Choice of Contour in Complex Analysis

[unchained1978]

Say you want to evaluate an integral over some domain, so one option is to write the integral as a contour integral in the complex plane. However, there can sometimes be several different contours that all cover the same domain, but may lead to different values in the event of singularities right? It seems that depending on the location of the singularities you could avoid them by a suitable contour or wrap your contour around them so as to use the residue theorem. I don't understand how the contour is to be chosen then if different routes may lead to different values, could someone please explain this to me? What conditions must the contour you choose satisfy?

 

[algebrat]

Usually it depends on #f# in #\int_C f(z) dz#. Usually your contour has some limiting behavior like letting some edge approach zero, or infinity, and/or another edge. In order for everything to work out, the particular #f# has to behave as desired, as the contour heads whichever way it is going. I have seen some applications in quantum field theory where it really did seem as arbitrary as you thought, but my guess is, for the problems you are considering, if you check the conditions for each contour technique, there is no ambiguity.

For instance, there is sometimes a choice of a half circle arc, being in the upper or lower half-plane. This usually relies on #f# going to zero at an appropriate speed, and often it won't if you pick the wrong half-plane