Inverse functions

All inverse circular functions and inverse hyperbolic functions can be defined in terms of the complex logarithm function.

For more information, see the paper “Reasoning about the elementary functions of complex analysis” by Robert M. Corless et al.

 

Inverse circular functions

\begin{align*} \sin^{-1}(z) &= -i \log \left( \sqrt{1 - z^2} + i z \right) \\ \cos^{-1}(z) &= \frac{2}{i} \log \left( \sqrt{\frac{1 + z}{2}} + i \sqrt{\frac{1 - z}{2}} \right) \\ \tan^{-1}(z) &= \frac{1}{2i} \bigl( \log(1 + i z) - \log(1 - i z) \bigr) \\ \cot^{-1}(z) &= \frac{1}{2i} \log \left( \frac{z + i}{z - i} \right) \\ \sec^{-1}(z) &= -i \log\left( \frac{1}{z} + i \sqrt{1 - \frac{1}{z^2}} \right) \\ \csc^{-1}(z) &= -i \log\left( \frac{i}{z} + \sqrt{1 - \frac{1}{z^2}} \right) \\ \end{align*}

 

Inverse hyperbolic functions

\begin{align*} \sinh^{-1}(z) &= \log \left( z + \sqrt{1 + z^2} \right) \\ \cosh^{-1}(z) &= 2 \log \left( \sqrt{\frac{z + 1}{2}} + \sqrt{\frac{z - 1}{2}} \right) \\ \tanh^{-1}(z) &= \frac{1}{2} \bigl( \log(1 + z) - \log(1 - z) \bigr) \\ \coth^{-1}(z) &= \frac{1}{2} \bigl( \log(-1 - z) - \log(1 - z) \bigr) \\ \text{sech}^{-1}(z) &= 2 \log \left( \sqrt{\frac{z + 1}{2z}} + \sqrt{\frac{1 - z}{2z}} \right) \\ \text{csch}^{-1}(z) &= \log \left( \frac{1}{z} + \sqrt{1 + \frac{1}{z^2}} \right) \\ \end{align*}