(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

It is about an example fromEssential Mathematical methods for physicistsby 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.

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# Homework Help: When complex integration depends on the method of evaluation

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