Complex Analysis: Theory and Applications · Problem 7.3.11

Taylor and Laurent Series — Howell & Matthews

Mathematics — Complex Analysis · laurent-series

Problem

Find the Laurent series for

\mathrm{Log}\!\left(\dfrac{z-a}{z-b}\right)

, where

a

and

b

are positive real numbers with

b > a > 1

, and state where the series is valid. Hint: For these conditions, show that

\mathrm{Log}\!\left(\dfrac{z-a}{z-b}\right) = \mathrm{Log}\!\left(1-\dfrac{a}{z}\right) - \mathrm{Log}\!\left(1-\dfrac{b}{z}\right)