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\centerline{\bf Preliminary Examination}

\centerline{\bf Complex Analysis}

\centerline{\bf January 1998}

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\item{1.} Suppose that $f$ is an entire function and $f({\bold{C}}) \cap \{w: \text{Re} \ w=0\} = \emptyset$.  Prove that $f$ is constant.

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\item{2.} Evaluate $\dsize\int^\pi_0 \frac{d\theta}{2+\cos \theta}$.

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\item{3.} Give an example of a function $f$ which is holomorphic in ${\bold{C}}\backslash\{z_0\}$ for some $z_0\neq 0$, has an essential singularity at $z_0$ and is continuous in $\{z: |z|\leq |z_0|\}$.  Show that the function given actually has these properties.

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\item{4.} Find the maximum value of $|g(z)|$ if $g(z) = \dsize\frac{z}{4z^2-1}$ and $z$ varies over the region $\{z: |z|\geq 1\}$.

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\item{5.} A. State carefully the Riemann Mapping Theorem.

\itemitem{B.} Let $D=\{z: |z|<1\}$, $\Omega=\{z: \text{Re} \  z>0\}$ and fix $\alpha \in \Omega$.  Find all conformal maps $g$ from $\Omega$ onto $D$ such that $g(\alpha)=0$.

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\item{6.} Suppose that $f$ is a holomorphic function in an open disk $D$, $f$ is continuous on ${\overline{D}}$ and $|f|$ is constant and nonzero on $\partial D$.  Prove that $f$ is a rational function.

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\item{7.} Let $P$ be a nonzero polynomial.  Suppose that $\dsize\int_{|z|=r} \frac{1}{P(z)} dz \neq 0$ whenever $r>0$ and the integral is defined. Show that $\deg P=1$.

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\item{8.} Suppose that $f$ is an entire function, and for $r>0$ let $M_f(r) = \sup\{|f(z)|: |z|\leq r\}$.  Assume that $0<\alpha<1$ and let
$$L(\alpha) = \lim_{r\to \infty} \frac{M_f(\alpha r)}{M_f(r)} \ .$$

(a) Determine $L(\alpha)$ in the case $f$ is a polynomial.

(b) Show that $L(\alpha)=0$ if $f$ is not a polynomial.

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