# UMD Analysis Qualifying Exam/Jan07 Complex

## Problem 2

 Compute the partial fractions decomposition of ${\displaystyle f(z)={\frac {z^{7}}{z^{8}+1}}}$ .

## Problem 4

 Consider the series ${\displaystyle g(z)=\sum _{n=1}^{\infty }{\frac {z^{2n}}{n!}}}$  (a) Find the domain ${\displaystyle D}$  where the series is convergent (b) Prove that for any ${\displaystyle n,k\in \mathbb {N} ,g(e^{2\pi ik/2^{n}}z)=g(z)+p(z)}$ , for some polynomial ${\displaystyle p}$ . (c) Prove that if ${\displaystyle g}$  has analytic extension from ${\displaystyle D}$  then it has analytic extension to ${\displaystyle \partial D}$ . (d) Prove that ${\displaystyle g}$  has no analytic extension from ${\displaystyle D}$ .