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 \(z \in \mathbb C\).

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Solutions

Solution to Problem 7 (coming soon)

Solution to Problem 8 (coming soon)

Solution to Problem 9 (coming soon)

Solution to Problem 10 (coming soon)

Solution to Problem 11 (coming soon)

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Real Analysis Exams

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