(REAL AND COMPLEX) ANALYSIS PRELIMINARY EXAMINATION August, 2005

Consider a measurable subset \(A\) of the circle \(\{z\in \mathbb C\mid |z|=1\}\). Fix a real number \(\theta \) so that \(\theta /\pi \) is irrational. Define \[ A_\theta =\{e^{i\theta }z\mid z\in A\}, \] the rotation of \(A\) through \(\theta \) radians. Prove that if \[ A=A_\theta \] then either \(A\) has measure zero or else \(A\) is the complement of a set of measure zero. (Hint: consider the Fourier series of \(\chi _A-\chi _{A_\theta }\); here \(\chi _S\) denotes the characteristic function of \(S\).)

Let \((\mathcal M,\mu ,X)\) be a positive measure space. Let \(\mathcal M'\) denote the collection of subsets (in \(\mathcal M\)) whose measures are finite. Prove that \[ L^2(X)\subset L^1(X) \] if and only if \[ \sup _{E\in \mathcal M'}\mu (E)<\infty . \]

Let \(H\) be a Hilbert space with inner product \((\cdot ,\cdot )\). Suppose \(\{x_n\}\) is a sequence of elements in \(H\) with the property that the sequence \(\{(x_n,y)\}\) converges for each \(y\in H\). Prove that there is an element \(x\in H\) such that \(\{(x_n,y)\}\) converges to \((x,y)\) for each \(y\in H\).

Construct a sequence of continuous real-valued functions \(f_n\) on \([0,1]\) such that \(0\le f_n\le 1\) and \[ \lim _{n\to \infty }\int _0^1 f_n(x)\,dx=0, \] but such that the sequence \(\{f_n(x)\}\) converges for no \(x\in [0,1]\).

Suppose \(f\) and \(g\) are two complex measurable functions on a measure space \(X\). Let \[ S=\{x\mid f(x)\ne g(x)\}. \] Prove or disprove: \(S\) is a measurable subset of \(X\).

Provide a rigorous computation of the sum of the following series: \[ \sum _{n=1}^{\infty }\frac {1}{n^4}. \] (Hint: one possible approach is to consider the integral of \(f(z)=\frac {\pi }{z^4}\cot (\pi z)\) over an appropriate sequence of contours.)

Determine the number of zeros of \[ p(z)=3z^3-2z^2+2iz-8 \] which lie in the annulus \(1<|z|<2\). Prove your answer is correct.

Fix positive integers \(k\) and \(l\). Describe the space of meromorphic functions on the Riemann sphere that have a have a pole of order at most \(k\) at \(\infty \) and a zero of order at least \(l\) at the origin.

The series \[ f(z)=\sum _{n=1}^{\infty }\frac {1}{n^2}z^n \] defines a holomorphic function near the origin. Describe a maximal open subset of the complex plane on which \(f\) can be analytically continued.

Determine the region of values of \(z\) for which \[ f(z)=\int _0^\infty \frac {e^{tz}}{1+t^2}\,dt \] defines a holomorphic function. (Hint: one approach is to use the holomorphicity criterion of Morera’s Theorem. If you pursue this approach, make certain to justify any application of Fubini’s theorem.)