Complex Analysis: Theory and Applications · Problem 5.3.4

Elementary Functions — Howell & Matthews

Mathematics — Complex Analysis · complex-exponents-principal-branch

Problem

For

z = re^{i\theta} \neq 0

, show that the principal branch of ((a))

z^i

is given by the equation

z^i = e^{-\theta}\left[(\ln r) + i\sin(\ln r)\right],

where

r > 0

and

-\pi < \theta \leq \pi

. ((b))

z^\alpha

(

\alpha

a real number) is given by the equation

z^\alpha = r^\alpha \cos\alpha\theta + ir^\alpha \sin\alpha\theta,

where

r > 0

and

-\pi < \theta \leq \pi