Analysis General Exam - August 2011

Closed book, closed notes. Please pledge. In each problem, justify all assertions, show calculations, and identify those theorems which you invoke in your arguments.

Problem 1

(a) Suppose that $f$ is analytic in an open set containing the closed disk $\{z$ : $|z| \leq R\}$ and that $a, b$ are two complex numbers with $|a|<R$ and $|b|<R$. Evaluate

$$\int_{\gamma_{R}} \frac{f(z)}{(z-a)(z-b)} d z$$

where $\gamma_{R}$ is the positively oriented circle centered at 0 with radius $R$.
(b) Using your work in (a), prove Liouville's theorem on bounded entire functions.

Problem 2

Let $a$ be a real number that is not an integer and let

$$f(z)=\frac{\pi \cos \pi z}{(z+a)^{2} \sin \pi z}$$

(a) Compute the residue of $f$ at each of its singularities.
(b) Consider the rectangle $\gamma_{n}$ as shown. Show that $\int_{\gamma_{n}} f(z) d z \rightarrow 0$ as $n \rightarrow \infty$. (You may give "order of magnitude" estimates in doing this.)

(c) Use your work in (a) and (b) to find a formula for

$$\sum_{k=-\infty}^{\infty} \frac{1}{(k+a)^{2}}$$

(d) What is

$$\sum_{k=-\infty}^{\infty} \frac{1}{(2 k+1)^{2}} ?$$

Problem 3

Suppose that $a_{1}, a_{2}, \cdots, a_{n}$ are $n$ points in the open unit disk $\mathbb{D}=\{z:|z|<1\}$. Set

$$F(z)=\prod_{m=1}^{n} \frac{z-a_{m}}{1-\overline{a_{m}} z}$$

Show that for each $c \in \mathbb{D}$, the equation

$$F(z)=c$$

has $n$ roots in $\mathbb{D}$ (counting multiplicities).

Problem 4

A function $f(z)$ that is analytic in the unit disk $\mathbb{D}=\{z:|z|<1\}$ is said to be subordinate to the analytic function $F(z)$ if $f(z)=F(g(z))$ for some function $g(z)$ that is analytic in $\mathbb{D}$ and satisfies $|g(z)| \leq|z|$ there. This is written $f \prec F$.
(a) Show that if $f \prec F$, then $\left|f^{\prime}(0)\right| \leq\left|F^{\prime}(0)\right|$.
(b) Suppose that $f$ is analytic in $\mathbb{D}, f(0)=0$ and $|\operatorname{Re} f(z)|<1$ for all $z \in \mathbb{D}$. Set

$$F(z)=\frac{2}{\pi} i \log \frac{1+z}{1-z}$$

Show that $f \prec F$. (Log denotes the principal branch.)
(c) Let $f$ be as in (b). Show that $\left|f^{\prime}(0)\right| \leq \frac{4}{\pi}$.

Problem 5

Let $(X, \Sigma, \mu)$ be a measure space, $\Sigma$ a $\sigma$-algebra of sets, $\mu$ a measure defined on $\Sigma$. Let $\mathcal{H}=L^{2}(X, \Sigma, \mu)$ be the Hilbert space of square-integrable $\Sigma$-measurable functions. Let $\Sigma_{0} \subset \Sigma$ be a sub- $\sigma$-algebra of $\Sigma$ and let $\mathcal{H}_{0}=L^{2}\left(X, \Sigma_{0}, \mu\right)$ be the subspace of $\mathcal{H}$ consisting of functions in $\mathcal{H}$ which are $\Sigma_{0}$-measurable.

Let $f$ be a function in $\mathcal{H}$. Show that there is a function $f_{0}$ in $\mathcal{H}_{0}$ (in particular $\Sigma_{0}$-measurable) such that

$$\int_{X} f_{0}(x) g(x) d \mu(x)=\int_{X} f(x) g(x) d \mu(x) \text { for all } g \in \mathcal{H}_{0}$$

Explain your reasoning.

Problem 6

(a) Let $F(x, y)$ be a continuous real-valued function defined on the closed square $[0,1] \times[0,1] \subset \mathbb{R}^{2}$. Show that

$$g(x) \equiv \sup _{y \in[0,1]} F(x, y)$$

is lower semi-continuous in the sense that for each $x \in[0,1]$ and $\epsilon>0$, there is a $\delta$ such that

$$g(z)>g(x)-\epsilon \text { for }|z-x|<\delta$$

(b) Let $f(x)$ be a continuous real-valued function defined on $[0,1]$, and for $\kappa>0$ define

$$A_{\kappa} \equiv\{x \in[0,1]:|f(x)-f(y)| \leq \kappa|x-y| \text { for all } y \in[0,1]\}$$

By considering $F(x, y) \equiv|f(x)-f(y)|-\kappa|x-y|$, show that the complement $A_{\kappa}^{c}$ of $A_{\kappa}$ is open, hence $A_{\kappa}$ is closed.
(c) Show that for $\kappa>0$,

$$B_{\kappa} \equiv\{x \in[0,1]:|f(y)-f(x)|<\kappa|x-y| \text { for all } y \in[0,1], y \neq x\}$$

is a Borel measurable set of the real line.

Problem 7

Consider the real-valued function $F(x), x \in \mathbb{R}$, defined by the (improper) Riemann integral

$$F(x)=\int_{0}^{\infty} \frac{\cos (x t) d t}{1+t}$$

Show that $F(x)$ is a continuous function of $x$ for $0<x<\infty$. Hint: First integrate by parts to obtain an absolutely convergent integral.

Problem 8

Let $\left\{a_{n}\right\}_{n=1,2, \ldots}$ be a square summable sequence of complex numbers such that $\sum_{n}\left|a_{n}\right|^{2}=1$, and set

$$f_{r}(x)=\sum_{n \geq 1} r^{n} a_{n} e^{i n x}$$

for $0 \leq r<1$.
(a) Show that for each $r \in[0,1)$, the series defining $f_{r}(x)$ converges for each $x$, and that $f_{r}(x)$ is a bounded function of $x$. In particular, $f_{r}(x)$ is in $L^{2}([0,2 \pi], d x)$, the space of square-integrable functions on the interval $[0,2 \pi]$ with Lebesgue measure.
(b) Let $\left\{r_{j}\right\}_{j=1,2, \ldots}$ be a sequence of numbers in $[0,1)$ such that $\lim _{j \rightarrow \infty} r_{j}=1$. Show that the sequence of functions $\left\{f_{r_{j}}\right\}$ is an $L^{2}$-Cauchy sequence.
(c) Let

$$f=\lim _{j \rightarrow \infty} f_{r_{j}}$$

the limit meaning in an $L^{2}$-sense. Show that

$$a_{m}=\frac{1}{2 \pi} \int_{[0,2 \pi]} e^{-i m x} f(x) d x$$

for all integers $m$.