反双曲函数

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反双曲函数双曲函数反函数。与反圆函数不同之处是它的前缀ar意即area(面积),而不是arc()。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[编辑]

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

主值[编辑]

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

反双曲函数的导数[编辑]


\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[编辑]

\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[编辑]


\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}

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[编辑]

参见[编辑]