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Math 246a, Homework 5
Christoph Thiele
Problem 1: Mean value property
From a qualifying exam: Let Un be a sequence of positive harmonic functions on a connected open set Ω containing the origin. Show that if
nlim→∞ Un(0) = 0
then
nlim→∞ ‖Un‖L∞(K)^ = 0
for all compact sets K in Ω. Hint: first prove that if Un converges to 0 at some z 0 , then it converges uniformly to 0 on a small ball about z 0. For this use the obvious variant of the mean value theorem for annuli.
Problem 2: Connectedness
A set M ⊂ R^2 is called connected, if there do not exist two disjoint open sets U 1 and U 2 in R^2 such that the sets M ∩ U 1 and M ∩ U 2 are nonempty and their union is M. A set M ⊂ R^2 is called path-connected, if for any two points z 1 and z 2 in M there is an interval [a, b] and a continuous map
γ : [a, b] → M
such that γ(a) = z 1 and γ(b) = z 2.
- Prove that a connected open set is path-connected.
- Prove that a path connected set is connected.
- Find a connected set that is not pathwise connected. Prove these assertions.
Problem 3: Radius of convergence
a) Given a power series ∑∞
n=
anzn
Prove that there is a uniqu R ∈ [0, ∞] (enmdpoints included) such that the powers series converges uniformly on compact subsets of the disc |z| < R and it does not converge at any point z for which |z| > R. (This is called the radius of convergence. b) Assume R > 0, then the power series defines a holomoprhic function on the disc D of radius R about the origin by a). Prove that there exists a point z on the boundary of D so that there does not exist an open neighborhood Ω of z and a holomorphic function g : Ω → R^2 such that g = f on Ω ∩ D. Hint: Prove that if no such point exists, then the functionf extends holomorphi- cally to a larger disc about the origin. Then write the Taylor coefficients of f in terms of integrals over circles (or regular n − gons if you want to use the techniques of the polygonal Cauchy theorem of the lecturenotes) and prove that the radius of convergence is larger than R.
Problem 4: Runge’s theorem
a) If (z 1 , z 2 ) is a line segment of length δ, z 3 is a point of distance less than δ from each of z 1 and z 2 , and assume f is continuous on (z 1 , z 2 ). Prove that outside the disc of radius 5δ about z 3 the function
g(z′) =
∫
z 1 ,z 2
f (z) z − z′^
dz
can be approximated uniformly by rational functions with pole at z 3 only. b) Prove that on any open (and bounded for simplcitiy) set Ω ⊂ C, any holo- morphic function can be approximated uniformly on comapct subsets by rational functions. Use the construction we did to prove the polygonal Cauchy theorem (see lecture and lecture notes) c) Prove that one can do the approximation of b) even if one allows at most one pole in every connected component of Ωc. (Hint: In c) you might want to use 6.4 from the last homework assignment and what follows therefrom by Moebius transforms. In c) you are allowed to use 6. without reproving it. In part a) of the current problem the same result could be used, but since a) is much easier you are only allowed to use 6.4 for a) if you have proved 6.4.)
Problem 5: More contour integrals
- From a qualifying exam Evaluate the integral lim N →∞
∫ (^) ∞
0
cos(x^2 ) dx
Justify your reasoning Hint: Work on eix 2 and integrate over a certain triangle with a 45 degree corner. The integral with e−x 2 is often done with a polar coordinate tick.
