WJD Open Notebook

2007 November 16

Instructions. Do as many problems as you can. Complete solutions (except for minor flaws) to 5 problems would be considered an excellent performance. Fewer than 5 complete solutions may still be passing, depending on the quality.

1. Let $G$ be a bounded open subset of the complex plane. Suppose $f$ is continuous on the closure of $G$ and analytic on $G$. Suppose further that there is a constant $c\geq 0$ such that $|f| = c$ for all $z$ on the boundary of $G$. Show that either $f$ is constant on $G$ or $f$ has a zero in $G$.

2. (a) State the residue theorem.
   (b) Use contour integration to evaluate $$\int_0^\infty \frac{x^2}{(x^2+1)^2} \ dx.$$ Important: You must carefully: specify your contours, prove the inequalities that provide your limiting arguments, and show how to evaluate all relevant residues.

3. (a) State the Schwarz lemma.
   (b) Suppose $f$ is holomorphic in $D = \{z: |z|< 1\}$ with $f(D) \subseteq D$. Let $f_n$ denote the composition of $f$ with itself $n$ times $(n= 2, 3, \dots)$. Show that if $f(0) = 0$ and $|f'(0)| < 1$, then ${f_n}$ converges to 0 locally uniformly on $D$.

4. Exhibit a conformal mapping of the region common to the two disks $|z|<1$ and $|z-1|<1$ onto the region inside the unit circle $|z| = 1$.

5. Let $\{f_n\}$ be a sequence of functions analytic in the complex plane $\mathbb C$, converging uniformly on compact subsets of $\mathbb C$ to a polynomial $p$ of positive degree $m$. Prove that, if $n$ is sufficiently large, then $f_n$ has at least $m$ zeros, counting multiplicities. (Do not simply refer to Hurwitz's theorem; prove this version of it.)

6. Let $(X, d)$ be a metric space.
   (a) Define what it means for a subset $K\subseteq X$ to be compact.
   (b) Using your definition in (a), prove that $K\subseteq X$ is compact implies that $K$ is both closed and bounded in $X$.
   (c) Give an example that shows the converse of the statement in (b) is false.