- #1
karlzr
- 131
- 2
Homework Statement
It is about an example from Essential Mathematical methods for physicists by Weber & Arfken, which describes a scattering process:
[tex]I(\sigma)=\int^{+\infty}_{-\infty}\frac{x \sin x dx}{x^2-\sigma^2}[/tex]2. The attempt at a solution
The straightforward way is to contruct a contour and one can find the result is [itex]\pi \cos\sigma[/itex]. Then the author said that since this is not an outgoing scattering wave, we should try a different technique. He moved both singular points off the real axis by letting [itex]\sigma→\sigma+i\gamma[/itex], and obtained [itex]\pi e^{-i\sigma}[/itex] or [itex]\pi e^{i\sigma}[/itex].
My question is, how can one integration has two results? Both methods make sense to me, but lead to totally different values. This problem has bothered me a lot when discussing Green function in quantum mechanics.