Math 336 Sample Problems

One notebook sized page of notes will be allowed on the test. The test will cover through section III.5 in Gamelin.

1. Suppose w ≠ 1, w ∈ ℂ is an n^{th} root of unity, w^{n} = 1. Prove that

1 + 2w + 3w^{2} + ⋯ + nw^{n−1} = \frac{n}{w − 1}.

2. Suppose Re(z_{j}) ≥ 0 for j ≥ 1 and suppose the series

z_{1} + z_{2} + ⋯ + z_{n} + …
z_{1}^{2} + z_{2}^{2} + ⋯ + z_{n}^{2} + …,

both converge. Prove that

|z_{1}|^{2} + |z_{2}|^{2} + ⋯ + |z_{n}|^{2} + …,

converges.

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

4. Let f(z) = u(z)+iv(z), u = Re(f(z)), v = Imf((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.

5. 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.

6. Let u be harmonic on W (assume u is twice continuously differentiable). Prove that f(z) = u_{x}(z) − iu_{y}(z) is analytic.

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

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

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

8. Suppose f = u + iv is analytic on {Re(z) > 0} and u_{x} + v_{y} = 0. Prove that there is a real number c

and complex number d so that

f(z) = icz + d.

9. Let f(z) = e^{−z^{−4}} if z ≠ 0, f(0) = 0. Prove that f is analytic at z if z ≠ 0 and that the Cauchy-Riemann equations are satisfied at 0. Is f analytic at 0?

10. 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.

11. Find a sequence of complex numbers z_{n} such that ∑_{n=1}^{∞} z_{n}^{k} converges for every k = 1, 2 … but ∑_{n=1}^{∞} |z_{n}|^{k} diverges for every k = 1, 2 . . .. Hint: Try z_{n} = \frac{e^{2πins}}{log(n+1)} for an appropriate real number s.

12. Suppose f is analytic on a connected open set. Assume f^{2} = \overline{f}. Prove that f is constant. What are the possible values of the constant?

13. 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 and trigonometric functions

(g) Complex logarithm and powers

(h) Linear fractional transformations

(i) Mean value property

(j) Maximum principle

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