1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Showing a sequence/series of holomorphic functions is holomorphic

  1. Nov 16, 2012 #1
    1. The problem statement, all variables and given/known data
    1) Suppose a sequence of functions [itex]f_n \in H(\Omega)\cap C(\overline{\Omega})[/itex] converges uniformly on [itex]\partial \Omega[/itex]. Prove that the sequence [itex]f_n[/itex] converges uniformly on [itex]\overline{\Omega}[/itex] to a function [itex]f \in H(\Omega)\cap C(\overline{\Omega})[/itex].

    2)Suppose [itex]f \in H(\mathbb{D})[/itex] and [itex]f(0)=0[/itex]. Show that [itex]\sum_{n=1}^{\infty}f(z^n)[/itex] defines a holomorphic function on [itex]\mathbb{D}[/itex].

    2. Relevant equations

    3. The attempt at a solution

    1) I think I need to use the uniform convergence on the boundary to find a common M>0 such that each [itex]|f_n| < M[/itex] by Maximum Modulus, from which it follows that I have uniform convergence on the disk.

    2) So f is analytic and has no constant term. I'm not quite sure how that helps or what I should even think about applying to this problem.
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted

Similar Discussions: Showing a sequence/series of holomorphic functions is holomorphic
  1. Holomorphic functions (Replies: 1)

  2. Holomorphic functions (Replies: 9)

  3. Holomorphic function (Replies: 1)