\input vanilla.sty
\magnification=1200
\baselineskip=20pt
\nopagenumbers

\centerline{\bf{Complex Analysis Preliminary Examination}}

\centerline{\bf{June 1995}}

\bigskip
$$U=\{z: |z|<1\}, \ {\overline{U}}=\{z: |z|\leq 1\}, \ \Gamma=\{|z|=1\}$$

\bigskip

\item{1.} A complex-valued function is said to be harmonic if both its real and imaginary parts are harmonic.  Suppose both $f(z)$ and $z \ f(z)$ are complex-valued harmonic functions on a region $R$.  Show that $f(z)$ is analytic on $R$.

\bigskip

\item{2.} Suppose $f$ is entire and the range of $f$ fails to meet some circle.  What can be said about $f$?  Justify your answer.

\bigskip

\item{3.} Present a contour integration argument to show that
$$\int^\infty_{-\infty} \frac{\cos x}{(x^2+1)^2} \ dx = \frac{\pi}{e}$$

\bigskip

\item{4.} Let $f$ be holomorphic in a neighborhood of ${\overline{U}}$.  Let $f(z) = \dsize\sum^\infty_{k=0} \ a_k \ z^k$ and $s_n(z)=\dsize\sum^n_{k=0} \ a_k \ z^k$.  Show that, among all polynomials $P$ of degree $n$ or less, the integral $\dsize\frac{1}{2\pi} \dsize\int^{2\pi}_0 \ |f(e^{i\theta})-p(e^{i\theta})|^2 \ d\theta$ attains its minimum for $p=s_n$.

\bigskip

\item{5.} A.  State Cauchy's integral formula for an open disk.

\item{ } B.  Let $f$ be in the disk algebra, i.e., let $f$ be continuous on ${\overline{U}}$ and holomorphic on $U$.  Use part (A) to prove that
$$f(z) = \frac{1}{2\pi i} \int_\Gamma \frac{f(w)}{w-z} \ dw \ \text{for} \ z\in U \ .$$

\bigskip

\item{6.} Let $f$ be holomorphic in a neighborhood of 0 and suppose that the series $\dsize\sum^\infty_{k=0} \ f^{(k)}(0)$ converges.  Show that $f$ can be extended to an entire function.

\bigskip

\item{7.} Let $f$ be meromorphic in a neighborhood of ${\overline{U}}$ with no poles on $\Gamma$.  Let \newline $|A| > \dsize\max_{z\in \Gamma} \ |f(z)|$.  Prove that in $U$, counting multiplicity and order, the number of solutions to $f(z)=A$ equals the number of poles of $f$.

\bigskip

\item{8.} Let ${\cal{F}}$ denote the family of functions $f(z)=\dsize\sum^\infty_{n=0} \ a_nz^n$ which are holomorphic in $U$ and satisfy $|a_n|\leq n$ for $n=0,1,2,\dots$ \ .  Prove that any sequence in ${\cal{F}}$ has a subsequence which converges uniformly on compact subsets of $U$.

\bigskip

\item{9.} Let $f(z)$ be nonconstant and holomorphic in a neighborhood of ${\overline{U}}$ with $f(0)=a_0$.  Let $M=\dsize\max_{z\in \Gamma} |f(z)|$.  Let $\lambda\in U$ and suppose $f(\lambda)=0$.  Prove that $|\lambda|\geq \dsize\frac{|a_0|}{M}$.  \newline [Hint:  Consider $g(z)=f\big(\dsize\frac{z+\lambda}{1+{\overline{\lambda}} \ z}\big)$.]



\bye