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[tex]\int_{-\infty}^{\infty}q\,dq\,\frac{e^{iq|\vec x-\vec x'|}-e^{-iq|\vec x-\vec x'|}}{q^2-k^2\mp i\varepsilon'}[/tex]

This equation has poles at [tex]q \simeq \pm k\pm i\varepsilon'[/tex], however when doing the residue calculation it seems that Sakurai only treats the poles [tex]k+i\varepsilon'[/tex] and [tex]k-i\varepsilon'[/tex], but not the companion poles poles [tex]-k-i\varepsilon'[/tex] and [tex]-k+i\varepsilon'[/tex].

Is there a physical reason for this I am missing or do I have a mathematical error? If included, it seems the other poles would give both the [tex]\psi^{(\pm)}[/tex] solutions over again?

Thanks!

Tom