# Thread: Complex Integration - Resiidue Theorm

1. ## 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

2. 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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