Complex Analysis: Theory and Applications · Problem 2.4.6

Analytic and Harmonic Functions — Howell & Matthews

Mathematics — Complex Analysis · branches-cube-root

Problem

Let

f_2(z) = r^{\frac{1}{3}}\cos\!\left(\frac{\theta+2\pi}{3}\right) + ir^{\frac{1}{3}}\sin\!\left(\frac{\theta+2\pi}{3}\right)

, where

r > 0

and

-\pi < \theta \leq \pi

. (a) Show that

f_2

is a branch of the multivalued cube root

f(z) = z^{\frac{1}{3}}

. (b) What is the range of

f_2

? (c) Where is

f_2

continuous? (d) What is the branch point associated with

f