Reference no: EM131105133

COMPLEX ANALYSIS HONORS EXAM 2013

1. Real Analysis

(1) Let K be the family of all bounded sets in R. For every K ∈ K, denote by

diam K = sup{|x - y| : x, y ∈ K}

its diameter, and, for K₁, K₂ ∈ K, denote by

u(K₁, K₂) = inf{|x - y| : x ∈ K₁, y ∈ K₂}

their naive distance (which may well be zero). Prove that

defines a metric on K.

(2) Let f: R × R → R be a continuous function. Assume additionally that:

• For every x ∈ R, the function f_x: R → R defined by f_x(y) = f(x, y) is a uniformly continuous function of y, and
• For every y ∈ R, the function f^y: R → R defined by f^y(x) = f(x, y) is a uniformly continuous function of x.

Must f be uniformly continuous?

(3) Let a < b, let I be an open interval containing [a, b], and let f: I → R be a continuous function such that, for every c ∈ [a, b], the limit

f'₊(c) = lim_x→c+(f(x) - f(c)/x - c)

exists (as a finite real number).

(a) If f'₊(c) > 0 for every c ∈ [a, b], then prove that then f(b) ≥ f(a). (Hint: Use the Axiom of Completeness, also known as the Axiom of Supremum.)

(b) Prove the previous claim under the weaker assumption that f'₊(c) ≥ 0 for every c ∈ [a, b]. (Hint: Construct another function that is very close to f in some sense and satisfies the condition of (a).)

(4) Let l¹ 1 be the set of all sequences a = (a_n)_n=1^∞ (with real terms a_n) such that

_n=1∑^∞|a_n| < ∞.

Then

d₁(a, b) = _n=1∑^∞|a_n - b_n| and       d₂(a, b) = _n=1∑^∞(|a_n - b_n|/2ⁿ)

are metrics on l¹. Let

L = {a ∈ l¹: a_n ≥ 0 for all n ∈ N, _n=1∑^∞a_n = 1}.

(a) Is L a closed set in the metric d₁? Is it bounded? Is it compact?

(b) Find a sequence in L that converges to 0 = (0, 0, . . .) in the metric d₂.

(c) Prove that, whenever (a_k)_k=1^∞ is a sequence of elements a_k = (a_kn)_n=1^∞ in L with lim_k→∞ d₂(a_k, 0) = 0 and (x_n)_n=1^∞ is a sequence of real numbers with lim_n→∞ x_n = x, then

lim_{k→∞ n=1}∑^∞a_knx_n = x.

2. Complex Analysis

(1) Let p be a nonzero polynomial with nonnegative coefficients, ? ⊂ C a bounded open region, ?¯ its closure, and  f: ?¯ → C a continuous function analytic on ?. Prove that the function p(|f(z)|) achieves a maximum on the boundary ∂?.

(2) Find the number of zeros of the function

f(z) = e^1/(2(z-1)) + 2z⁴ - z

inside the annulus {z ∈ C: ½ < |z| < 1}.

(3) Let U = {z ∈ C: |z| < 1} and U¯ = {z ∈ C: |z| ≤ 1}. Prove that the series

f(z) = _n=1∑^∞(z^{3^n}/3ⁿ)

converges for all z ∈ U¯, and that it defines a continuous function f : U¯ → C which is analytic on U but does not admit an analytic continuation to any connected open region V such that U ⊂ V and U ≠ V .

(4) Let T = ABC be a closed equilateral triangle in the complex plane. Denote by T^o its interior and by ∂T its boundary (the union of closed segments AB, BC, and CA), so that T = T^o ∪ ∂T.

Let ? be an open region containing T and let F? be the set of all analytic functions f: ? → ? such that f({A, B, C}) = {A, B, C} and f(∂T) ⊆ ∂T. Find the cardinality of the set F? and describe explicitly all functions in this set.