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\centerline{\bf Complex Analysis Prelim}

\centerline{\bf January, 2005}

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Throughout let $D= \{z: |z|<1\}$ and ${\bold{C}}$ = complex plane.

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\item{1.} Let $u(x,y) = \dsize\frac{y}{x^2+y^2}$.  Show that $u$ is harmonic in the punctured plane ${\bold{C}} \backslash \{0\}$ in 2 ways:

A.  From the definition of harmonic.

B.  By finding a function $f$, analytic in ${\bold{C}} \backslash \{0\}$ with $u$ = Ref.

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\item{2.} Construct a conformal map of the region $D \backslash \{z: |z+ \dsize\frac{1}{2}| \leq \dsize\frac{1}{2}\}$ onto the region \newline ${\bold{C}} \backslash \{z: Re \ z \leq 0\}$.

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\item{3.} Let $\alpha$ be a real number and consider the integral $\dsize\int^\infty_{-\infty} \dsize\frac{dx}{x^2-2x+ \alpha}$.  Determine for what $\alpha$ the integral converges and, in those cases, determine its value.  Include the details of your contour argument.

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\item{4.} Let $f(z) = \cos(i \ z^3)$.  Determine $Z(f) = \{z: f(z)=0\}$.  Indicate with a picture where the solutions lie in ${\bold{C}}$.

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\item{5.} Let $p(z) = 3z^{15} + 4z^8 + 6z^5 + 19z^4 + 3z+1$.  Show that $p(z)$ has 4 zeros for $|z|<1$ and 11 zeros for $1< |z| <2$.

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\item{6.} Let $f: D\to D$ be analytic and satisfy $f(\dsize\frac{1}{2}) = \dsize\frac{1}{2}$ and $f'(\dsize\frac{1}{2}) = -1$.  Find an explicit formula for $f$.

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\item{7.} Let $f$ and $g$ be analytic in a nonempty connected open set $U$ and satisfy $|f| = |g|$ there.  What else can you deduce about the relationship between $f$ and $g$.  Justify your answer.

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\item{8.} Let $f$ be analytic in $D$ and satisfy $|f(z)| \leq\dsize\frac{1}{1-|z|}$ there.  Show that $|f'(0)| \leq 4$.

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\item{9.} Let $\{b_n\}$ be a sequence of complex numbers such that $\limsup |b_n|^{\frac{1}{n}} = 1$.  Let ${\cal{F}}$ be the family of function $f(z) = \dsize\sum^\infty_{n=0} a_n z^n$ which are analytic in $D$ and satisfy $|a_n| \leq |b_n|$, $n=0,1,2, \dots$  Prove that ${\cal{F}}$ is a compact family in the topology of uniform convergence on compact sets in $D$.

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\item{10.} A.  State carefully the Riemann Mapping Theorem.

\itemitem{B.} Let $f$ be a conformal map from $D$ onto $D$ satisfying $f(0)=0$ and $|f'(0)|=1$.  Using only the Riemann Mapping Theorem show that $f(e^{i \theta}z) = e^{i \theta} f(z)$ for every real number $\theta$.

\itemitem{C.} Deduce that there is a real number $\theta_0$ such that $f(z) = e^{i \theta_0} z$.

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