Complex Analysis: Theory and Applications · Problem 7.2.19

Taylor and Laurent Series — Howell & Matthews

Mathematics — Complex Analysis · taylor-series

Problem

Consider the function

f(z) = \begin{cases} \dfrac{1}{1-z}, & \text{when } z \neq \tfrac{1}{2}; \\ 0, & \text{when } z = \tfrac{1}{2}. \end{cases}

((a)) Use Theorem 7.4, Taylor's theorem, to show that the Maclaurin series for

f(z)

equals

\displaystyle\sum_{n=0}^{\infty} z^n

. ((b)) Obviously, the radius of convergence of this series equals

1

(ratio test). However, the distance between

0

and the nearest singularity of

f

equals

\tfrac{1}{2}

. Explain why this condition does not contradict Corollary 7.3.