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/(x2+y2) and f(x) = log(x2+y2)
on R2− {(0,0)}satisfy the Laplace equation ∆f= 0, and find their harmonic conjugates if they
exist.
2. Let P(x, y) = Pd
k=0 akxkyd−k(ak∈C,x, y ∈R) be a homogeneous polynomial of degree d. Show
directly that Pis an analytic function of z=x+iy if and only if P=adzd. Conclude that a
polynomial in Re(z) and Im(z) is an analytic function of ziff it is a polynomial in z.
3.∗i) Show that two polynomials P(z), Q(z) satisfy P0Q=PQ0if 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 without multiplicity, is at least k+ 1. (Hint:
consider A0B−AB0=B0C−BC0.)
iii) (“Fermat’s Last Theorem” for polynomials.) Show that if A(z), B(z), C(z) are polynomials
satisfying An+Bn=Cnfor 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 an+bn=cn. 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 fbe a nonconstant analytic function on (some region containing) the closed unit disc D={z:
|z| ≤ 1}, and a∈Da complex number at which |f(z)|attains its maximum in D. Prove that
f0(a)6= 0. Construct a counterexample to show that this is no longer true if Dis replaced by the
square {x+iy :|x| ≤ 1,|y| ≤ 1}.
6. Let fbe a real-valued C2function on an open set in Rncontaining the closed ball B={x:|x−x0| ≤ r}
relative to the standard (Euclidean) metric on Rn. Suppose ∆f(x) = 0 for x∈B. Prove that f(x0)
equals the average of f(x) over the sphere {x:|x−x0|=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 Rnsatisfy the maximum principle.
This problem set is due Wednesday, October 8, at the beginning of class.