反双曲函数

反双曲函数是双曲函数的反函数。与反圆函数不同之处是它的前缀是ar意即area（面积），而不是arc（弧）。因为双曲角是以双曲线、通过原点直线以及其对x轴的映射三者之间所夹面积定义的，而圆角是以弧长与半径的比值定义。

符号${\displaystyle \mathrm {sinh} ^{-1},\mathrm {cosh} ^{-1}$等常用于${\displaystyle \mathrm {arsinh} ,\mathrm {arcosh} }$等。但是这种符号有时在${\displaystyle \mathrm {sinh} ^{-1}x}$和${\displaystyle {\frac {1}{\mathrm {sinh} x}$之间易造成混淆。

下表列出基本的反双曲函数。

名称	常用符号	定义	定义域	值域	图像
反双曲正弦	${\displaystyle y=\mathrm {arsinh} x}$	${\displaystyle \ln(x+{\sqrt {x^{2}+1})}$	${\displaystyle \mathbb {R} }$	${\displaystyle \mathbb {R} }$
反双曲余弦	${\displaystyle y=\mathrm {arcosh} x}$	${\displaystyle \ln(x\pm {\sqrt {x^{2}-1})}$[註 1]	${\displaystyle [1,+\infty )}$	${\displaystyle [0,+\infty )}$
反双曲正切	${\displaystyle y=\mathrm {artanh} x}$	${\displaystyle {\frac {1}{2}\ln \left({\frac {1+x}{1-x}\right)}$	${\displaystyle (-1,1)}$	${\displaystyle \mathbb {R} }$
反双曲余切	${\displaystyle y=\mathrm {arcoth} x}$	${\displaystyle {\frac {1}{2}\ln \left({\frac {x+1}{x-1}\right)}$	${\displaystyle (-\infty ,-1)\cup (1,+\infty )}$	${\displaystyle (-\infty ,0)\cup (0,+\infty )}$
反双曲正割	${\displaystyle y=\mathrm {arsech} x}$	${\displaystyle \ln \left({\frac {1}{x}+{\frac {\sqrt {1-x^{2}{x}\right)}$	${\displaystyle (0,1]}$	${\displaystyle [0,+\infty )}$
反双曲余割	${\displaystyle y=\mathrm {arcsch} x}$	${\displaystyle \ln \left({\frac {1}{x}+{\frac {\sqrt {1+x^{2}{\left|x\right|}\right)}$	${\displaystyle (-\infty ,0)\cup (0,+\infty )}$	${\displaystyle (-\infty ,0)\cup (0,+\infty )}$

${\displaystyle {\begin{aligned}{\frac {d}{dx}\operatorname {arsinh} \,x&{}={\frac {1}{\sqrt {1+x^{2}\\{\frac {d}{dx}\operatorname {arcosh} \,x&{}={\frac {1}{\sqrt {x^{2}-1},\qquad x>1\\{\frac {d}{dx}\operatorname {artanh} \,x&{}={\frac {1}{1-x^{2},\qquad |x|<1\\{\frac {d}{dx}\operatorname {arcoth} \,x&{}={\frac {1}{1-x^{2},\qquad |x|>1\\{\frac {d}{dx}\operatorname {arsech} \,x&{}={\frac {-1}{x{\sqrt {1-x^{2},\qquad x\in (0,1)\\{\frac {d}{dx}\operatorname {arcsch} \,x&{}={\frac {-1}{|x|{\sqrt {1+x^{2},\qquad x{\text{ ≠ }0\\\end{aligned}$

求导范例： 设θ = arsinh x，则：

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

${\displaystyle \operatorname {arsinh} \,x}$

${\displaystyle =x-\left({\frac {1}{2}\right){\frac {x^{3}{3}+\left({\frac {1\cdot 3}{2\cdot 4}\right){\frac {x^{5}{5}-\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right){\frac {x^{7}{7}+\cdots }$

${\displaystyle =\sum _{n=0}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}\right){\frac {x^{2n+1}{(2n+1)},\qquad \left|x\right|<1}$

${\displaystyle \operatorname {arcosh} \,x}$

${\displaystyle =\ln 2x-\left(\left({\frac {1}{2}\right){\frac {x^{-2}{2}+\left({\frac {1\cdot 3}{2\cdot 4}\right){\frac {x^{-4}{4}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right){\frac {x^{-6}{6}+\cdots \right)}$

${\displaystyle =\ln 2x-\sum _{n=1}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}\right){\frac {x^{-2n}{(2n)},\qquad x>1}$

${\displaystyle \operatorname {artanh} \,x=x+{\frac {x^{3}{3}+{\frac {x^{5}{5}+{\frac {x^{7}{7}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n+1}{(2n+1)},\qquad \left|x\right|<1}$

${\displaystyle \operatorname {arcsch} \,x=\operatorname {arsinh} \,x^{-1}$

${\displaystyle =x^{-1}-\left({\frac {1}{2}\right){\frac {x^{-3}{3}+\left({\frac {1\cdot 3}{2\cdot 4}\right){\frac {x^{-5}{5}-\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right){\frac {x^{-7}{7}+\cdots }$

${\displaystyle =\sum _{n=0}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}\right){\frac {x^{-(2n+1)}{(2n+1)},\qquad \left|x\right|<1}$

${\displaystyle \operatorname {arsech} \,x=\operatorname {arcosh} \,x^{-1}$

${\displaystyle =\ln {\frac {2}{x}-\left(\left({\frac {1}{2}\right){\frac {x^{2}{2}+\left({\frac {1\cdot 3}{2\cdot 4}\right){\frac {x^{4}{4}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right){\frac {x^{6}{6}+\cdots \right)}$

${\displaystyle =\ln {\frac {2}{x}-\sum _{n=1}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}\right){\frac {x^{2n}{2n},\qquad 0<x\leq 1}$

${\displaystyle \operatorname {arcoth} \,x=\operatorname {artanh} \,x^{-1}$

${\displaystyle =x^{-1}+{\frac {x^{-3}{3}+{\frac {x^{-5}{5}+{\frac {x^{-7}{7}+\cdots }$

${\displaystyle =\sum _{n=0}^{\infty }{\frac {x^{-(2n+1)}{(2n+1)},\qquad \left|x\right|>1}$

${\displaystyle \operatorname {arcosh} (2x^{2}-1)=2\operatorname {arcosh} x}$

${\displaystyle \operatorname {arcosh} (2x^{2}+1)=2\operatorname {arsinh} x}$

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

使用分部积分法和上面的简单导数很容易得出它们。

1. ^ 双曲余弦函数是偶函数，所以对于一个y值（y>1），都有两个x值与之对应，取反的时候只取一个（通常是正的）即可。

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• 双曲函数