Complex Analysis: Theory and Applications · Problem 7.3.2

Taylor and Laurent Series — Howell & Matthews

Mathematics — Complex Analysis · laurent-series

Problem

Show that

f(z) = \dfrac{1}{1-z} = \dfrac{1}{1-i}\left(\dfrac{1}{1-\frac{z-i}{1-i}}\right)

has a Laurent series representation about the point

z_0 = i

given by

f(z) = \frac{1}{1-z} = -\sum_{n=1}^{\infty}\frac{(1-i)^{n-1}}{(z-i)^n}, \quad\text{valid for } |z-i| > \sqrt{2}.