Complex Analysis: Theory and Applications · Problem 3.2.8

Sequences, Series, and Fractals — Howell & Matthews

Mathematics — Complex Analysis · analytic-functions-polar-form

Problem

Use Theorem 3.5 with regard to the following. (a) Let

f(z) = f(re^{i\theta}) = \ln r + i\theta

, where

r > 0

and

-\pi < \theta < \pi

. Show that

f

is analytic in the domain indicated and that

f'(z) = \frac{1}{z}

. (b) Let

f(z) = f(re^{i\theta}) = (\ln r)^2 - \theta^2 + i2\theta \ln r

, where

r > 0

and

-\pi < \theta \leq \pi

. Show that

f

is analytic for

r > 0

-\pi < \theta < \pi

, and find

f'(z)