# 反双曲函数

## 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}}, \qquad x>1\\ \frac{d}{dx} \operatorname{artanh}\, x & {}= \frac{1}{1-x^2}, \qquad |x| <1\\ \frac{d}{dx} \operatorname{arcoth}\, x & {}= \frac{1}{1-x^2}, \qquad |x| >1\\ \frac{d}{dx} \operatorname{arsech}\, x & {}= \frac{-1}{x\sqrt{1-x^2}}, \qquad x \in (0,1)\\ \frac{d}{dx} \operatorname{arcsch}\, x & {}= \frac{-1}{|x|\sqrt{1+x^2}}, \qquad x \text{ ≠ }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

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

## 反双曲函数的不定积分

\begin{align} \int \operatorname{arsinh}\,x\,dx &{}= x\,\operatorname{arsinh}\,x - \sqrt{x^2+1} + C\\ \int \operatorname{arcosh}\,x\,dx &{}= x\,\operatorname{arcosh}\,x - \sqrt{x^2-1} + C,\qquad x >1\\ \int \operatorname{artanh}\,x\,dx &{}= x\,\operatorname{artanh}\,x + \frac{1}{2}\ln\left|1-x^2\right| + C,\qquad |x| <1\\ \int \operatorname{arcoth}\,x\,dx &{}= x\,\operatorname{arcoth}\,x + \frac{1}{2}\ln\left|1-x^2\right| + C,\qquad |x| >1\\ \int \operatorname{arsech}\,x\,dx &{}= x\,\operatorname{arsech}\,x - \arcsin\,x + C,x \in (0,1)\\ \int \operatorname{arcsch}\,x\,dx &{}= x\,\operatorname{arcsch}\,x + \left(\sgn\,x\right) \operatorname{arsinh}\,x + C \end{align}