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Typically, you need an integral over a scalar field each time you want to know the total amount of an extensive value fiven in the field: total mass, total load, etc from the density field... To compute how a light ray is absorbed as traversing a semi-opaque media, you how have to write the integral giving the optical depth along the ray. etc.

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Write $d^3 q = dq q^2 d\theta d\phi \sin\theta$ and integrate over the angular variables. The only angular dependence in the integrand is in $e^{i \vec{q} \cdot ( \vec{x}-\vec{y}) } = e^{i q r \cos\theta}$ where I've defined $r = | \vec{x} - \vec{y} |$. Then, we have $$\int_0^{2\pi} d\phi \int_0^\pi d\theta \sin\theta e^{i q r \cos\theta}$$ There is no ...

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I'm basically at a loss how to simplify the integral. I am not sure this needs to be done. In spherically symmetric cases, the volume element, $dV$ is quite simple to deal with. In your case, you have: \begin{align} W & = \frac{1}{2} \int dV \ \rho\left(r\right) \ \Phi\left(r\right) \\ & = \frac{4 \pi \ C}{2} \int dr \ r^{2} \left( r^{-2} ... 1 The idea here is to use a stationary phase / saddle point approximation, that needs to be slightly adapted. For example, if you rewrite the integrand of the first integral in the form e^{-it\left(E+\frac{i}{2t}\ln\left(E^2-m^2\right)\right)},  you will find two stationary points $E_\pm=\pm m +\mathcal{O}(t^{-1})$. Only $+m$ can contribute to the ...

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