How to perform a integral in momentum space

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The discussion focuses on the process of performing integrals in momentum space, specifically addressing a transition between two lines in an integral expression. A participant questions the validity of the integral's transformation, suggesting an alternative form involving a factor of $$\frac{4 \pi}{(2 \pi)^3}$$ and an integral over momentum. The presence of a ##\cos(\theta)## term in the dot product is highlighted as a key factor that influences the integration process. This term leads to differences in the exponential functions when integrating over ##d(\cos(\theta))##. Understanding these nuances is crucial for accurately performing integrals in momentum space.
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I am not sure how does the integral was did here. More preciselly, How to go from the first line to the second line? Shouldn't it be $$\frac{4 \pi}{(2 \pi)^3} \int _{0} ^{\infty} p^2 e^{ip*r}/(2 E_p)$$ ? (x-y is purelly spatial)
 
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There is a ##\cos(\theta)## in the dot product which brings out the ipr and causes the difference in the exponentials when you integrate over ##d(\cos(\theta))##
 
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