Complex Analysis: Theory and Applications · Problem 7.3.17

Taylor and Laurent Series — Howell & Matthews

Mathematics — Complex Analysis · laurent-series-fourier

Problem

Suppose that the Laurent expansion

f(z) = \displaystyle\sum_{n=-\infty}^{\infty} a_n z^n

converges in the annulus

A(0, r_1, r_2)

, where

r_1 < 1 < r_2

. Consider the real-valued function

u(\theta) = f(e^{i\theta})

and show that

u(\theta)

has the Fourier series expansion

u(\theta) = f(e^{i\theta}) = \sum_{n=-\infty}^{\infty} a_n e^{in\theta},

where

a_n = \frac{1}{2\pi}\int_0^{2\pi} e^{-in\varphi}f(e^{i\varphi})\,d\varphi.