has isolated singularity at

Now,

where is a closed path around

(This is directly from Residue Theorem)

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

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

Can I say ?

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