UMD Analysis Qualifying Exam/Jan07 Complex

Problem 2

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

Problem 4

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