2.10.3 不同面积函数间的关系

$$\mathrm{Arsinh}x=\left(\mathrm{sign}x\right)\mathrm{Arcosh}\sqrt{x^2+1}=\mathrm{Artanh}\frac{x}{\sqrt{x^2+1}}$$$$\begin{array}{}\text{(2.215)}& =\mathrm{Arcoth}\frac{\sqrt{x^2+1}}{x}\left(\left|x\right|<\mathrm{\infty }\right),\end{array}$$$$\mathrm{Arcosh}x=\mathrm{Arsinh}\sqrt{x^2-1}=\mathrm{Artanh}\frac{\sqrt{x^2-1}}{x}$$$$\begin{array}{}\text{(2.216)}& =\mathrm{Arcoth}\frac{x}{\sqrt{x^2-1}}\left(\left|x\right|\ge 1\right),\end{array}$$$$\mathrm{Artanh}x=\mathrm{Arsinh}\frac{x}{\sqrt{1-x^2}}=\mathrm{Arcoth}\frac{1}{x}$$$$\begin{array}{}\text{(2.217)}& =\left(\mathrm{sign}x\right)\mathrm{Arcosh}\frac{1}{\sqrt{1-x^2}}\left(\left|x\right|<1\right),\end{array}$$$$\mathrm{Arcoth}x=\mathrm{Artanh}\frac{1}{x}=\left(\mathrm{sign}x\right)\mathrm{Arsinh}\frac{1}{\sqrt{x^2-1}}$$$$\begin{array}{}\text{(2.218)}& =\left(\mathrm{sign}x\right)\mathrm{Arcosh}\frac{\left|x\right|}{\sqrt{x^2-1}}\left(\left|x\right|>1\right).\end{array}$$