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\topmatter
\title
\bf \bf Ph.D. Comprehensive Examination\\
Complex Analysis Section %Complex Analysis Qualifying Exam
\endtitle
\author
\bf January 1995
\endauthor
\endtopmatter
\document

%\s{NOTE}
%{Please state on the outside of your examination book which
%problems  you want graded.
%All reasoning should be justified.}
\vglue\baselineskip

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

\s{I.1}
{Suppose $f$ is analytic in a domain $D$, and let $a$ be a point in $D$. Let $\{z_n\}$, $\{w_n\}$ be two sequences of points in $D$ such that $z_n\ne w_n$ for all $n$, and  $\lim_{n\to\infty} z_n=a$, $\lim_{n\to\infty} w_n=a$. Show that
$$\lim_{n\to\infty}\frac{f(w_n)-f(z_n)}{w_n-z_n}=f'(a).$$
}

\s{I.2}
{Evaluate $\int_C {e^z}{(z+1)^{-4}}dz$ where $C$ is the imaginary axis from $-i\infty$ to $+i\infty$.}

\s{I.3}
{If $f$ is entire and $f(-z)=f(z)$ for all $z$, then there is an entire function
$g$ satisfying $f(z)=g(z^2)$ for all $z$. }


\s{I.4}
{Let $D=\{z:0<\text{arg}z<3\pi/2\}$. Find a function $u$ which is continuous on $\bar D\smallsetminus\{0\}$, harmonic in $D$, and satisfying $u(x,0)=1$ for $x>0$ and $u(0,y)=0$ for $y<0$.}

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

\s{II.1}
{Let $H=\{z:\text{Im}(z)\ge0\}$. Suppose $F:H\to H$ is analytic and $a\in H$. Prove that
$$|F'(a)|\le\frac{\text{Im} F(a)}{\text{Im}a}.$$ }

\s{II.2}
{Suppose $f$ is a polynomial of degree $n\ge1$ and satisfies $|f(z)|\le1$ on the unit disc. Show that $|f(z)|\le|z|^n$ if $|z|\ge1$.}


\s{II.3}
{Suppose $f$ is holomorphic in the unit disk $D$ and continuous on $\bar D$ and thus
$$f(z)=\sum_{n=0}^\infty c_nz^n,\qquad z\in D.$$
If $f$ has exactly $m$ zeroes in $D$, show that
$$\min\{|f(z)|:|z|=1\}\le|c_0|+|c_1|+\dots+|c_m|.$$
 }

\enddocument