1991 Nov 21
Instructions. In each of sections A, B, and C, do all but one problem.
Time Limit. 2 hours
SECTION A (Do 3 of the 4 problems.)

Where does the function $f(z) = z\ \mathrm{Real} z + \bar{z}\ \mathrm{Imag} z + \bar{z}$ have a complex derivative? Compute the derivative wherever it exists.

(a) Prove that any nonconstant polynomial with complex coefficients has at least one root.
(b) From (a) it follows that every nonconstant polynomial $P$ has the factorization $$P(z) = a \prod_{n=1}^N (z  \lambda_n),$$ where $a$ and each root $\lambda_n$ are complex constants. Prove that if $P$ has only real coefficients, then $P$ has a factorization $$P(z) = a \prod_{k=1}^K (z  r_k) \prod_{m=1}^M (z^2  b_m z + c_m),$$ where a and each $r_k, b_m, c_m$ are real constants. 
Use complex residue methods to compute the integral $$\int_0^\pi \frac{1}{5 + 3 \cos \theta} \ d\theta.$$
 (a) Explain how to map an infinite strip (i.e., the region strictly between two parallel lines) onto the unit disk by a onetoone conformal mapping.
(b) Two circles lie outside one another except for common point of tangency. Explain how to map the region exterior to both circles (including the point at infinity) onto an infinite strip by a onetoone conformal mapping.
SECTION B (Do 3 of the 4 problems.)

Suppose that $f$ is analytic in the annulus $1< z<2$, and that there exists a sequence of polynomials converging to $f$ uniformly on every compact subset of this annulus. Show that $f$ has an analytic extension to all of the disk $z<2$. (See also 1996 April (8).)

Let $f$ be analytic in $z<2$, with the only zeros of $f$ being the distinct points $a_1, a_2, \ldots, a_n$, of multiplicities $m_1, m_2, \dots, m_n$, respectively, and with each $a_j$ lying in the disk $z<1$. Given that $g$ is analytic in $z<2$, what is $$\int_{z=1} \frac{f'(z) g(z)}{f(z)}\ dz \quad \text{?}$$ (Verify your answer.)

Let $\{f_n\}$ be a sequence of analytic functions in the unit disk $D$, and suppose there exists a positive constant $M$ such that $$\int_C f_n(z)\ dz \leq M$$ for each $f_n$ and for every circle $C$ lying in $D$. Prove that $\{f_n\}$ has a subsequence converging uniformly on compact subsets of $D$.
 State and prove:
(a) the mean value property for analytic functions
(b) the maximum principle for analytic functions.
SECTION C (Do 2 of the 3 problems.)

Let $X$ be a Hausdorff topological space, let $K$ be a compact subset of $X$, and let $x$ be a point of $X$ not in $K$. Show that there exist open sets $U$ and $V$ such that $$K \subset U, ; x\in V, ; U\cap V = \emptyset.$$

A topological space $X$ satisfies the second axiom of countability. Prove that every open cover of $X$ has a countable subcover.

Let $X$ be a topological space, and let $U$ be a subset of $X$.
(a) Show that if an open set intersects the closure of $Y$ then it intersects $Y$.
(b) Show that if $Y$ is connected and if $Y\subset Z \subset \bar{Y}$, then $Z$ is connected.