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Thread: Complex Integration - Resiidue Theorm

  1. #1
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    Complex Integration - Resiidue Theorm

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

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

    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 $\displaystyle z_0$

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

    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
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    Let $\displaystyle f(z)$ have a simple pole at $\displaystyle z_0$ and $\displaystyle \gamma$ be an arc of a circle of radius $\displaystyle r$, and angle $\displaystyle \alpha$ then:

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

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

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