# Analysis Exercises

#### Real Analysis

• Let $C$ be the Cantor set. Let $f$ be a bounded real function on $[0,1]$ which is continuous at every point outside of $C$. Prove that $f \in \mathcal{R}$ on $[0,1]$.
• If $f(x) = 0$ for all irrational $x$, $f(x) = 1$ for all rational $x$, prove that $f \notin \mathcal{R}$ on $[a,b]$ for any $a < b$.
• Let $I = [0,1]$ be the closed unit interval. Suppose $f$ is a continuous mapping of $I$ into $I$. Prove that $f(x) = x$ for at least one $x \in I$.
• If $f$ is defined on $E$, the graph of $f$ is the set of points $(x,f(x))$, for $x \in E$. In particular, if $E$ is the set of real numbers, and $f$ is real-valued, the graph of $f$ is a subset of the plane. Suppose $E$ is compact, and prove that $f$ is continuous if and only if its graph is compact.
• Associate to each sequence $a = {\alpha_{n}}$, in which $\alpha_{n}$ is $0$ or $2$, the real number $x(a) = \sum_{n = 1}^{\infty}\frac{\alpha_{n}}{3^{n}}$. Prove that the set of all $x(a)$ is precisely the Cantor set.
• Suppose $f(x)f(y) = f(x+y)$ for all real $x$ and $y$. Assuming that $f$ is differentiable and not zero, prove that $f(x) = e^{cx}$. Prove the same thing, assuming only that $f$ is continuous.
• A function $f:\mathbb{R} \to \mathbb{R}$ is said to have infinite derivative everywhere if for all $x \in \mathbb{R}$, $\lim\inf_{y \to x}(f(x)-f(y))/(x-y) = +\infty$. Show that no functions have infinite derivative everywhere.
• Find a sequence of functions ${f_{n}:\mathbb{R} \to \mathbb{R}}$ converging to $0$ almost everywhere but not converging to $0$ in $L^{1}$.
• Find a sequence of function ${f_{n}:\mathbb{R} \to \mathbb{R}}$ converging to $0$ pointwise but not converging to $0$ in $L^{1}$.
• Let $f:\mathbb{R} \to \mathbb{R}$ be any function such that if $x$ is in the domain then so is $1/x$ and $f(x)+f(1/x) = x$. What is the largest domain of all such functions?
• Let $A$ be a closed set of $\mathbb{R}$. Show that there exists a continuous function $f:\mathbb{R} \to \mathbb{R}$ such that $f(x) = 0$ if and only if $x \in A$.
• Let $f:[0,1] \to [0,1]$ be a continuous function. Prove that the arc length of $f$ is well-defined, $2$ is an upper bound for the arc length, and give an example of a well-known continuous function $f$ with arc length $2$.

#### Complex Analysis

• Write three terms of the Laurent expansion of $f(x) = \frac{1}{z(z-1)(z+1)}$ in the annulus $1 < |z|$.
• Show that $\sum_{n = 1}^{\infty}\frac{z^{n}}{n}$ converges at all points on the unit circle $|z| = 1$ except $z =1$.
• Show that a holomorphic function $f$ with $|f(z)| = 1$ for all $z$ is constant.
• Let $f$ be holomorphic and bounded on the upper half-plane $\mathbb{H}$ and takes real values on $\mathbb{R}$. Prove $f$ is constant.
• Evaulate $\int_{-\infty}^{\infty}\frac{e^{ix}}{1+x^{2}}\,dx$.
• Prove that a function which is analytic in the whole plane and satisfies an inequality $|f(z)| < |z|^{n}$ for some $n$ and all sufficiently large $|z|$ reduces to a polynomial.
• Show that a function which is analytic in the whole plane and has a nonessential singularity at $\infty$ reduces to a polynomial. Use this to show that the functions $e^{z}$, $\sin{z}$ and $\cos{z}$ have essential singularities at $\infty$.
• How many roots does the equation $z^{7}-2z^{5}+6z^{3}-z+1 = 0$ have in the disk $|z| < 1$?
• Find the poles and residues of $\frac{1}{\sin^{2}{z}}$.
• Evaluate the integral $\int_{0}^{\infty}\frac{x^{2}}{(x^{2}+a^{2})^{3}}\,dx$ by the method of residues.

#### Fourier Analysis

• Let $f \in C^{1}(\mathbb{T})$. Prove that $\sum_{n = -\infty}^{\infty}|\hat{f}(n)|n^{2} < \infty$.
• Use the Fourier coefficients of the Sawtooth function $s(x) = x-\pi$ on $[0,2\pi]$ to prove $\sum_{n = 1}^{\infty}\frac{1}{n^{2}} = \frac{\pi^{2}}{6}$.
• For any measurable $E \subset [0,1]$, show that $\lim_{n \to \infty}\int_{E}e^{2\pi inx}\,dx = 0$.