Math 213a: Complex analysis Problem Set #2 (29 September 2003): Analytic functions, cont’d; Cauchy applications, I

Polynomial and rational functions:

1. Show directly that the functions

f(x, y) = x/(x^{2} + y^{2}) and f(x) = log(x^{2} + y^{2})

on R^{2} − {(0, 0)} satisfy the Laplace equation Δf = 0, and find their harmonic conjugates if they exist.

2. Let P (x, y) = ∑_{k=0}^{d} a_{k}x^{k}y^{d−k} (a_{k} ∈ C, x, y ∈ R) be a homogeneous polynomial of degree d. Show directly that P is an analytic function of z = x + iy if and only if P = a_{d}z^{d}. Conclude that a polynomial in Re(z) and Im(z) is an analytic function of z iff it is a polynomial in z.

3.^{∗} i) Show that two polynomials P (z), Q(z) satisfy P′Q = P Q′ if and only if aP + bQ = 0 for some

complex a, b, not both zero. ii) (“ABC conjecture” for polynomials.) Let A(z), B(z), C(z) be three polynomials without common

factors such that A + B + C = 0. Let k = max(deg(A), deg(B), deg(C)), and assume that k > 0. Show that the number of zeros of ABC, counted \underline{without} multiplicity, is at least k + 1. (Hint: consider A′B − AB′ =B′C − BC′.)

iii) (“Fermat’s Last Theorem” for polynomials.) Show that if A(z), B(z), C(z) are polynomials satisfying A^{n} + B^{n} = C^{n} for some integer n ≥ 3 then A, B, C are of the form aP, bP, cP for some polynomial P (z) and complex numbers a, b, c with a^{n} + b^{n} =c^{n}. Give an example showing that this result is no longer true for n=2.

Maximum principle(s):

4.^{∗} [Blaschke; Ahlfors 2.1, Ex. 4 (p.46)] What is the general form of a rational function that has absolute value 1 on the unit circle |z|=1? In particular, how are the zeros and poles related to each other?

5. Let f be a nonconstant analytic function on (some region containing) the closed unit disc D = {z : |z| ≤ 1}, and a ∈ D a complex number at which |f(z)| attains its maximum in D. Prove that f′(a) ≠ 0. Construct a counterexample to show that this is no longer true if D is replaced by the square {x + iy : |x| ≤ 1, |y| ≤ 1}.

6. Let f be a real-valued C^{2} function on an open set in R^{n} containing the closed ball B = {x : |x−x_{0}| ≤ r} relative to the standard (Euclidean) metric on R^{n}. Suppose Δf(x) = 0 for x ∈ B. Prove that f(x_{0}) equals the average of f(x) over the sphere {x : |x − x_{0}| = r}. [When n = 2 this is a consequence of Cauchy’s theorem for an analytic function whose real part is f.] Conclude that harmonic functions on regions in R^{n} satisfy the maximum principle.

This problem set is due Wednesday, October 8, at the beginning of class.