Math 336 Final Exam, 8:30 am, June 8, 2009

Name:

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

1. Let f be analytic on a connected open set Ω and suppose \overline{𝔻} = \{z : |z| ≤ 1\} ⊂ Ω. Suppose that f is real on \{z : |z| = 1\}. Prove that f is constant on Ω.

Final 2

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

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. Suppose f is analytic on H = {z = x + iy : y > 0} and suppose

|f(z)| ≤ 1 on H and f(i) = 0. Prove

|f(z)| ≤ |\frac{z − i}{z + i}| .

5. Let ζ(s) be the Riemann zeta function. Let N be the set of positive integers and P be the set of primes. Prove that

log(ζ(s)) = ∑_{m∈N,p∈P} \frac{1}{m}p^{−ms}

where log(z) denotes the branch of the logarithm that is purely real on the positive real axis and the convergence is absolute and uniform on compact subsets of {Re(s) > 1}. You should prove that log(ζ(s)) makes sense. Prove the result for s > 1 and go from there.

Final 6

6. Suppose a, b, c, d are complex numbers, not negative integers, and a +

b = c + d. Prove that

∏_{n=1}^{∞} \frac{(a + n)(b + n)}{(c + n)(d + n)}

converges absolutely and express the answer using Γ functions.

7. Name five mathematicians prominently associated with the Prime Number Theorem and give approximate dates for their work on the PNT.