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\centerline{\bf Preliminary Examination}

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\centerline{\bf Complex Analysis}

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\centerline{\bf June 1997}

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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.} Find a conformal mapping of the open unit disk onto the domain
$$\Omega = \left\{w: |w+\frac{1}{2}| > \frac{1}{2}\right\} \bigcap \{w: |w|<1\} \ .$$

\item{3.} Suppose that $f$ is a holomorphic function in an open disk $D$, $f$ is continuous in ${\overline{D}}$ and $|f(z)|$ is constant for $z\in \partial D$. Prove that $f$ is a rational function.

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\item{4.} Determine all polynomials $P$ such that $I(r) = \dsize\int_{|z|=r} \dsize\frac{1}{P(z)} dz$ has the property that $I(r)\neq 0$ for all $r>0$ for which $I(r)$ is well-defined.

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\item{5.} 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{6.} Suppose that the function $f$ is holomorphic in $\{z: |z|<R\}$, and for each \newline $r$ $(0<r<R)$ let $L(r)$ denote the length of the curve $w=f(re^{i\theta})$, $0\leq \theta \leq 2\pi$.  Show that $L(r)\geq 2\pi r|f'(0)|$ and determine all functions for which equality holds.

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\item{7.} Suppose that the function $f$ is holomorphic in $\{z: |z|<R\}$ for some $R>0$.  Prove that $f(z) = \dsize\frac{1}{2\pi} \dsize\int^{2\pi}_0 \dsize\frac{re^{i\theta}+z}{re^{i\theta}-z} \text{Re}\{f(re^{i\theta})\}d\theta + i \ \text{Im} \ f(0)$ for $|z|<r <R$.

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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.

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


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