GENERAL EXAM - ANALYSIS

January 2010
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

Let $w_{1}, w_{2}, \cdots, w_{n}$ be $n$ points on the unit circle $\mathbb{T} \equiv\{z \in \mathbb{C}:|z|=1\}$. Show that there exists a point $z \in \mathbb{T}$ so that the product of the distances from $z$ to $w_{j}$ (i.e., $\prod_{j=1}^{n}\left|z-w_{j}\right|$ ) is at least 1 . Then show there there is a point $v \in \mathbb{T}$ so that the product of the distances from $v$ to the points $w_{j}$ is exactly 1 .

Problem 2

Suppose that $f$ is analytic in an open set containing the closed unit disk $\overline{\mathbb{D}}=\{z:|z| \leq 1\}$, except for a simple pole at $z_{0}$ where $\left|z_{0}\right|=1$. Show that if

$$\sum_{n=0}^{\infty} a_{n} z^{n}$$

is the power series for $f$ in $\mathbb{D}$, then

$$\lim _{n \rightarrow \infty} \frac{a_{n}}{a_{n+1}}=z_{0}$$

Problem 3

Suppose that $f$ and $g$ are analytic in an open set containing the closed unit disk $\overline{\mathbb{D}}= \{z \in \mathbb{C}:|z| \leq 1\}$. Suppose that $f$ has a simple zero at $z=0$ and no other zero in $\overline{\mathbb{D}}$. Set

$$f_{\epsilon}(z)=f(z)+\epsilon g(z)$$

Show that if $\epsilon$ is sufficiently small, then $f_{\epsilon}$ has a unique zero in $\overline{\mathbb{D}}$.

Problem 4

Evaluate

$$\int_{0}^{\infty} \frac{d x}{1+x^{n}}$$

for $n=2,3,4, \ldots$ by integrating over the boundary of an appropriately chosen "pie-shaped" region.

Problem 5

Let $\mathcal{B}\left(\mathbb{R}^{1}\right)$ be the collection of Borel sets on the real line and $\mathcal{B}\left(\mathbb{R}^{2}\right)$ those of the plane.
(a) Show that sections of Borel sets in the plane, e.g., sets of the form $B_{y}=\{x \in \mathbb{R}:(x, y) \in \mathcal{B}\}$ with $B \in \mathcal{B}\left(\mathbb{R}^{2}\right)$ are in $\mathcal{B}\left(\mathbb{R}^{1}\right)$.
(b) Suppose that $f(x, y)$ is a Borel measurable function on the plane, $\mathbb{R}^{2}$. Show that for fixed $y, f_{y}(x)=f(x, y)$ is a Borel function on $\mathbb{R}$.

Problem 6

Let $0<a<1<b$ be real constants. Define the function $S(x)$ on $[0, \infty)$ by the equation

$$S(x)=\sum_{n=1}^{\infty} a^{n} \chi_{\left[0, b^{n}\right]}(x)$$

where $\chi_{\left[0, b^{n}\right]}(x)$ is the characteristic function for $\left[0, b^{n}\right]$, equal to 1 on this interval and zero otherwise.
(a) Show that $S(x) \in L^{p}([0, \infty), d x)$,ie., is $L^{p}$-integrable (with Lebesgue measure), provided $a b^{1 / p}<1$.
(b) Show that

$$S(x) \leq \frac{x^{(\ln a / \ln b)}}{1-a}$$

Problem 7

Let $\left\{\phi_{n}\right\}$ be a complete orthonormal set of functions in the real Hilbert space $L^{2}([0,1])$ with inner product

$$\langle f, g\rangle=\int_{0}^{1} f(x) g(x) d x$$

For $0 \leq a \leq 1$, set

$$h(x)=\sum_{n=1}^{\infty} \phi_{n}(x) \int_{0}^{a} \phi_{n}(y) d y$$

(a) Show that the series converges in an $L^{2}$-sense to an $L^{2}$ function.
(b) Show that for any $b, 0 \leq b \leq 1$,

$$\int_{0}^{b} h(x) d x=\sum_{n=1}^{\infty} \int_{0}^{b} \phi_{n}(x) d x \int_{0}^{a} \phi_{n}(y) d y$$

(c) Show that

$$\int_{0}^{b} h(x) d x=\min (a, b)$$