\input vanilla.sty \magnification=1200 \baselineskip=10pt \nopagenumbers \centerline{\bf{Ph.D. Preliminary Examination}} \bigskip \centerline{\bf{Complex Analysis}} \bigskip \centerline{\bf{September 3, 1993}} \bigskip \subheading{\underbar{Notations}} \bigskip \item{(a)} $\Delta=\{z\in {\bold{C}}: |z|<1\}$ is the open unit disk. \bigskip \item{(b)} ${\overline{\Delta}}=\{z \in {\bold{C}}: |z|\leq 1\}$ is the closed unit disk. \bigskip \item{(c)} ${\overline{\bold{C}}}={\bold{C}}\cup \{\infty\}$ is the extended complex plane. \midspace{.25in} \item{1.} Let $f: \Delta\to \Delta$ be analytic. Suppose there exists $z_0\in \Delta$ with $f(z_0)=z_0$ and $f'(z_0)=1$. Prove that $f(z)\equiv z$. \bigskip \item{2.} Consider the elementary functions: \bigskip \item\item{(1)} $\cos: {\bold{C}}\mapsto {\bold{C}}$; \bigskip \item\item{(2)} $\tan: {\bold{C}}\mapsto {\overline{\bold{C}}}$. \bigskip A. Is $\tan$ continuous on ${\bold{C}}$? \bigskip B. Determine \bigskip \item\item{(a)} the range of $\cos z$; \bigskip \item\item{(b)} the range of $\tan$; \bigskip \item\item{(c)} $\cos^{-1} \{\dsize\frac{5}{4}\}$; \bigskip \item\item{(d)} $\tan^{-1} \{\infty\}$. \bigskip \item{3.} Let $L$ denote the line segment joining $-i$ and $i$, and let $\Omega={\overline{\bold{C}}} \backslash L$. \bigskip \item\item{A.} Map $\Omega$ conformally onto $\Delta$. \bigskip \item\item{B.} Deduce that if $f$ is an entire function such that $f({\bold{C}})\cap L=\phi$ then $f(z)$ is constant. \newpage \item{4.} Let $z_0$ be a simple pole of a function $f(z)$. \bigskip A. Prove $$\lim_{r\to 0^+} \int_{\gamma_r} f(z)dz=(b-a)i \ \text{Res}[f,z_0] \ ,$$ where $$\gamma_r=\{z_0+re^{it}: a\leq t\leq b\} \ .$$ \bigskip B. Use this result to evaluate $\int^\infty_{-\infty} \dsize\frac{\sin x}{x}dx$. \bigskip \item{5.} Suppose $f$ is a nonconstant analytic function in $\Delta$. Show that the function $$M(r)=\max_{|z|=r} |f(z)|, \ 0