Complex Analysis, Spring 2011.

Instructor: Dmitry Ryabogin

Assignment II.

1. Problem 1.

a) Describe curves defined by the following equations: ℵ) z = t + i/t; ℶ) z = t^{2} + it^{4}, −∞ < t < ∞; ℷ) z = −t + i\sqrt{1 − t^{2}}, −1 ≤ t ≤ 1.

b) For a map w = z^{2} find the images of the lines x = C, y = C, x = y, |z| = R, argz = α. Are these curves transformed one-to-one?

c) Find the pre-images (on z-plane) of u = C, v = C, (w = z^{2} = u + iv).

d) Find the domain where f(z) = |x^{2} − y^{2}| + 2i|xy| is analytic.

2. Problem 2.

a) If g(w) and f(z) are analytic functions, show that g(f(z)) is also analytic.

b) Show that an analytic function cannot have a constant absolute value without reducing to a constant.

c) Prove rigorously that f(z) and \overline{f(z̄)} are simultaneously analytic.

3. Problem 3.

a) Find the most general harmonic polynomial of the form ax^{3} + bx^{2}y + cxy^{2} + dy^{3}. Determine the conjugate harmonic function and the corresponding analytic function by integration and by the formal method.

b) Prove that the functions u(z) and u(z̄) are simultaneously harmonic.

c) Show that a harmonic function satisfies the formal differential equation

\frac{∂^{2}u}{∂z∂z̄} = 0.

d) Are |f(z)|, argf(z), ln |f(z)| harmonic, provided f(z) is analytic?

e) Let u be harmonic. Is u^{2} harmonic?

4. Problem 4. Following the steps outlined below prove the Lucas’s theorem: the smallest convex polygon that contains the zeros of a polynomial P (z) also contains the zeros of the derivative P′(z).

a) Observe that it is enough to prove that if all zeros of P (z) lie in a half plane, then all zeros of P′(z) lie in the same half plane.

b) To prove a) show at first that

\frac{P′(z)}{P (z)} = ∑_{j=1}^{n} \frac{1}{z − α_{j}} .

c) Assume that all α_{k} are in the half plane H defined as Im(z − a)/b < 0 but a root z of P′(z) is not there. To get a contradiction look at Im(bP′(z)/P (z)). What can you say about Im(z − α_{k})/b?