I am working on Schwartz QFT book problem 14.3, particularly part (c).(adsbygoogle = window.adsbygoogle || []).push({});

Basically, it asks us to evaluate the following integration.

[tex] \int \cfrac{d^3 p}{2\pi^3} \omega_p e^{i \vec{p} \cdot (\vec{x}-\vec{y})}[/tex] where

[tex] \omega_p = \sqrt{p^2 + m^2} [/tex]

I could perform the angular integration, and the result is

[tex] \cfrac{2}{(2\pi)^2 |x-y|} \int_0^{\infty} dp \,\,p \sqrt{p^2+m^2} \sin{(p|x-y|)} = \cfrac{2}{(2\pi)^2 |x-y|} \textrm{Im}\left( \int_0^{\infty} dp \,\,p \sqrt{p^2+m^2} e^{i p |x-y|} \right) [/tex]

The textbook says it should be expressed as some sort of Hankel functions, but I am not sure how I get to that point. Do you guys have any suggestion? Thank you.

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# Schwartz QFT book, Problem 14.3

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