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\centerline{\bf Preliminary Exam}

\centerline{\bf Complex Analysis}

\centerline{\bf August 1999}

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\item{1.} Find an analytic function $f(z)$ whose real part is $(z=x+iy)$.
$$\text{Re} \ f(z) = xy-10 \ .$$
Does such a function exist?  Justify your answer.

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\item{2.} Find the general form of an entire function $f(z)$ satisfying
$$|f(z)| \leq A+B|z|^{3/2}, \ \text{where} \ A \ \text{and} \ B \ \text{are constants} \ .$$

\item{3.} Find the general form of a function $f(z)$ which is analytic inside the ellipse $D$ $(z=x+iy)$
$$\frac{x^2}{16} + \frac{y^2}{9}=1 \ ,$$
continuous in ${\overline{D}}$, and
$$\text{Im} \ f(z) = -5 \ \ \ (z\in \partial {\bold{D}})$$

\item{4.} Find a conformal mapping from ${\bold{C}} \backslash \{[0, + \infty)\}$ to the unit disk.

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\item{5.} 1. Prove that for any polynomial $p$ and any $a\in \Delta$
$$p(a) = \frac{1}{2\pi} \ \int^{2\pi}_0 \frac{p(e^{i\theta})}{1-e^{-i\theta}a} d\theta$$

2. Deduce from 5.1 that
$$|p(a)| \leq \left[\frac{1}{(1-|a|^2)} \frac{1}{2\pi} \int^{2\pi}_0 |p(e^{i\theta})|^2 d\theta\right]^{1/2}$$

\item{6.} Let $f$ be analytic in the unit disk and map the unit disk into itself given $f(1/2)=0$.

Prove that $|f'(1/2)| \leq \dsize\frac{4}{3}$.

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\item{7.} Let
$$f(z) = \frac{1}{z} \cdot \frac{1-2z}{z-2} \cdot \dots \cdot \frac{1-10z}{z-10}$$
Find $\dsize\int_{|z|=100} f(z)dz$.

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\item{8.} Let $f(z) \not\equiv 0$ be a meromorphic function in ${\bold{C}}$ such that
$$|f(z)|=1 \ \ \ (|z|=1)$$
and
$$f\left(\frac{1}{2}\right)=0 \ .$$
Can $f$ be an entire function?


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