I am trying to understand how to go from the first line to the next:(adsbygoogle = window.adsbygoogle || []).push({});

##\frac{1}{(2\pi)^{3}}\int d^{3}p\ e^{-it\sqrt{{\bf{p}}^{2}+m^{2}}}\ e^{i {\bf{p}}({\bf{x}}-{\bf{x}}_{0})} = \frac{1}{2\pi^{2}|{\bf{x}}-{\bf{x}}_{0}|} \int_{0}^{\infty} dp\ p\ sin(p|{\bf{x}}-{\bf{x}}_{0}|)\ e^{-it\sqrt{{p}^{2}+m^{2}}}##.

As an initial guess, I'm thinking that the vector ##\bf{p}## is decomposed in spherical polar coordinates, evidenced by the facts that ##\frac{1}{(2\pi)^{3}}## has been multiplied by ##4\pi## and that the volume element is afterwards ##dp\ p## with limits from ##0## from ##\infty##. Also, I'm thinking that firstly the exponential ##e^{i {\bf{p}}({\bf{x}}-{\bf{x}}_{0})}## is broken using Euler's relation into its sine and cosine components.

I wonder what's happened to the cosine component, the factor of ##i## in the sine component, how the polar and azimuthal angles have been separated out from the vector expression ##-it\sqrt{{\bf{p}}^{2}+m^{2}}## and how the factor of ##|{\bf{x}}-{\bf{x}}_{0}|## in the denominator outside the integral appears.

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# Integration of complex exponentials

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