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

\centerline{\bf{June 6, 1994}}

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\item{1.} Show that every function that is meromorphic on
the extended complex plane is rational.

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\item{2.} Show that if $u$ is a real-valued harmonic
function in a domain $\Omega\subset {\bold{C}}$ such that
$u^2$ is harmonic in $\Omega$, then $u$ is constant.

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\item{3.} By using complex integration verify the formula
$\dsize\int^{2\pi}_0 \dsize\frac{1}{a+\sin \theta} \ d\theta
= \dsize\frac{2\pi}{\sqrt{a^2-1}}$ where $a>1$.

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\item{4.} Let $f(z)=e^z+z$ and for $0\leq \theta<2\pi$ let
$R(\theta)=\{z: z=re^{i\theta}, \ r\geq 0\}$.  Show that
$\dsize\lim_{z\to \infty \atop {z\in R(\theta)}} \ |f(z)|=\infty$ for all $\theta$.  Does this imply that $\dsize\lim_{z\to \infty} \dsize\frac{1}{f(z)}=0$?  Explain.

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\item{5.} Suppose that $f$ is an analytic function in ${\cal{H}}=\{z: \text{Im} \ z>0\}$ and Im $f(z)>0$ for $z\in {\cal{H}}$.  Show that $|f'(z)|\leq \dsize\frac{\text{Im} \ f(z)}{\text{Im} \ z}$ for $z\in {\cal{H}}$, and determine when equality holds in this inequality.

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\item{6.} Let $\Delta$ denote the open unit disk in ${\bold{C}}$ and let $A=\{z: \dsize\frac{3}{4}<|z|<1\}$.  Show that the function $f(z)=\dsize\frac{1}{z-\frac{1}{2}}$ cannot be approximated uniformly on compact subsets of $A$ by functions analytic in $\Delta$.

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\item{7.} For $|z|<1$ let $f(z)=\dsize\frac{1}{1-z} \exp\big[- \dsize\frac{1}{1-z} \big]$, and for $0\leq \theta<2\pi$ let $\ell_\theta=\{z: z=re^{i\theta}, 0\leq r<1\}$.  Show that $f$ is bounded on each set $\ell_\theta$.  Is $f$ bounded in $\Delta$?  Explain.

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\item{8.} Find all conformal automorphisms of the annulus $\{z: 1<|z|<2\}$.


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