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\topmatter
\title
\bf \bf Ph.D. Comprehensive Examination\\
Complex Analysis Section %Complex Analysis Qualifying Exam
\endtitle
\author
\bf January 1996
\endauthor
\endtopmatter
\document


\t{Part I}{Do three (3) of these problems.}


 \s{I.1}{ Let ${\Bbb R}^{-}=\{ x\in{\Bbb R}:\ x\leq 0\}$. Suppose $f(z)$ is
analytic on ${\Bbb C}\setminus{\Bbb R}^{-}$ and $f(x)=x^{x}$ for $x\in{\Bbb R}$,
$x>0$. Find $f(i)$ and $f(-i)$.}

 \s{I.2} {Let $f(z)$ be an analytic function on an open connected subset
$G\subset{\Bbb C}$. Suppose that $f(z)$ maps $G$ onto a subset of a straight
line. Show that $f(z)$ is a constant.}

 \s{I.3} {Find a conformal mapping from the region $\{z\in{\Bbb C}:\ \vert
z-1\vert >1 \text{ and }\vert z+1\vert >1\}$ onto the punctured disc
$D=\{z\in{\Bbb  C}:\ \vert z\vert <1\}\backslash 0$. Hint: Apply $T(z)={1\over
z}$ first.}

 \s{I.4} {Evaluate $\int_{-\infty}^{\infty}{\cos x\over x^{2}+1}dx$ using
residues.}


\vglue\baselineskip
\t{Part II}{Do two (2) of these problems.}


 \s{II.1}{ Let $G_{1}$ and $G_{2}$ be two bounded simply connected regions, and
let $z_{0}\in G_{1}$ and $w_{0}\in G_{2}$. Show that there exists a bijective
analytic mapping $f(z)$ from $G_{1}$ to $G_{2}$ such that $f(z_{0})=w_{0}$.
}

 \s{II.2} {Let $\Gamma(z)=\int_{0}^{\infty}t^{z-1}e^{-t}dt$, $Re(z)>0$. Show that
$\Gamma(z+1)=z\Gamma(z)$, use this formula to obtain a meromorphic continuation
of $\Gamma(z)$ to the entire complex plane, and find the poles of $\Gamma(z)$ on
$\Bbb C$, their orders and residues.}

 \s{II.3} {\roster
 \runinitem "i)" Let $u(x,y)$ be a harmonic function on the disc $D=\{ z:
\vert z-z_{0}\vert <R\}$. Show that for any $r<R$, $u(z_{0})={1\over
2\pi}\int_{0}^{2\pi}u(z_{0}+re^{i\theta})d\theta$.
 \item "ii)" Let $u(x,y)$ be a
harmonic function on a bounded region $G$ that is continuous on the closure
$\overline G$ of $G$. Show that $u(x,y)$ achieves its maximum and minimum values
on the boundary of $G$. 
 \endroster}
 \enddocument