Math 336 Sample Problems

One notebook sized page of notes (both sides) will be allowed on the test. The test will be comprehensive. The final exam is at 8:30 am, Monday, June 8, in the regular classroom.

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

(a)

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

(b)

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

2. 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.

3. Let D_{2} = {z : |z| < 2} and I = {x ∈ R : −1 ≤ x ≤ 1}. Find a bounded harmonic function u, defined in D_{2} − I such that u does not extend to a harmonic function defined in all of D_{2}.

4. Find a conformal map from the region between the two lines y = x and y = x + 2 to the upper half plane, which sends 0 to 0.

5. 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.

6. Find a function, h(x, y), harmonic in {x > 0, y > 0}, such that

h(x, y) = \begin{cases}0 & if 0 < x < 2, y = 0, \\ 1 & if x > 2, y = 0, \\ 2 & if x = 0, y > 0 \end{cases}

7. Suppose that u is harmonic on all of ℂ and u ≥ 0. Prove that u is constant.

8. 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}| .

9. 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.

10. Let u(z) be harmonic in \{z : 0 < |z| < 1\}. Let P = ∫_{|z|=r} \frac{∂u}{∂n}ds, where

0 < r < 1. Show that P does not depend on r. Prove that

u(z) = \frac{P}{2π} log |z| + Re(f(z))

where f is analytic in {z : 0 < |z| < 1}.

11. Prove that if |z| < 1

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

12. Find the Green’s function for the region {z : |z − 2| < 3} with pole at 4.

13. There may be homework problems or example problems from the text on the final. Don’t forget previous sample problems.