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 fx: R → R defined by fx(y) = f(x, y) is a uniformly continuous function of y, and
- For every y ∈ R, the function fy: R → R defined by fy(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) = limx→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 = (an)n=1∞ (with real terms an) such that
n=1∑∞|an| < ∞.
Then
d1(a, b) = n=1∑∞|an - bn| and d2(a, b) = n=1∑∞(|an - bn|/2n)
are metrics on l1. Let
L = {a ∈ l1: an ≥ 0 for all n ∈ N, n=1∑∞an = 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 (ak)k=1∞ is a sequence of elements ak = (akn)n=1∞ in L with limk→∞ d2(ak, 0) = 0 and (xn)n=1∞ is a sequence of real numbers with limn→∞ xn = x, then
limk→∞ n=1∑∞aknxn = 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) = e1/(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∑∞(z3^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 To its interior and by ∂T its boundary (the union of closed segments AB, BC, and CA), so that T = To ∪ ∂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.