反双曲函数

反雙曲函數示意圖

幾個反雙曲函數的圖形。

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。

數學符號

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

主值

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

反双曲函数的导数

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

求导范例： 设θ = arsinh x，则：

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

幂级数展开式

$\operatorname {arsinh} \,x$

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

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

$\operatorname {arcosh} \,x$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

反双曲函数的不定积分

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

使用分部积分法和上面的简单导数很容易得出它们。

外部链接

• Inverse trigonometric functions at MathWorld

参见

• 双曲函数