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I Complex analysis proof

  1. May 26, 2017 #1
    Hello! I have this Proposition: "A harmonic function is infinitely differentiable". The book gives a proof that uses this theorem: "Suppose u is harmonic on a simply-connected region G. Then there exists a harmonic function v in G such that ##f = u + iv## is holomorphic in G. ". In the proof they present in the book they begin with: "Suppose u is harmonic in G and ##z_0 ∈ G##. Let ##r > 0## such that the disk ##D[z_0, r]## is contained in G. " and as a disk is simply connected the conclusion follows from the theorem. My question is, how can you make sure that for any ##z_0 \in G## you can have a disk around ##z_0##? (For example if ##G=\mathbb{C}##\##\mathbb{R}_{<0}## and ##z_0=0##, you can't find such a disk. What am I missing?
     
  2. jcsd
  3. May 26, 2017 #2

    FactChecker

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    Typically, these things are only considered in the interior of the set G. Statements about points on the boundary of G would require a lot of conditions and restrictions.
     
  4. May 26, 2017 #3
    they evidently assume ##G## to be an open set:
     
  5. May 26, 2017 #4

    martinbn

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    Region usually means open and connected.
     
  6. May 26, 2017 #5
    by the way, what is a harmonic function at such a point ##z_0##?
     
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