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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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