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Thread: Complex Integration - Resiidue Theorm

  1. #1
    Super Member
    Apr 2009

    Complex Integration - Resiidue Theorm

    f(z) has isolated singularity at z=z_0
    Now, \oint _Cf(z).dz = 2\pi i Res_{z0}f(z)
    where C is a closed path around z_0
    (This is directly from Residue Theorem)

    My question is what is this integral if C is rather an open semi-circular arc around z_0

    Do we have any formula/theorem for this?
    May in some special case - (I'm more interested in finding the limit of this integral when the semi-circular arc in getting closer and closer to z_0

    Can I say \oint _Cf(z).dz = \pi i Res_{z0}f(z)?

    Is this result ever applicable?

    Sorry, if my questions are vague but don't have a clear idea to ask a specific question at this moment
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  2. #2
    Super Member
    Aug 2008
    Let f(z) have a simple pole at z_0 and \gamma be an arc of a circle of radius r, and angle \alpha then:

    \lim_{r\to 0} \int_{\gamma} f(z)dz=\alpha i \mathop\text{Res}_{z=z_0} f(z)

    Also, something can be said about poles of other orders. See:

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