Examination 9¶

2007 Nov¶

Instructions. Do as many problems as you can. Complete solutions (except for minor flaws) to 5 problems would be considered an excellent performance. Fewer than 5 complete solutions may still be passing, depending on the quality.

Problem 53

Let \(G\) be a bounded open subset of the complex plane. Suppose \(f\) is continuous on the closure of \(G\) and analytic on \(G\). Suppose further that there is a constant \(c\geq 0\) such that \(|f| = c\) for all \(z\) on the boundary of \(G\). Show that either \(f\) is constant on \(G\) or \(f\) has a zero in \(G\).

Problem 54

State the residue theorem.
Use contour integration to evaluate

\[\int_0^\infty \frac{x^2}{(x^2+1)^2} \, dx.\]

Important: You must carefully: specify your contours, prove the inequalities that provide your limiting arguments, and show how to evaluate all relevant residues.

Problem 55

State Schwarz’s lemma.
Suppose \(f\) is holomorphic in \(D = \{z: |z|< 1\}\) with \(f(D) \subset D\). Let \(f_n\) denote the composition of \(f\) with itself \(n\) times \((n= 2, 3, \dots)\). Show that if \(f(0) = 0\) and \(|f'(0)| < 1\), then \(\{f_n\}\) converges to 0 locally uniformly on \(D\).

Problem 56

Exhibit a conformal mapping of the region common to the two disks \(|z|<1\) and \(|z-1|<1\) onto the region inside the unit circle \(|z| = 1\).

Problem 57

Let \(\{f_n\}\) be a sequence of functions analytic in the complex plane \(\mathbb{C}\), converging uniformly on compact subsets of \(\mathbb{C}\) to a polynomial \(p\) of positive degree \(m\). Prove that, if \(n\) is sufficiently large, then \(f_n\) has at least \(m\) zeros (counting multiplicities). Do not simply refer to Hurwitz’s theorem; prove this version of it.

Problem 58

Let \((X, d)\) be a metric space.

Define what it means for a subset \(K\subset X\) to be compact.
Prove (using your definition in (a)) that \(K\subset X\) is compact implies that \(K\) is both closed and bounded in \(X\).
Give an example that shows the converse of the statement in (b) is false.