Math 336 Sample Problems

One notebook sized page of notes (both sides) will be allowed on the test. The test will cover up to §5.1 in the text and will be comprehensive. The final exam is at 8:30 am, Monday, June 9, in the regular classroom.

1. 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.

2. 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}.

3. 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.

4. 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}

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

6. The function

f(z) = \frac{1}{(z + 1)(z + 3)}

has three different Laurent series expansions in powers of z. Find them all and specify the annulus for each series.

7. Let

f(z) = exp(z + \frac{1}{z})

Prove that

Res[f, 0] = ∑_{k=0}^{k=∞} \frac{1}{k!(k + 1)!}

Carefully justify all steps in your computation.

8. Suppose that u is a harmonic function and v is its conjugate harmonic function. Prove that uv is harmonic.

9. Let f(z) = 1 − \frac{1}{2^{z}} + \frac{1}{3^{z}} − \frac{1}{4^{z}} + . . .. Prove that the series converges absolutely and uniformly on compact sets to an analytic function if Re(z) > 1. Prove that f(z) = (1 − 2^{1−z})ζ(z) if Re(z) > 1. Prove that the series converges conditionally when z = x is real and 0 < x < 1. Prove that this statement is still true on the infinite strip 0 < Re(z) < 1.

10. 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.

11. 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}.

12. Prove that if |z| < 1

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

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

14. The characteristic function, χ_{S} of a set is defined by

χ_{S}(x) = \begin{cases}1, if x ∈ S \\ 0, if x ∉ S \end{cases}

(a) Let I = [a, b] ⊂ R be a compact interval. Prove that lim_{x→±∞} χ̂_{I} (x) = 0 by directly computing χ̂_{I} (x) and then computing the limit.

(b) Let {I_{j}}_{j=1}^{n} be a finite collection of compact intervals in R and

let {c_{j}}_{j=1}^{n} be a finite collection of constants. Let g = ∑_{j=1}^{n} c_{j}χ_{I_{j}}.

Using part (a), prove that lim_{x→±∞} ĝ(x) = 0. (This can be used to give an alternate proof of the Riemann-Lebesgue lemma.

15. There may be homework problems or example problems from the text on the final. Don’t forget previous sample problems. The final will be comprehensive.