Complex Analysis: Theory and Applications · Problem 4.4.12

Elementary Functions — Howell & Matthews

Mathematics — Complex Analysis · power-series-representation

Problem

Show that, for

|z - i| < \sqrt{2}

\frac{1}{1-z} = \sum_{n=0}^{\infty} \frac{(z-i)^n}{(1-i)^{n+1}}.

Hint:

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

. Now use Theorem 4.12.