Honors Exam in Complex Analysis 2007
1. Suppose K is a subset of Rn. Prove that K is compact if and only if every continuous function f: K → R is bounded.
2. Suppose f: (0, 1] → R is differentiable and satisfies |f'(x)| < 1 there. Show that the sequence {f(1/n)} converges.
3. Suppose f: [0,∞) → R is of class C2, and f(x) → 0 as x → ∞.
(a) If f'(x) → b as x → ∞, show that b = 0.
(b) If f'' is bounded, show that f'(x) → 0 as x → ∞.
(c) Give an example of such an f for which f'(x) does not converge as x → ∞.
4. Suppose f: [0, 1] → R is upper semicontinuous: This means that for every x ∈ [0, 1] and every ε > 0, there exists δ > 0 such that |y -x| < δ implies f(y) < f(x) + ε. Prove that f is bounded above and achieves its maximum value at some x ∈ [0, 1].
5. (a) Suppose f: C → C is an entire function and for all z ∈ C. Show that f is identically zero.
(b) Suppose that f is complex analytic for 0 < |z| < 1 and Re f(z) is bounded there. Show that f has a removable singularity at the origin.
6. Using basic principles (e.g., considerations of uniform convergence, periodicity, and Liouville's theorem), prove that
π2/sin2 πz = n=-∞∑∞ 1/(z - n)2.
7. Let E = {(x, y):0 < y < x}. Use conformal mapping techniques to find a harmonic function u on E with the following boundary values:
8. Use the residue theorem to evaluate the integral
0∫2π (dθ/a + cos θ),
where a is a real number such that a > 1.