If we equip the set e of all continuous functions f 0 1

Question

Honors Exam Complex Analysis 2010

Part I - Real Analysis

1. If we equip the set E of all continuous functions f: [0, 1] → R with the metric d(f, g) = max_x∈[0,1]|f(x) - g(x)|, E becomes a complete metric space. Define S = {f ∈ E: f'(x) exists for all x ∈ (0, 1)}.

Prove that S is neither open nor closed in E.

2. Let K denote the set of nonempty compact subsets of R², and write d for the Euclidean metric on R². Given A ∈ K and r ≥ 0, write

A_r = {x ∈ R²: d(x, y) ≤ r for some y ∈ A}.

Observe that A₀ = A, and that A_s ⊂ A_t whenver s ≤ t.

a) Given A, B ∈ K, define

µ(A, B) = min{r ≥ 0 : B ⊂ A_r}.

Show that this definition makes sense - that is, that the above minimum in fact exists.

b) The Hausdorff metric on K is defined as

ρ(A, B) = max(µ(A, B), µ(B, A)).

Prove that ρ is in fact a metric.

c) Show, with an explicit example, that µ: K×K → R is not a metric. (Hint: Consider the case B ⊂ A.)

3. Let E be the space [0, 1] equipped with the discrete metric:

a) Show that any function f: E → R, where R has its usual metric, is continuous.

b) Is E connected?

c) Is E compact?

4. a) Prove that

lim_n→∞ n!/nⁿ = 0.

b) Does

_k=1∑^∞ k!/k^k

converge?

5. Given a > 0 and two continuous functions f: R → R and g: R → R, let us define the function h: [0, a] → R by the formula

h(x) = _g(0)∫^g(x) f(s) ds.

a) Show that h is continuous on [0, a].

b) Show via an explicit example that h need not be differentiable on (0, a). (Hint: try making f a constant function.)

c) Can you impose an additional condition on g or on f that guarantees that h(x) is differentiable on (0, a)?

Part II - Complex Analysiss

1. Expand the function f(z) = 1/1-z about 0 and about -1 to show that

_k=0∑^∞1/3^k = _k=0∑^∞2^k-1/3^k.

(As part of your proof, be sure to explain why both of the above series converge.)

2. Suppose that f: C → C is an entire nonconstant function.

a) Show that the image of f is dense - that is, given any w ∈ C and any ∈ > 0, there is a z ∈ C such that |f(z) - w| <  ∈.

b) Show that if |f(z)| → ∞ as |z| → ∞, then f is onto - that is, f(z) = w has a solution for every w ∈ C.

c) Give an explicit example that shows that, if the condition in part b) fails, f need not be onto.

3. Compute

_-∞∫^∞ 1/2x² + 1 dx.

4. Show that e^z + z² - 5 has exactly one root in the left half plane. (Hint: consider a region whose boundary is of the form

[-ri, ri] ∪ { re^iθ : θ ∈ [π/2, 3π/2]},

where r > 0.)

5. Suppose that f: C → C is entire, and that the image f(C) of f contains no nonpositive real numbers - that is,

f(C) ∩ {a + 0i : a ≤ 0} = ∅.

Prove that f is constant. (Hint: find analytic functions g and h such that g o h o f is entire with bounded image.)