Reference no: EM131107498
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) = maxx∈[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
Ar = {x ∈ R2: d(x, y) ≤ r for some y ∈ A}.
Observe that A0 = A, and that As ⊂ At whenver s ≤ t.
a) Given A, B ∈ K, define
µ(A, B) = min{r ≥ 0 : B ⊂ Ar}.
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:
![1211_Figure.png](https://secure.expertsmind.com/CMSImages/1211_Figure.png)
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
limn→∞ n!/nn = 0.
b) Does
k=1∑∞ k!/kk
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∑∞1/3k = k=0∑∞2k-1/3k.
(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 ez + z2 - 5 has exactly one root in the left half plane. (Hint: consider a region whose boundary is of the form
[-ri, ri] ∪ { reiθ : 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.)