Inverse hyperbolic functions.

Theorem 4.8.1   For any complex number $z$ :

$$\sinh^{-1} z = \log (z + (z^2+1)^{1/2})$$

$$\cosh^{-1} z =\log (z +(z^2-1)^{1/2})$$

$$\tanh^{-1} z = \frac 12 \log \frac {1+z}{1-z}$$

Example 4.8.2  

$$\cosh^{-1} (1) =\log (1 +0^{1/2}) = \log 1 = 2k\pi i, \; k \in \mathbb{Z}.$$

$$\tanh^{-1}(i) = \left( \frac {\pi}{4}+ 2k \pi \right) \; i$$