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MATH 606
{\bf 1.} Let $\Omega$ be a domain in$\Bbb C$ , $a\in \Omega$ and $B_R (a) = {z; |z - a| < R}$ $\subset$ $\Omega$. Prove the following mean-value property for any $f \in A (\Omega)$ :
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$$\item i) $f (a) = \frac {1}{2 \pi} \int_0^2\pi f (a + Re^i\theta) d \theta.$$
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\item ii) $f (a) = \frac{1}{\pi R ^2} \quad \iint\limits_{B_R (a)}\ \ f (z) d \ x\ d\ y.$
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{\bf 2.} Let $\Omega$ be a simply connected domain in $\Bbb C$ and U is a real harmonic function in $\Omega$.
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\item i) Prove that there exists an f which is holomorphic in $\Omega$ and U = Ref.
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\item ii) Show that this fails in every region which is not simply connected.
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\item iii) Show that $U^2$ cannot be harmonic in $\Omega$ unless U is a constant.
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{\bf 3.) Let $F$ be the family of all functions \{f\} such that $f$ is holomorphic in the open unit disc\ \ $D = \{z;\ \ |z| < \}$\ and $f$ (0) = 1,\ \ Re f \ $>$ 0. Show that $F$ is a normal family.
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{\bf 4. }
\item i) Let $\{fn (z) \}$ \ be a sequence of entire functions which converges uniformly on compact subsets of $\Bbb C$ to $f (z)$. Prove that $f (z)$ is entire.
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\item 2) Show that \quad $\sum\limits_{n = 1}^\infty$\ \ $\frac{(sin z)^n}{N^n}$ \quad defines an entire function.
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{\bf 5.}
\item i) State and prove the Mittag-Leffler theorem for the whole plane without using Rumge's theorem.
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\item ii) State the Rumge's theorem for a bounded domain $\Omega$ and apply the Rumge's theorem to prove the Mittag-Leffler theorem for bounded domain $\Omega$.
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{\bf 6.}
\item a) Let $a \epsilon \ \Bbb C$ and $\Omega = \Bbb C\\ \setminus\{a\}$. Can one map $\Omega$ onto the unit disc holomorphically? Can one map $\Omega$ onto an annulus holomorphically?
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\item b) Let $\Omega$ = $\Bbb C$ $\setminus$\ \${ (0, \infty) \}$. Can one map $\Omega$ conformally onto the unit disc? If the answer for a) or b) is yes, find such a map.
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