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I've asked this question in Mathematics Stack Exchange, but unfortunately there is no answer yet. I repost it because this integral comes from QFT and maybe someone here did it before or could help me. I merely copy this post.

I want to calculate this integral in $n$-dimensional euclidean space.

$$I(x)=\int_{\mathbb{R}^n}\frac{d^n k}{(2\pi)^n}\frac{e^{i(k\cdot x)}}{k^2+a^2},$$ where $k^2=(k\cdot k)$, $k=(k_1,\ldots,k_n)\in\mathbb{R}^n$, $x=(x_1,\ldots,x_n)\in\mathbb{R}^n$,$a\in \mathbb{R}$.

I've done this integral for $n=3$ by spherical coordinates and residue theorem. I have

$$I(r)=\frac{1}{4\pi r}e^{-ar},$$ where $r=|x|$

But in $n$-dimensions I failed in using spherical coordinates, because I have never done it before. Also I see that this integral is Fourier transform of $\frac{1}{k^2+a^2}$, but I failed here too, because I can't find Fourier pair in my reference books.

If someone could guide me in this integration it would be great.

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  • $\begingroup$ Hint: Fourier transform. $\endgroup$ – user3657 Feb 4 '14 at 17:40
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    $\begingroup$ @William As you can see, he mentioned it: "Also I see that this integral is Fourier transform of <...>, but I failed here too, because I can't find Fourier pair in my reference books." $\endgroup$ – xxxxx Feb 4 '14 at 17:43
  • $\begingroup$ @xxxxx: Ah, yes, should have read more closely. $\endgroup$ – user3657 Feb 4 '14 at 17:44
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WARNING: The function is not absolutely integrable for $n>1$, so the integral strongly depends on how you decide to compute it if you break the integration into iterated integrals.

Use instead cylindric coordinates. $k = (z, \vec{r})$, where $\vec{r} \in \mathbb R^{n-1}$ and $z\in \mathbb R$. You have this way, assuming that $x$ is directed along $z$: $$I(x) = \frac{1}{(2\pi)^n}\int_{\mathbb R^{n-1}} d\vec{r} \int_{\mathbb R} dz \frac{e^{i|x|z}}{\vec{r}^2 + z^2 +a^2}=\frac{\omega_{n-1}}{(2\pi)^n}\int_{0}^{+\infty} dr \int_{\mathbb R} dz \frac{e^{i|x|z}r^{n-2} }{r^2 + z^2 +a^2}$$ So: $$I(x) = \frac{2\omega_{n-1}}{(2\pi)^n}\int_{0}^{+\infty} dr \int_0^{+\infty} dz \frac{r^{n-2}\cos(|x|z) }{r^2 + z^2 +a^2}$$ where $\omega_{n-1} = \frac{2\pi^{(n-1)/2}}{\Gamma((n-1)/2)}$ is the measure of the surface of the unit sphere in $\mathbb R^{n-1}$.

The internal integral can be found in several books e.g. identity 3.723(2) in Gradshteyn - Ryzhik book (seventh edition). Performing it one has:

$$I(x) = \frac{\pi\omega_{n-1}}{(2\pi)^n}\int_{0}^{+\infty} dr \frac{r^{n-2}e^{-|x|\sqrt{r^2 +a^2}} }{\sqrt{r^2 +a^2}} $$ The remaining integral, passing to integrate in $d(r^2/a^2)$, can be computed in terms of Bessel functions $K_\nu$ using identity 3.479(1) in Gradshteyn - Ryzhik book (seventh edition).

Please check everything since, as usual, I am not confident in my computations!

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  • $\begingroup$ Thank you! I checked your calculations and everything is fine. $\endgroup$ – Oiale Feb 5 '14 at 10:16

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