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\centerline{\bf Complex Analysis Preliminary Exam}

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\centerline{\bf June 4, 1999}

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\item{1.} Let $f$ be a complex-valued harmonic function in a domain $\Omega\subset {\bold{C}}$.  Prove that if \newline $|f|$ = const in $\Omega$, then $f$ = const.

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\item{2.} Let $f$ be a holomorphic function in the unit disk which is continuous up to the boundary of the disk ${\bold{T}}=\partial \Delta=\{z\in {\bold{C}}: |z|=1\}$.  Prove that if $|f(z)|=1$ for all $z\in {\bold{T}}$, then $f$ is a rational function.

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\item{3.} Let $f$ be an entire function such that Re$(f(z))\leq 0$ for all $z\in {\bold{C}}$.  Prove that \newline $f$ = const.

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\item{4.} For each real $t$ compute the integral $\varphi(t) = \dsize\int^\infty_{-\infty} \frac{e^{itx}}{1+x^2} dx$.

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\item{5.}  Construct a conformal mapping of the unit disk onto the crescent
$$\{z\in {\bold{C}}: |z|<1, \ \ \Bigg|z-\frac{1}{2}\Bigg| \geq \frac{1}{2}\} \ .$$

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\item{6.} How many complex solutions does the equation
$$z= \cos z$$
have? Justify your answer.

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{\underbar{Hint}}.  Use the following fact:

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If an entire function $F(z)$ has no zeros and satisfies
$$|F(z)| \leq C_1 e^{C_2|z|} \ \ (z\in {\bold{C}})$$

then $F(z) = e^{az+b}$.

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\item{7.} Let $f$ be a bounded analytic function in the right half-plane.  Prove that if
$$f(n)=0 \ \text{for} \ n=1,2,3,\dots \ ,$$
then $f\equiv 0$.

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\item{8.} Let $f_1,f_2$ be entire functions, and let $J$ be the set of all combinations
$$A_1f_1 + A_2f_2,$$
where $A_1$ and $A_2$ are entire functions.  Show that there exists an entire function $f$ such that $J$ consist of all entire functions $Af$, where $A$ is entire.

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{\underbar{Hint}}:  Use the result of Problem \#9.

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\item{9.} Let $\{a_n\}$ be a sequence in ${\bold{C}}$, $\dsize\lim_{n\to \infty} a_n=\infty$.  Prove that for any sequence $\{b_n\}$ of complex numbers there exists an entire function $f$ such that $f(a_n)=b_n$.

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