Riley hobson bence · Problem 3.28

Complex numbers and hyperbolic functions

Mathematics — Methods · complex-logarithm-inverse-trig

Problem

The principal value of the logarithmic function of a complex variable is defined to have its argument in the range

-\pi < \arg z \leq \pi

. By writing

z = \tan w

in terms of exponentials show that

\tan^{-1} z = \frac{1}{2i} \ln \left( \frac{1 + iz}{1 - iz} \right).

Use this result to evaluate

\tan^{-1}\left( \frac{2\sqrt{3} - 3i}{7} \right).