Math 336 𝔉𝔦𝔫𝔞𝔩𝔈𝔵𝔞𝔪, 8:30 am, June 10 , 2013

Name: One notebook-size page of notes is allowed (each side may be used).

1. Prove that ∑_{1}^{∞} \frac{sin nz}{2^{n}} represents an analytic function on |ℑ(z)| < log 2.

Final 2

2. Is there an analytic function f that maps |z| < 1 into |z| < 1 such that f(\frac{1}{2}) = \frac{3}{4}, f(\frac{1}{4}) = \frac{1}{3}?

3. Suppose that u is a harmonic function and v is its conjugate harmonic function. Suppose u^{2} + v^{2} is never 0. Prove that log(u^{2} + v^{2}) is harmonic.

Final 4

4. Prove that there is no entire function f with f(\frac{i}{n}) = \frac{1}{n+1} for all n. Hint: show that this implies that f(z) = \frac{z}{z+i}.

Final 5

5. (a) Prove that if ∏(1 + z^{k}) converges uniformly on compact subsets

of {|z| < 1}.

(b) Give an example to show that convergence of ∑ a_{k} does not imply convergence of ∏(1 + a_{k}).

Final 6

6. Let f and g be entire functions so that satisfy f^{2} + g^{2} = 1. Prove that there is an entire function h so that f = cos(h), g = sin(h).

7. Let u(x, y), v(x, y) be continuously differentiable as functions of (x, y) in a domain Ω. Let f(z) = u(z)+iv(z). Suppose that for every z_{0} ∈ Ω there is an r_{0} (depending on z_{0}) such that

∫_{|z−z_{0}|=r} f(z)dz = 0,

for all r with r < r_{0}. Prove that f is analytic in Ω. Hint: Show that f satisfies the Cauchy-Riemann equations in Ω.

Final 8

8. Does there exist a function f, which is analytic in a neighborhood of 0, for which:

(a) f(\frac{1}{n}) = f(−\frac{1}{n}) = \frac{1}{n^{2}} for all large integers n?

(b) f(\frac{1}{n}) = f(−\frac{1}{n}) = \frac{1}{n^{3}} for all large integers n? In each case, either give an example or prove no such example exists.