Department of Mathematics University of Utah Real and Complex Analysis Preliminary Examination

Let \((X,\mathcal M,\mu )\) be a measure space with \(\mu (X)<\infty \), and let \(1\le s<t\le \infty \). Show that \(L^t(X)\subset L^s(X)\).

Let \(H\) be a Hilbert space, \(V\subset H\) a closed subspace, and \(F\subset H\) a finite-dimensional subspace, with \(F\cap V=\{0\}\). Show that there is a positive constant \(\lambda <1\) such that \[ |\langle u,v\rangle |\le \lambda \,\lVert u\rVert \,\lVert v\rVert \] for all \(u\in F\) and \(v\in V\).

Consider the interval \([-1,1]\) with a finite Borel measure \(\mu \) (not assumed to be the Lebesgue measure). Let \(C([-1,1])\) be the space of continuous functions endowed with the \(L^2\) norm, and let \(T:C([-1,1])\to \mathbb R\) be the operator \(T(f)=f(0)\). Show that \(T\) is a bounded operator (with respect to the \(L^2\) norm) if and only if \(\mu (\{0\})>0\).

Let \((X,\mathcal M,\mu )\) be a measure space, and suppose that \(A_n\in \mathcal M\), \(n=1,\ldots \), with \[ \sum _{n=1}^{\infty }\mu (A_n)<\infty . \] Show that the set of \(x\) belonging to infinitely many \(A_n\) has measure zero.

For \(f:\mathbb R\to \mathbb R\) and \(y\in \mathbb R\), define \(\tau _y f(x)=f(x-y)\) for \(x\in \mathbb R\). Assume that \(\mathbb R\) is equipped with the Lebesgue measure. For \(f\in L^1(\mathbb R)\), show that the map \(\mathbb R\to L^1(\mathbb R)\) defined by \(y\mapsto \tau _y f\) is uniformly continuous.

Let \(f(z)\) be an analytic function. Show that the successive derivatives of \(f(z)\) at a point can never satisfy \[ |f^{(n)}(z)|>n!n^n \] for all \(n\).

Let \(\wp (z)\) be the Weierstrass \(\wp \) function. Prove that \[ \wp (z+u)+\wp (z)+\wp (u) =\frac 14\left (\frac {\wp '(z)-\wp '(u)}{\wp (z)-\wp (u)}\right )^2. \]

Prove that every 1-1 conformal mapping of a disk onto another is given by a linear transformation.

Evaluate the integral by the method of residues: \[ \int _0^\infty \frac {x^2}{x^4+5x^2+6}\,dx. \]

The Bernoulli numbers \(B_k\) are defined in terms of the Laurent series \[ \frac {1}{e^z-1} =\frac 1z-\frac 12+\sum _{k=1}^{\infty }(-1)^{k-1}\frac {B_k}{(2k)!}z^{2k-1}. \]

(a) Find the Laurent series of \(\cot \pi z\) in terms of the Bernoulli numbers.

(b) Using (a) and the fact on partial fractions \[ \pi \cot \pi z=\frac 1z+\sum _{n=1}^{\infty }\frac {2z}{z^2-n^2} \] to prove \[ \sum _{n=1}^{\infty }\frac {1}{n^{2k}} =2^{2k-1}\frac {B_k}{(2k)!}\pi ^{2k} \] for any positive integer \(k\).