Math 336 Sample Problems

One notebook sized page of notes will be allowed on the test. The test will cover up to section 2.5 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. Let f(z) = u(x, y)+iv(x, y) be 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.

4. Suppose ∑_{0}^{∞} |a_{n}|^{2} converges. Prove that f(z) = ∑_{0}^{∞} a_{n}z^{n} is analytic

for |z| < 1. Prove lim_{r→1} ∫_{0}^{2π} |f(re^{it})|^{2}dt exists and compute it.

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. Find the Laurent series of the function \frac{1}{z} in the annulus D = \{z : 2 < |z − 1| < ∞}.

9. Let f(z) = \frac{p′(z)}{zp(z)}, where p(z) = ∏_{j=1}^{n}(z −z_{j}) and the z_{j} are distinct and

different from 0. Find all the poles of f and compute the residues of f at these poles.

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

11. 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.

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

(a) Absolute Value (Modulus) and Argument of a complex number (b) Complex Derivative

(c) Cauchy-Riemann equations

(d) Harmonic Conjugate

(e) Complex Analytic

(f) Power Series

(g) Complex Exponential Function

(h) Complex Logarithm (i) Cauchy’s Integral Theorem and Formula (j) Morera’s Theorem

(k) Liouville’s Theorem

(l) Isolated Singularities (types)

(m) Residues

(n) Laurent Series

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