I am just curious about how the way one chooses to solve an integral with residue theorem (specifically how repositioning poles during the process of solving the integral; and afterwards taking a limit to position them back where they initially were) could change the final results.
The example that lead me to this question was the integral on page 469, example 7.1.5 Quantum Mechanical Scattering, in Arfken and Weber's "Mathematical Methods For Physicists" (pdf: https://msashigri.wordpress.com/wp-content/uploads/2016/11/methods-of-mathemacial-for-physicists.pdf), which is of the form:
$ \int_{-\infty}^{\infty}{\frac{z\sin(z)}{z^2-\sigma^2}dz}, \space \sigma\in\mathbb{R}^+. $
The authors use the complex exponential form of $\sin(z)$ and split the integral up into two. This they do in both presented solutions. In the first solution, they then use two semicircles each with little indents to avoid one pole and include another. Evaluating the integral in this manner then offers as an solution the following: $\pi \cos(\sigma)$. In the second solution, they let $\pm\sigma \to \pm\sigma \pm i\gamma \space$ where $\gamma$ is small and positive and then use two semicircles, this time without any indentations because the way they changed $\sigma$ already positions the poles s.t. for each contour one is included and the other excluded. They then evaluate the integrals and let $\gamma$ go to $0$, yielding the final result: $\pi e^{i \sigma}$.
How exactly such a repositioning of the poles lead to different results after taking the limit that puts it back where it originally was? What happens in the background mathematically while proceeding in such a manner?
