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Cauchy integral theorem

  1. Feb 28, 2008 #1
    Hi, so suppose f(z) is a complex function analytic on {z|1<|z|} (outside the unit circle). Also, we know that limit as z-->infinity of zf(z) = A. Now I need to show that for any circle [tex]C_{R}[/tex] centered at origin with radius R>1 and counterclockwise orientation, that

    [tex]\oint f(z)dz = 2\pi iA[/tex]

    Any ideas? I'm trying to use Cauchy integral theorem somehow but it's not working.
  2. jcsd
  3. Feb 28, 2008 #2
    Are you familiar with the Riemann sphere?

    In this picture, every closed curve in C can be considered to go around infinity (which is just one point on the sphere). If you choose a circle P with radius larger than R then your function is analytic in the connected component of C\P which contains infinity.

    The residue of your function at infinity is A so the Residue Theorem implies your assertion at once.
    Last edited: Feb 28, 2008
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