# Examination 2¶

## 1990 Mar¶

Instructions. There are 5 questions: Each question will be graded on a scale of 0-10. A passing score at the Ph.D. level is 30. A passing score at the M.A. level is 20.

Notation. The symbol $$D$$ stands for the unit disc $$\{z \in \mathbb C \mid | z | < 1\}$$.

Problem 7
1. State the Riemann mapping theorem (including the relevant topological definitions).

2. Present a conformal mapping $$f$$ of $$D$$ onto $$\Omega = \{z= x + i y \mid x > 0, y > 0\}$$ such that $$f(0)= 1 + i$$.

Problem 8

Evaluate $$\int_{-\infty}^\infty \frac{dx}{1+x^2}$$ by use of the residue theorem.

Problem 9

Suppose that $$f$$ is holomorphic on $$D$$, $$|f(z)| < 1$$ for all $$z\in D$$, and $$f(1/2) = 1/2$$.

1. Prove that $$|f'(1/2)|\leq 1$$.

2. Determine $$f(z)$$ if $$f'(1/2) = 1$$.

Problem 10

Let $$\{f_n(z)\}$$ be a sequence of nonvanishing analytic functions on a region $$\Omega$$. Suppose that $$\{f_n(z)\}$$ converges uniformly on every compact subset of $$\Omega$$ to a function $$f(z)$$. If $$f(z_0) \neq 0$$ for some $$z_0 \in \Omega$$, then prove that $$f(z)\neq 0$$ for all $$z\in \Omega$$.

Problem 11

Suppose that $$U(z)$$ is a real valued harmonic function defined for all $$z \in \mathbb C$$. If

$\lim_{y\to 0+}\frac{u(x+iy)}{y} = O$

for all $$x \in \mathbb R$$, then prove that $$u(z) = 0$$ for all $$z \in \mathbb C$$.