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\centerline{\bf{Ph.D. Prelim in Complex Analysis}}

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\centerline{\bf{January 18, 1994}}

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\item{1.} Let $f$ be analytic in the unit disk
${\bold{D}}$.  Use Cauchy's integral formula to establish
the power series representation of $f$ in ${\bold{D}}$.  Obtain both an integral formula and a derivative formula for the $n$-th coefficient.

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\item{2.} Let $\Omega$ be a region and let ${\cal{F}}=\{f: f \ \text{is analytic in} \ \Omega$ and $|f(z)|\leq 1, \ \forall z \in \Omega\}$.  Fix $z_0\in \Omega$ and show that $\exists \ g \in {\cal{F}}$ such that Re $g'(z_0) \geq$ Re $f'(z_0)$, $\forall f \in {\cal{F}}$.

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\item{3.} Let $f$ be analytic and nonconstant in a region
$\Omega$ with $\mu=$ Ref and $v=$ Imf $f$.

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\item\item{(a)} Show that $|f'(z)|^2=u^2_x + u^2_y=v^2_x+v^2_y$.

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\item\item{(b)}  Determine all real numbers $a$ and $b$ such that $au^2+bv^2$ is harmonic in $\Omega$.

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\item{4.} Let $\Omega=\{z: |z-i|<1\}$ and $H=\{z: \text{Im}
\ z>0\}$.  Map $H\backslash {\overline{\Omega}}$ conformally
onto $\Omega$.

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\item{5.} If $p$ is a polynomial, prove that the series
$\dsize\sum^\infty_{n=0} p(n)z^n$ defines a rational
function.  {\bf{HINT}}:  Note that any linear combination of
rational functions is a rational function.

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\item{6.} (a) Let $f$ be analytic in the unit disk
$\bold{D}$ with
$$\lim_{|z|\to 1-} \ f(z)=0 \ .$$
Prove $f\equiv 0$.

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\item\item{(b)} Let $g$ be analytic in $\bold{D}$.  Prove
that the statement
$$\lim_{|z|\to 1-} \ g(z)=\infty$$
is impossible.

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\item{7.} Let $f$ be meromorphic in $\bold{C}$ and bounded
outside of some circle.  Determine the form of $f$ as
completely as possible.

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\item{8.} Let $\Gamma=\{z: |z|=1\}$.

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\item\item{(a)} Show that the mapping $z\longmapsto (z+1)^2$
takes $\Gamma$ onto the cardioid $r=2(1+\cos \theta)$.
Sketch this cardioid.

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\item\item{(b)} Let $g(w)= \int_\Gamma
\dsize\frac{z(z+1)}{z^2+2z-w} dz$ ($\Gamma$ traversed once
counterclockwise).  Use the result of part (a) to sketch a
domain containing 0 on which $g$ is analytic.

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\item\item{(c)} Determine $g(0)$ and $g'(0)$.


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