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Homework Help: Proof of Liouville's Theorem

  1. Oct 31, 2013 #1
    1. The problem statement, all variables and given/known data
    Prove Liouville's theorem directly using the Cauchy Integral formula by showing that f(z)-f(0)=0.

    2. Relevant equations
    f(a) = [itex]\frac{1}{2πi}[/itex][itex]\oint\frac{f(z)}{z-a}dz[/itex]

    3. The attempt at a solution
    So the thing is, I know how to prove Liouville's theorem using CIF, but it doesn't show f(z)-f(0)=0, or at least not directly, and I've tried looking up other methods of proving it this way, but can't find any.
    Last edited: Oct 31, 2013
  2. jcsd
  3. Oct 31, 2013 #2
    The proof I know expands f into a Taylor's series at zero , and shows that each ##a_k## has to be zero except for k = 0. We know ##a_0## = f(0). Are you familiar with this approach?
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