2.10.2 利用自然对数对面积函数的确定

由双曲函数 ((2.166) (2.171), 参见第 115 页 2.9.1) 的定义可知, 面积函数可

由对数函数来表示:

$$\begin{array}{}\text{(2.211)}& \text{Arsinh}x=\ln \left(x+\sqrt{x^2+1}\right)\text{,}\end{array}$$$$\begin{array}{}\text{(2.212)}& \mathrm{Arcosh}x=\ln \left(x+\sqrt{x^2-1}\right)=\ln \left(\frac{1}{x-\sqrt{x^2-1}}\right)\left(x\ge 1\right),\end{array}$$$$\begin{array}{}\text{(2.213)}& \mathrm{Artanh}x=\frac{1}{2}\ln \frac{1+x}{1-x}\left(\left|x\right|<1\right),\end{array}$$$$\begin{array}{}\text{(2.214)}& \mathrm{Arcoth}x=\frac{1}{2}\ln \frac{x+1}{x-1}\left(\left|x\right|>1\right).\end{array}$$