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Homework Help: Limit of a Complex Integral

  1. Mar 20, 2010 #1
    1. The problem statement, all variables and given/known data
    Let U be open in C, f : U -> C continuous.
    Prove that

    [tex]
    \lim_{R\rightarrow 0} \int_0^{2\pi} f(Re^{it}) dt = 2\pi f(0)
    [/tex]

    2. Relevant equations

    [tex]
    \lim_{R\rightarrow 0} f(Re^{it}) dt = f(0)
    [/tex]

    Also

    [tex]
    \int_0^{2\pi} \lim_{R\rightarrow 0} f(Re^{it}) dt = \int_0^{2\pi} f(0) = 2\pi f(0)
    [/tex]

    3. The attempt at a solution
    The question then just resolves to passing the limit inside the integral.
    I would think then we'd use either uniform convergence or DCT, since the integral of a complex function is simply the sum of the integrals of its real and imaginary parts.

    Would this be the correct way of doing it, or is there some other way from complex analysis instead?
     
  2. jcsd
  3. Mar 20, 2010 #2
    I don't know any complex analysis, but for DCT, the fact that the integral of a complex function is the sum of the integrals of its real and imaginary parts is irrelevant right? The dominating function would just be the upper bound guaranteed by continuity of f right?
     
  4. Mar 21, 2010 #3
    How would continuity imply boundedness in this case?
    I would understand that if U was compact.
     
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