WJD Open Notebook

1991 April

1. (a) State the homology form of the Cauchy Integral Therem and explain the concepts involved.
   (b) State a geeral form of the Residue Theorem and explain the concepts involved in this theorem.

2. (a) State the Riemann Mapping Theorem and explain the relevant concepts.
   (b) Find an explicit mapping of the unit disk $D$ onto the strip $\Omega = \{x + iy : 0 < x < 1, -\infty < y < \infty\}$.

3. Suppose that $f(x)$ is an analytic function on $\Omega = \{z \in \mathbb C : 0 < |z| < 1\}$. Assume that for $z \in \Omega$, $|f(z)| \leq 1$. Prove that there is a function $F(z)$, analytic for $|z|<1$ and satisfying $F(z) = f(z)$ for all $z \in \Omega$.

4. Let $\{P_n(z)\}$ be a sequence of polynomials in the variable $z$ and let $f(z)$ be a function defined for $\Omega = \{z : 1 < |z| < 2\}$. Suppose that $$\lim_{n\to \infty} \left[\sup_{z\in \Omega} |f(z)- P_n(z)|\right] = 0.$$ Prove that there is a function $F(z)$, analytic for $|z| < 1$, that satisfies $F(z) = f(z)$ for all $z \in \Omega$.

5. Suppose that $f(z)$ is an analytic function for $|z|< 1$ and that $f(z)$ is continuous for $|z| \leq 1$. Determine $f(z)$ explicitly, given that
   (i) $f(z)\neq 0$ for all $0< |z| < 1$, and
   (ii) $|f(z)| = 1$ for all $|z| = 1$.

6. Suppose that $u(z)$ is a continuous function defined on the open subset $\Omega \subset \mathbb C$. Assume that for each disk $D(a,r) = \{z \in \mathbb C : |z - a| \leq r\}$ contained in $\Omega$ we have $$u(a) = \frac{1}{\pi r^2}\iint_{D(a,r)}u(z)\ dx\ dy.$$ Prove that $u$ is a harmonic function on $\Omega$.