Complex Analysis: Theory and Applications · Problem 7.3.15

Taylor and Laurent Series — Howell & Matthews

Mathematics — Complex Analysis · laurent-series-fourier

Problem

Consider the real-valued function

u(\theta) = \dfrac{1}{5-4\cos\theta}

. ((a)) Use the substitution

\cos\theta = \frac{1}{2}\left(z+\frac{1}{z}\right)

and obtain

u(\theta) = f(z) = \frac{-z}{(z-2)(2z-1)} = \frac{1}{3}\left(\frac{1}{1-\frac{z}{2}}\right) - \frac{1}{3}\left(\frac{1}{1-2z}\right).

((b)) Expand the function

f(z)

in part (a) in a Laurent series that is valid in the annulus

A\!\left(0,\tfrac{1}{2},2\right)

. ((c)) Use the substitutions

\cos(n\theta) = \frac{1}{2}(z^n + z^{-n})

in part (b) and obtain the Fourier series for

u(\theta)

, where

u(\theta) = \frac{1}{3} + \frac{1}{3}\displaystyle\sum_{n=1}^{\infty} 2^{-n+1}\cos(n\theta)