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\centerline{\bf Preliminary Examination in Complex Analysis}

\centerline{\bf June 4, 1996}

\centerline{Notation:  $\Delta = \{z\in {\bold{C}}: |z|<1\}$ is the open unit disk}

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\item{1.} Let $f$ be analytic in a nonempty connected open set $U$.  Let $F$ be a nonconstant entire function.  Show that if $F(f(z))=0$ for all $z$ in a neighborhood of some $z_0\in U$, then $f$ is constant in $U$.

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\item{2.} (a) Find all constants $c_1$ and $c_2$ so that the functions
$$f_1(z)=c_1z \ \ \text{and} \ \ f_2(z)=\frac{c_2}{z}$$
define conformal self-maps of the annulus ${\cal{A}}=\{z\in {\bold{C}}: a<|z|<b\}$ ($0<a<b$ are given constants).

(b) Prove that there are no other conformal self-maps of ${\cal{A}}$.

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\item{3.} Evaluate
$$\int_\gamma \ \frac{1-\cos z}{(e^z-1) \sin z} \ dz$$
where the path $\gamma$ is the circle $|z|=e$ traversed once counterclockwise.

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\item{4.}  For $n\in {\bold{N}}$ show that
$$\int_\Delta \Bigg|\frac{1-z^n}{1-z}\Bigg|^2 dxdy = \pi\bigg(1+\frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}\Bigg)$$

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\item{5.} Let $f$ be analytic in $\Delta$, and let $f(\Delta) \subseteq \Delta$.  Prove that if $f(0)=0$ and $f(a)=a$ for some $a\neq 0$, then $f(z)=z$.

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\item{6.} Let $f$ be analytic in $\Delta$.  Show that
$$\sup_{z\in \Delta} \ (1-|z|^2) \ |f'(z)| \leq \sup_{z\in \Delta} \ |f(z)| \ .$$

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\item{7.} Let $f(z)$ be analytic in $\Delta$.  Suppose
$$\lim_{r\uparrow 1} \int^{2\pi}_0 |f(re^{i\theta})|d\theta = 0  \ .$$
Show that $f\equiv 0$.

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\item{8.} Prove that the zero set ${\cal{S}}$ of $e^z+z$:
$${\cal{S}} = \{z\in {\bold{C}}: e^z+z=0\}$$
is nonempty:  ${\cal{S}}\neq \emptyset$.

\underbar{Bonus}.  Prove that ${\cal{S}}$ is an infinite set.

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\item{9.} Find $w=f(z)$ that maps $\Delta$ conformally onto the strip $|$Im $w| < \dsize\frac{\pi}{2}$ so that $f(0)=0$ and $f'(0)>0$.

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