Math 336 Sample Problems

One notebook sized page of notes will be allowed on the test. The test will cover through chapter IV, section 5 in the text.

1. Suppose that v is the harmonic conjugate of u and u is the harmonic conjugate of v. Show that u and v must be constant.

2. Let f(z) = u(x, y) + iv(x, y). Suppose that f is analytic. Prove that

det (\begin{matrix}u_{x} & u_{y} \\ v_{x} & v_{y} \end{matrix}) = |f′(z)|^{2}.

3. Compute ∫_{|z|=2} \frac{dz}{(z − 1)(z − 3)(z − 4)}.

4. Let f(z) = u(x, y) + iv(x, y) be twice continuously (real) differentiable on an open set. Suppose that the real and imaginary parts of f(z) and zf(z) are harmonic. Prove that f is analytic.

5. Let a be a complex number and suppose |a| < 1. Let f(z) = \frac{z − a}{1 − \overline{a}z}. Prove the following statments.

(a) |f(z)| < 1, if |z| < 1.

(b) |f(z)| = 1, if |z| = 1.

6. Let z_{j} = e^{\frac{2πij}{n}} denote the n roots of unity. Let c_{j} = |1 − z_{j}| be the n − 1 chord lengths from 1 to the points z_{j}, j = 1, ….n − 1. Prove that the product c_{1} · c_{2} ⋯ c_{n−1} = n. Hint: Consider z^{n} − 1.

7. Let f(z) = x + i(x^{2} − y^{2}). Find the points at which f is complex differentiable. Find the points at which f is complex analytic.

8. Let f(z) = u(z) + iv(z), u = Re(z), v = Im(z) be analytic on an open connected set Ω. Suppose there are real numbers a, b, c with a^{2}+b^{2} ≠ 0 and au(z) + bv(z) = c for all z ∈ Ω. Prove that f is constant.

9. Let f be analytic within and on a simple closed curve Γ. Prove that Re (∫_{Γ} \overline{f}(z)f′(z)dz) = 0.

10. Compute ∫_{|z|=r} \frac{|dz|}{|z − a|^{2}} , where |a| ≠ r. Use the fact that on

\{|z| = r\}, |dz| = −ir \frac{dz}{z}; and then use the Cauchy integral formula.

11. You will need to know the definitions of the following terms and statements of the following theorems.

(a) Modulus (absolute value) and argument of a complex number (b) Complex derivative

(c) Complex analytic function

(d) Cauchy-Riemann equations

(e) Harmonic functions and harmonic conjugate

(f) Complex exponential function

(g) Complex logarithm

(h) Cauchy’s integral theorem and formula

(i) Maximum principle

(j) Linear fractional transformations

13. There may be homework problems or example problems from the text on the midterm.