反雙曲函數示意圖

`<graph>`标签因安全原因被临时禁用。

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
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符号{\displaystyle \mathrm {sinh} ^{-1},\mathrm {cosh} ^{-1}}等常用于\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。

主值
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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 | {\displaystyle \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 | {\displaystyle \ln \left({\frac {1}{x}}+{\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
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\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}
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
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\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
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{\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。

註釋
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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。

外部链接
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* Inverse trigonometric functions（页面存档备份，存于） at MathWorld

参见
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* 双曲函数