Math 336 Final Exam, 8:30 am, June , 2010

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

1. Let m ≥ 0 be an integer. Compute the residue of Γ(z) at −m.

Final 2

2. Let g(z, t) be a continuous function of z, t where z ∈ D, t ∈ K, D is an open subset of ℂ and K is a compact subset of ℝ^{n}. Suppose that g is harmonic as a function z. Prove that G(z) = ∫_{K} g(z, t)dt is harmonic.

Final 3

3. Let f(z) = \frac{z − a}{1 − āz}, where |a| < 1. Let D = \{z : |z| < 1\}. Prove that

\frac{1}{π} ∫_{D} |f′(z)|^{2}dxdy = 1.

Final 4

4. Decide whether or not the following products converge.

(a) ∏_{j=1}^{∞} e^{(−1)^{j}/\sqrt{j}}

(b) ∏_{j=1}^{∞} (1 + i\frac{(−1)^{j}}{\sqrt{j}} )

Final 5

5. Prove that all conformal maps from the upper half plane to the unit

disk have the form

α\frac{z − β}{z − β̄},

where |α| = 1, Im(β) > 0.

Final 6

6. (a) Find a bounded harmonic function u defined on D = {z : |z| < 1}

so that

lim_{z→e^{it}} u(z) = \begin{cases}1, & if 0 < t < π/2 \\ 0, & if π/2 < t < 2π \end{cases}

(b) Find an unbounded harmonic function with the same property.

Final 7

7. Prove that if |z| < 1

∏_{k=0}^{k=∞}(1 + z^{2^{k}}) = \frac{1}{1 − z}