Complex Analysis: Theory and Applications · Problem 3.1.14

Analytic and Harmonic Functions — Howell & Matthews

Mathematics — Complex Analysis · nth-root-derivative

Problem

Let

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

denote the multivalued ``

n

th root function,'' where

n

is a positive integer. Use the chain rule to show that, if

g(z)

is any branch of the

n

th root function, then

g'(z) = \frac{1}{n}\frac{g(z)}{z}

in some suitably chosen domain (which you should specify).