Honors Exam 2011 Complex Analysis

Part I: Real Analysis

1. A topological space is called separable if it contains a countable dense subset.

(a) Show that euclidean space Rⁿ is separable.

(b) Show that every compact metric space is separable.

(c) Let l_∞ be the space of bounded sequences a = {aj}^∞_j=1. We make l_∞ into a metric space by declaring

d(a, b) = sup_j|a_j - b_j|

Show that l_∞ is not separable.

2. Let {f_n} be a uniformly bounded sequence of continuous functions on [a, b]. Let

F_n(x) = _a∫^xf_n(t)dt

(a) Show that there is a subsequence of {F_n} that converges uniformly on [a, b].

(b) Each F_n is differentiable. Show by an example that the uniform limit in part (a) need not be differentiable.

3. Let {a_n}^∞_n=1 be a positive decreasing sequence a_n ≥ a_n+1 ≥ 0.

(a) Show that _n=1∑^∞ a_n converges if and only if _k=0∑^∞2^ka_2^k converges.

(b) Use the result in part (a) to show that the harmonic series _n=1∑^∞(1/n) diverges.

(c) Use the result in part (a) to show that the series _n=2∑^∞(1/n(log n)^p) converges for p > 1.

4. Let f be a continuous function on the closed interval [a, b].

(a) Show that

lim_p→∞ (_a∫^b|f(x)|^pdx)^1/p = max_x∈[a,b]|f(x)|

(b) Give an example of a continuous function f on (a, b) where the improper integrals

_a∫^b|f(x)|^pdx

exist (i.e. are finite) for all 1 ≤ p < ∞, but

lim_p→∞ (_a∫^b|f(x)|^pdx)^1/p = ∞

Part II: Complex Analysis

1. Let f(z) be an entire function. Suppose there is a constant C and a positive integer d such that |f(z)| ≤ C|z|^d for all z with |z| sufficiently large. Show that f(z) is a polynomial of degree at most d.

2. Evaluate the integral ₀∫^∞dx/x⁵ + 1.

3. Let D ⊂ C be the unit disk, and

Q = {z ∈ C: |Im z| < π/2}

(a) Find a conformal mapping f: D → Q with f(0) = 0, f'(0) = 2.

(b) Use the conformal mapping in part (a) to show that if g: D → Q is analytic with g(0) = 0, then |g'(0)| ≤ 1/2.

4. For a complex parameter λ, |λ| < 2, consider solutions to the equation

z⁴ - 4z + λ = 0                                     (1)

(a) Show that there is exactly one solution z(λ) to eqn. (1) with |z(λ)| < 1.

(b) Show that the map λ |→ z(λ) is analytic for |λ| < 2.

(c) What is the order of vanishing of z(λ) at λ = 0?

5. The Bernoulli numbers B_n are defined by the equation

z/e^z - 1 = _k=0∑^∞B_k(z^k/k!)

(a) Compute B₀, B₁, and B₂.

(b) Show that

πz cot(πz) = _k=0∑^∞(-1)^kB_2k((2πz)^2k/(2k)!)

and that B_2n+1 = 0 for n ≥ 1.

(c) Compute the residues

Res_z=0(πz^-2n cot(πz))

for n = 1, 2, 3, . . ., in terms of Bernoulli numbers.