UMD Analysis Qualifying Exam/Jan07 Complex

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

Consider the series ${\displaystyle g(z)=\sum _{n=1}^{\infty }{\frac {z^{2^{n}}}{n!}}}$

(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 ${\displaystyle D}$.