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
1. State the residue theorem.

2. 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
1. State Schwarz’s lemma.

2. 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.

1. Define what it means for a subset $$K\subset X$$ to be compact.

2. Prove (using your definition in (a)) that $$K\subset X$$ is compact implies that $$K$$ is both closed and bounded in $$X$$.

3. Give an example that shows the converse of the statement in (b) is false.

Solutions¶

(coming soon)

Real Analysis Exams