The most common problem is with complex exponentials (or sines and cosines which can be written as such).
Calculate the integral (which falls slowly for large x!)
(A.14) |
We shall analyse this for .
If we substitute , we find
The problem can be seen on substitution of , for fixed (as a bove)
For the integrand goes to zero very quickly with , but for =0 we enter a grey territory, where the integrand decays like . If we move the original integral up by just a little bit ( ) we are OK, since doesn’t become zero. Thus
(A.15) |
The residue is easily seen to be , and thus
(A.16) |
In the same way we can show that for we must close the contour in the lower plane (since must be negative)
(A.17) |
since no pole is enclosed inside the contour.