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\centerline{\bf Complex Prelim, September 1998}
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\centerline{Do any six problems}

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\item{1.} Suppose $f$ is analytic in $|z|<1$ and
$$f({1\over n^2})={1\over n}$$
for all $n>2$. Show that $f$ is identically zero in $|z|<1$.
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\item{2.} Suppose $f$ is entire and $|f(z)|\le\log(1+|z|)$ for all
$z$. Show that $f$ is identically zero.
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\item{3.} Show that the function
$$u(x,y)=\arctan(y/x)$$
is harmonic in the (open) right half plane. Find its harmonic conjugate
there.
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\item{4.} Suppose $\{f_n\}$ is a sequence of analytic functions
in $|z|<1$. If
$$\lim_{n\to\infty}\int_{|z|<1}|f_n(z)|\,dA(z)=0,$$
where $dA$ is area measure on $|z|<1$, show that
$$\lim_{n\to\infty}f_n(z)=0$$
for all $|z|<1$.
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\item{5.} Evaluate the integeral
$$I=\int_C{\sin\zeta\over\zeta(\pi-6\zeta)^2}\,d\zeta,$$
where $C$ is the positively oriented unit circle.
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\item{6.} Suppose $f$ is entire and
$$|f(ne^{i\theta})|\le\exp(n\cos\theta)$$
for all $n\ge1$ and $\theta\in[0,2\pi]$. Show that $f(z)=ce^z$ for
some constant $c$ with $|c|\le1$.
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\item{7.} Find a conformal map from the unit disk $|z|<1$ onto the
region $|arg z|<\pi/3$.

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