Math 336 Makeup Exam, Summer, 2008

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

1. Prove that Γ(z) has a simple pole at −3 and compute its residue at this point.

Final 2

2. Find the Green’s function for the region {z : |z − 1| < 3}, with pole at 2.

3. 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.

4. Suppose that u is harmonic on all of ℂ and u(z) ≥ 0 for all z ∈ ℂ. Prove that u is constant. Is it true that a positive harmonic function on {(x, y) : x + y > 0} is constant?

Final 5

5. Let

g(x) = \{ \begin{matrix}1, if |x| < 1 \\ 0, if |x| > 1; \end{matrix}

h(x) = \{ \begin{matrix}|x|, if |x| < 1 \\ 0, if |x| > 1; \end{matrix}

f(x) = g(x) − h(x).

Compute the Fourier transform (my definition in class) of g, h and hence of f.

Final 6

6. Suppose f_{n}(z) is a sequence of analytic functions on an open set Ω and that |f_{n}(z)| < 1 for all z ∈ Ω. Suppose ∑ f_{n}(z) converges uniformly and absolutely on compact subsets of Ω. Prove that

f(z) = ∏(1 + f_{n}(z))

defines an analytic function and f(z) ≠ 0 (f is never zero.)

7. Let u(z) be harmonic in all of C. Suppose |u(z)| ≤ c|z|^{n} for some positive constant c. Prove that u is the real part of a complex polynomial of degree n.