Analysis General Exam August 20, 2014

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To get credit for a problem, you must show all of your reasoning and calculations.

Problem 1

Suppose $f:[0,1] \rightarrow \mathbb{C}$ is continuous. Find

$$\lim _{k \rightarrow \infty} \int_{0}^{1} k x^{k-1} f(x) d x$$

Prove your result.

Problem 2

(a) Show that any open subset of $\mathbb{R}$ is a countable disjoint union of open intervals $(a, b)$ (where $-\infty \leq a<b \leq \infty$ ).
(b) Show that the $\sigma$-algebra generated by the open subsets of $\mathbb{R}$ is the same as the $\sigma$-algebra generated by the intervals $[a, b)$ with $-\infty<a<b<\infty$.

Problem 3

Suppose ( $X, \Sigma$ ) is a measurable space, and $\nu$ and $\mu$ are two measures on the $\sigma$-algebra $\Sigma$ with $\nu$ absolutely continuous with respect to $\mu$. Suppose $\nu$ is a finite measure. Show that $\nu(E) \rightarrow 0$ as $\mu(E) \rightarrow 0$. (In other words, given $\epsilon>0$ there exists $\delta>0$ such that if $E \in \Sigma$ with $\mu(E)<\delta$, then $\nu(E)<\epsilon$.)
[Hint: If not show $\exists \epsilon>0$ and $E_{j} \in \Sigma$ such that $\bigcup_{j=1}^{\infty} E_{j}<\infty$ and $\nu\left(E_{j}\right) \geq \epsilon$. Proceed from there.]

Problem 4

(a) Use the Monotone Convergence Theorem and $\int_{1}^{t} \frac{d x}{x}=\log t$ to show

$$\lim _{n \rightarrow \infty} n \log \left(1+\frac{t}{n}\right)=t \text { for } t \geq 0$$

(b) Show $\lim _{n \rightarrow \infty} \int_{0}^{n}\left(1+\frac{t}{n}\right)^{n} e^{-2 t} d t=1$.
(c) Let $\Gamma(x)=\int_{0}^{\infty} t^{x-1} e^{-t} d t$ for $x>0$. Show that

$$\begin{aligned} \Gamma(x) & =\lim _{n \rightarrow \infty} \int_{0}^{n}\left(1-\frac{t}{n}\right)^{n} t^{x-1} d t \\ & =\lim _{n \rightarrow \infty} n^{x} n!(x(x+1) \cdots(x+n))^{-1} \end{aligned}$$

Problem 5

(a) State the Mean Value Property for analytic functions, then use the Cauchy Integral Formula to prove it.

(b) TRUE or FALSE: If $u$ is a harmonic function on a domain in $\mathbb{R}^{2}$, then $u$ has a harmonic conjugate.
(c) Find a fractional linear transformation (also called a Möbius transformation) $f$ that takes the first quadrant to the top half of the unit disk and satisfies $f(2)=i$. (You must explain some comprehensible procedure and not simply produce an $f$ out of thin air.) Under your map, what is the image of the vertical ray $\{\operatorname{Re} z= c>0, \operatorname{Im} z>0\} ?$

Problem 6

Let $f(z)$ be a bounded analytic function in the upper half-plane that extends continuously to the real axis. If $|f(z)| \leq M$ for real $z$, show that $|f(z)| \leq M$ for all $z$ in the upper half-plane.
[Suggestion: for the top half of an arbitrary disk centered at the origin, consider an appropriate branch of the function $(z+i)^{-\varepsilon} f(z)$ for small enough $\varepsilon>0$.]

Problem 7

Use the argument principle to determine the number of roots of $p(z)=z^{9}+4 z^{5}- 3 z^{4}+4 z+\alpha$ in the right half-plane. The answer may depend on the value of $\alpha$, which is assumed real.

Problem 8

Let $a$ and $b$ be unequal positive numbers. By integrating an appropriate branch of $\frac{(\log z)^{2}}{(z+a)(z+b)}$ around a keyhole contour, find

$$\int_{0}^{\infty} \frac{\log x}{(x+a)(x+b)} d x$$