Complex Analysis Exam

Part of the phd exam in analysis

Reading time: about 1 minute (372 words).

2003 April 23

Instructions. Each of the following $7$ problems will be scored from $0$ to $10$. We us the following notations: $\mathbb N = \{1, 2, 3, \dots\}$, $\mathbb C$ is the set of complex numbers and $\mathrm{Re}\ z$ denotes the real part of $z\in \mathbb C$; $D = \{z \in \mathbb C : |z| < 1\}$.

  1. Does there exist a function having both of the following properties:
    (i) $f$ is holomorphic on $D$;
    (ii) $\lim_{n\to \infty} |f(z_n)| = \infty$ co whenever $\{z_n\}$ is a sequence of elements of $D$ and $\lim_{n\to \infty} |z_n| = 1$?
    Justify your statement.

  2. Does there exist sequence of functions $\{f_n\}$ having all of the following properties:
    (i) $f_n$ is holomorphic on $D$ for all $n \in \mathbb N$;
    (ii) $|f_n(z)| < 1$ for all $n \in \mathbb N$ and all $z \in D$;
    (iii) $\lim_{n\to \infty} |f_n(x)| = 0$ for all $x \in (0,1)$;
    (iv) $\lim_{n\to \infty} |f_n(x)| = 1$ for all $x \in (-1,0)$.
    State a major theorem from complex analysis and use that theorem to justify your answer.

  3. (a) Expand $\sqrt{4-z^2}$ in a power series centered at $0$. (The indicated function is to have the value $2$ when $z = 0$.)
    (b) Give the Laurent series expansion of $\frac{1}{z(z-1)}$ in the region $A = \{z \in \mathbb C : 2 < |z+2| < 3\}$.

  4. Describe the image of the strip $S = \{z \in \mathbb C : -\pi/2 < \mathrm{Re}\ z < \pi/2\}$ under the mapping $z\mapsto \sin(z)$; include a description of the boundary correspondence.

  5. Suppose that $f\colon D \to \mathbb C$, $f$ is holomorphic, $|f(z)| \leq 1$ for all $z \in D$, and $f(1/2)=0$. Prove that $|f(0)|\leq 1/2$.

  6. State the homology version of the residue theorem. Use the residue theorem to prove that $$\int_0^\pi \frac{1}{4+\cos\theta}\ d\theta = \frac{\pi}{\sqrt{15}}.$$

  7. Suppose that $(X, d)$ is a metric space, $K$ is a compact subset of $X$, and $y \in X$. Prove that there exists a point $p\in K$ such that $d(p, y) \geq d(k, y)$ for all $k \in K$.