Reading time: about 18 minutes (3705 words).

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.

  1. (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 Cauchy-Riemann 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 $$f'(z_0) = \lim_{z\rightarrow z_0} \frac{f(z) - f(z_0)}{z-z_0}$$ 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 $T$.)
  1. 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, well-known theorems and principles such as Liouville's Theorem, the Argument Principle, the Maximum Principle, Rouche's Theorem, and/or the Open Mapping Theorem.

  2. (a) State and prove the Casorati-Weierstrass 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 Casorati-Weierstrass Theorem directly for a specific analytic function of your choice, with a suitable singularity.

  3. (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.

  1. (a) Suppose that $f$ is analytic on a region $G\subset \mathbb C$ and $\{z\in \mathbb C: |z-a|\leq R\} \subset G$. Show that if $|f(z)| \leq M$ for all $z$ with $|z-a|=R$, then for any $w_1, w_2\in \{z\in \mathbb C: |z-a|\leq \frac{1}{2}R\}$, 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 Arzela-Ascoli Theorem to prove Montel's Theorem asserting the normality of any locally bounded family $F$ of analytic functions on a region $G$.