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

\centerline{\bf{Ph.D. Preliminary Examination}}

\centerline{\bf{Complex Analysis}}

\centerline{\bf{August 26, 1994}}

\bigskip

\item{1.} Find an explicit conformal map from the region
$$\{z: |z|<1\} - \{x\in {\bold{R}}: x\leq 0\}$$
onto the upper halfplane \{Im $z > 0\}$.

\bigskip

\item{2.} Find the explicit Laurent series of the function
$$f(z)=\frac{1}{z(z-3)}$$
on the annulus $\{z: 1 < |z-1|<2\}$ centered at 1.

\bigskip

\item{3.} Let $D\subset {\bold{C}}$ be open and connected, and fix $z_0\in D$; set $A(D,z_0)=\{|f'(z_0)|: f$ holomorphic on $D$ and $|f(z)|<1$ for $z\in D\}$.  Prove that $A(D,z_0)$ is a compact subset of ${\bold{R}}$.  What is $A({\bold{C}},z_0)$?

\bigskip

\item{4.} Let $f$ be holomorphic in the connected region $\Omega\subset {\bold{C}}$, and assume that there exists a nonempty open set $U\subset \Omega$, such that $|f(z)|=1$ for all $z\in U$.  Show that $f$ is constant in $\Omega$.

\bigskip

\item{5.} Suppose $f(z)=\dsize\sum^\infty_{n=0} a_nz^n$ is holomorphic on the closed unit disc.  Prove that
$$\int^{2\pi}_0 |f(e^{i\theta})|^2d\theta = 2\pi \sum^\infty_{n=0} |a_n|^2 \ .$$

\newpage

\item{6.} Suppose $h$ is holomorphic in a neighborhood of $\{z: |z|\leq R\}$, and that $h(z)\neq 0$ for $|z|=R$.

\item\item{(a)} Use the Theorem of Residues to show that
$$\oint_{|z|=R} \frac{h'(z)}{h(z)} dz=2\pi i \ Z_R(h) \ ,$$
where $Z_R(h)$ is the number of zeroes of $h$ in $\{|z|<R\}$, counted with multiplicities.

\item\item{(b)} Use (a) to prove that if $f$ and $g$ satisfy the same hypotheses as $h$, and if
$$|f-g|<|f| \ \text{on} \ \{|z|=R\} \ ,$$
then $Z_R(f)=Z_R(g)$.

\bigskip

\item{7.} Use the Theorem of Residues for appropriate contours to evaluate
$$\int^\infty_{-\infty} \frac{\sqrt{x+i}}{1+x^2} dx \ ,$$
where on \{Im $z>0\}$, we choose the branch of $\sqrt{z+i}$ whose value at 0 is $e^{\pi i/4}$.  Describe your method carefully, and include verification of all relevant limit statements.

\bigskip

\item{8.} Find an \underbar{explicit} series representation for a meromorphic function on ${\bold{C}}$, which is holomorphic on ${\bold{C}}-\{1,2,3,\dots\}$, and which has at each point $z=n\in {\bold{N}}$ a simple pole with residue $n$.  Include proofs of all required convergence statements.

\bigskip

\item{9.} Prove that all holomorphic automorphisms of ${\bold{C}}$ (i.e. holomorphic maps $f: {\bold{C}}\to {\bold{C}}$ which are one-to-one and onto) are precisely  the linear functions $f(z)=a+bz$ for arbitrary $a,b\in {\bold{C}}$.


\bye