=e^{y}+e^{-y}$，因此有二次方程$\displaystyle e^{2y}-2xe^{y}+1=0$

$y=\ln \left( x+\sqrt{x^{2}-1} \right)$

### 其它反双曲函数

$\operatorname{artanh}\, x = \ln\frac{\sqrt{1 - x^2}}{1-x} = \frac{1}{2} \ln\frac{1+x}{1-x},\operatorname{arcoth}\, x = \ln\frac{\sqrt{x^2 - 1}}{x-1} = \frac{1}{2} \ln\frac{x+1}{x-1}, \operatorname{arsech}\, x = \pm \frac{1}{2} \ln\frac{1 + \sqrt{1 - x^2}}{1 - \sqrt{1 - x^2}} , \operatorname{arcsch}\, x = \begin{cases} \ln\frac{1 - \sqrt{1 + x^2}}{x}, & \mbox{for }x < 0\!\, \\ \ln\frac{1 + \sqrt{1 + x^2}}{x}, & \mbox{for }x > 0\!\, \end{cases}$

## 反双曲函数的导数

\begin{align} \frac{d}{dx} \operatorname{arsinh}\, x & {}= \frac{1}{\sqrt{1+x^2}}\\ \frac{d}{dx} \operatorname{arcosh}\, x & {}= \frac{1}{\sqrt{x^2-1}}\\ \frac{d}{dx} \operatorname{artanh}\, x & {}= \frac{1}{1-x^2}\\ \frac{d}{dx} \operatorname{arcoth}\, x & {}= \frac{1}{1-x^2}\\ \frac{d}{dx} \operatorname{arsech}\, x & {}= \frac{-1}{x(x+1)\,\sqrt{\frac{1-x}{1+x}}}\\ \frac{d}{dx} \operatorname{arcsch}\, x & {}= \frac{-1}{x^2\,\sqrt{1+\frac{1}{x^2}}}\\ \end{align}

\begin{align} \frac{d}{dx} \operatorname{arsech}\, x & {}= \frac{\mp 1}{x\,\sqrt{1-x^2}}; \qquad \Re\{x\} \gtrless 0\\ \frac{d}{dx} \operatorname{arcsch}\, x & {}= \frac{\mp 1}{x\,\sqrt{1+x^2}}; \qquad \Re\{x\} \gtrless 0 \end{align}

$\frac{d\,\operatorname{arsinh}\, x}{dx} = \frac{d \theta}{d \sinh \theta} = \frac{1} {\cosh \theta} = \frac{1} {\sqrt{1+\sinh^2 \theta}} = \frac{1}{\sqrt{1+x^2}}$

## 幂级数展开式

$\operatorname{arsinh}\, x$
$= x - \left( \frac {1} {2} \right) \frac {x^3} {3} + \left( \frac {1 \cdot 3} {2 \cdot 4} \right) \frac {x^5} {5} - \left( \frac {1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6} \right) \frac {x^7} {7} +\cdots$
$= \sum_{n=0}^\infty \left( \frac {(-1)^n(2n)!} {2^{2n}(n!)^2} \right) \frac {x^{2n+1}} {(2n+1)} , \qquad \left| x \right| < 1$
$\operatorname{arcosh}\, x$
$= \ln 2x - \left( \left( \frac {1} {2} \right) \frac {x^{-2}} {2} + \left( \frac {1 \cdot 3} {2 \cdot 4} \right) \frac {x^{-4}} {4} + \left( \frac {1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6} \right) \frac {x^{-6}} {6} +\cdots \right)$
$= \ln 2x - \sum_{n=1}^\infty \left( \frac {(-1)^n(2n)!} {2^{2n}(n!)^2} \right) \frac {x^{-2n}} {(2n)} , \qquad x > 1$
$\operatorname{artanh}\, x = x + \frac {x^3} {3} + \frac {x^5} {5} + \frac {x^7} {7} +\cdots = \sum_{n=0}^\infty \frac {x^{2n+1}} {(2n+1)} , \qquad \left| x \right| < 1$
$\operatorname{arcsch}\, x = \operatorname{arsinh}\, x^{-1}$
$= x^{-1} - \left( \frac {1} {2} \right) \frac {x^{-3}} {3} + \left( \frac {1 \cdot 3} {2 \cdot 4} \right) \frac {x^{-5}} {5} - \left( \frac {1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6} \right) \frac {x^{-7}} {7} +\cdots$
$= \sum_{n=0}^\infty \left( \frac {(-1)^n(2n)!} {2^{2n}(n!)^2} \right) \frac {x^{-(2n+1)}} {(2n+1)} , \qquad \left| x \right| < 1$
$\operatorname{arsech}\, x = \operatorname{arcosh}\, x^{-1}$
$= \ln \frac{2}{x} - \left( \left( \frac {1} {2} \right) \frac {x^{2}} {2} + \left( \frac {1 \cdot 3} {2 \cdot 4} \right) \frac {x^{4}} {4} + \left( \frac {1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6} \right) \frac {x^{6}} {6} +\cdots \right)$
$= \ln \frac{2}{x} - \sum_{n=1}^\infty \left( \frac {(-1)^n(2n)!} {2^{2n}(n!)^2} \right) \frac {x^{2n}} {2n} , \qquad 0 < x \le 1$
$\operatorname{arcoth}\, x = \operatorname{artanh}\, x^{-1}$
$= x^{-1} + \frac {x^{-3}} {3} + \frac {x^{-5}} {5} + \frac {x^{-7}} {7} +\cdots$
$= \sum_{n=0}^\infty \frac {x^{-(2n+1)}} {(2n+1)} , \qquad \left| x \right| > 1$
$\operatorname{arcosh}(2x^2-1) = 2\operatorname{arcosh} x$
$\operatorname{arcosh}(2x^2+1) = 2\operatorname{arsinh} x$