Complex Integration - Resiidue Theorm

**aman_cc** · October 2nd 2009, 11:00 AM

$f(z)$ 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

**shawsend** · October 2nd 2009, 02:13 PM

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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