While deriving the Helmholtz Green function in Sakurai we come across the integral(adsbygoogle = window.adsbygoogle || []).push({});

[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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Poles in the Lippmann-Schwinger Equation

**Physics Forums | Science Articles, Homework Help, Discussion**