TITLE: Why do we use only upper half plane to do Residue Integration?
QUESTION [10 upvotes]: In the class of mathematics my professor showed me that "how to use the residue integration method?". And i doubt at almost the last step that professor did. he made a contour over the upper half plane of complex plan. and i didn't know that why did we must focus only upper half(that will cover poles on only upper plane and ignore another on lower plane)
My question is:
I doubt that why do we use only upper half of complex plane to do Residue Integration?
What's happened if i use lower plane or both lower and upper plane?

REPLY [6 votes]: This depends entirely on the function which you are integrating.  Let's say that you are taking a Fourier transform:
$$\int_{-\infty}^{\infty} dx \, f(x) e^{i k x}$$
If you wish to use a contour integral to evaluate this integral, then your choice of whether you use the upper or lower half plane makes a huge difference.  For $k \gt 0$, for example, the contour integral simply does not converge on a semicircle in the lower half plane as the radius of that semicircle goes to $\infty$.  Thus, you would use the upper half plane.  The opposite goes for $k \lt 0$.