Suppose f 0 1 rarr r is upper semicontinuous this means

Question

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.