Prove the casorati-weierstrass theorem suppose f has an

Question

2008 Honors Exam in Complex Analysis

Part I - Real Analysis

1. Find a metric d(·, ·) on the real line R that makes R into a bounded set: i.e. there is some M > 0 such that d(x, y) ≤ M for any x, y ∈ R. Verify that d is in fact a metric.

2. Let A be the line segment {(x, 0) : -1 < x < 1 } and let B be the line segment {(0, y) : -1 < y < 1}. Suppose that f: A ∪ B → A is a continuous function (view A ∪ B and A as subsets of the plane, with its usual metric). Prove that f cannot be one-to-one.

3. Suppose that X is a subset of a metric space E. Define the following number:

α(X) = inf(δ : X can be covered by finitely many open balls in E of radius δ }.

a) Show that, if X is compact, then α(X) = 0.

b) Show that, if X is not closed, X need not be compact even if α(X) = 0.

(If E is complete and X is closed, then α(X) = 0 if and only if X is compact. This is a version of the so-called Bolzano-Weierstrass theorem. The function α(X) is called a measure of noncompactness.)

4. 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. Let X be the closed unit ball in this space:

X = {continuous functions f: [0, 1] → R such that |f(x)| ≤ 1 for all x ∈ [0, 1]}.

Show that α(X) = 1, where α is as in problem 3. Conclude that X, while closed and bounded, is not compact.

5. Suppose that f: R → R is a continuous function. Choose and fix some closed interval [a, b]. For each k ∈ N, define the function

g_k(x) = k _x-(1/k)∫^xf(t) dt.

Show that each g_k(x) is differentiable for all x, and that g_k → f uniformly on [a, b] as k → ∞. You may use the fundamental theorem of calculus.

Part II - Complex Analysis

6. Find a meromorphic function f on C with the feature that, if γ is any smooth closed curve intersecting neither the point a = 1 + 0_i or b = -1 + 0_i in the complex plane,

∫_γ f dz

is the number of times that γ winds around a plus the number of times that γ winds around b (both in the positive - i.e. counterclockwise - direction).

7. Prove the Casorati-Weierstrass Theorem: Suppose f has an isolated essential singularity at p. Then the image of any neighborhood of p under f is dense in C. Otherwise put, given w ∈ C, there is a sequence {z_n} converging to p such that {f(z_n)} converges to w. Do not use the Picard theorems. (Hint: imagine that the image is not dense. This means that there is some w such that the function f(z) - w has modulus greater than some  for z near p; this in turn implies that

1/f(z) - w is bounded near p.)

8. Suppose that a sequence {f_k} of analytic functions converges normally (i.e. uniformly on compact subsets) to f on a domain D ⊂ C, and that f_k(z) ≠ 0 for every k ∈ N and every z ∈ D. Prove that f must either have no zeros or must be identically zero.

9. Let H be the upper half plane: H = {a + b_i : b > 0}. Let f: H → H be an analytic function such that f(i) = i. Suppose that b and c are two real numbers greater than 1 and that f(bi) = ci. Show that c ≤ b.