$\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