2001 November 21
Instructions. Make a substantial effort on all parts of the following problems. If you cannot completely answer Part (a) of a problem, it is still possible to do Part (b). Partial credit is given for partial progress. Include as many details as time permits. Throughout the exam, $z$ denotes a complex variable, and $\mathbb C$ denotes the complex plane.
 (a) Suppose that $f(z) = f(x+iy) = u(x,y) + i v(x,y)$ where $u$ and $v$ are $C^1$ functions defined on a neighborhood of the closure of a bounded region $G\subset \mathbb C$ with boundary which is parametrized by a properly oriented, piecewise $C^1$ curve $\gamma$. If $u$ and $v$ obey the CauchyRiemann equations, show that Cauchy's theorem $\int_\gamma f(z) \ dz = 0$ follows from Green's theorem, namely $$\int_\gamma P\ dx + Q\ dy = \int_G \left(\frac{\partial Q}{\partial x}\frac{\partial P}{\partial y}\right)\ dx\ dy,$$ for $C^1$ functions $P$ and $Q$.
(b) Suppose that we do not assume that $u$ and $v$ are $C^1$, but merely that $u$ and $v$ are continuous in $G$ and $$f'(z_0) = \lim_{z\rightarrow z_0} \frac{f(z)  f(z_0)}{zz_0}$$ exists at some (possibly only one!) point $z_0 \in G$. Show that given any $\epsilon >0$, we can find a triangular region $\Delta$ containing $z_0$, such that if $T$ is the boundary curve of $\Delta$, then $$\left\int_T f(z)\ dz\right = \frac{\epsilon L^2}{2},$$ where $L$ is the length of the perimeter of $\Delta$. (Hint: Note that part (a) yields $\int_T (az+b) \ dz =0$ for $a, b \in \mathbb C$, which you can use here in (b), even if you could not do Part (a). You may also use the fact that $\left\int_T g(z)\ dz\right \leq L \cdot \sup\{g(z):z\in T\}$ for $g$ continuous on $T$.)

Give two quite different proofs of the Fundamental Theorem of Algebra that if a polynomial with complex coefficients has no complex zero, then it is constant. You may use independent, wellknown theorems and principles such as Liouville's Theorem, the Argument Principle, the Maximum Principle, Rouche's Theorem, and/or the Open Mapping Theorem.

(a) State and prove the CasoratiWeierstrass Theorem concerning the image of any punctured disk about a certain type of isolated singularity of an analytic function. You may use the fact that if a function $g$ is analytic and bounded in the neighborhood of a point $z_0$, then $g$ has a removable singularity at $z_0$.
(b) Verify the CasoratiWeierstrass Theorem directly for a specific analytic function of your choice, with a suitable singularity. 
(a) Define $\gamma : [0,2\pi] \rightarrow \mathbb C$ by $\gamma(t) = \sin (2t) + 2i \sin (t)$. This is a parametrization of a "figure 8" curve, traced out in a regular fashion. Find a meromorphic function $f$ such that $\int_\gamma f(z) \ dz = 1$. Be careful with minus signs and factors of $2\pi i$.
(b) From the theory of Laurent expansions, it is known that there are constants $a_n$ such that, for $1<z<4$, $$\frac{1}{z^2  5z + 4} = \sum_{n=\infty}^\infty a_n z^n.$$ Find $a_{10}$ and $a_{10}$ by the method of your choice.
 (a) Suppose that $f$ is analytic on a region $G\subset \mathbb C$ and $\{z\in \mathbb C: za\leq R\} \subset G$. Show that if $f(z) \leq M$ for all $z$ with $za=R$, then for any $w_1, w_2\in \{z\in \mathbb C: za\leq \frac{1}{2}R\}$, we have $$f(w_1)  f(w_2) \leq \frac{4M}{R} w_1  w_2.$$ (b) Explain how Part (a) can be used with the ArzelaAscoli Theorem to prove Montel's Theorem asserting the normality of any locally bounded family $F$ of analytic functions on a region $G$.