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I am trying to find the Fourier series of a 3D function, $e^{-\alpha(x^2 + y^2 + z^2)}$ with bounds $-\ell_1 < x < \ell_1$, $-\ell_2 < y < \ell_2$, $-\ell_3 < z < \ell_3$. I have found a reference, however I am confused about some of their mathematical notation with regards to (what looks like) summation over a vector, $\vec{\mathbf{k}}$.

How do I expand the summation into something that I would be able to compute?

$$\sum_{\vec{\mathbf{k}}} c_k e^{i(m_1 2\pi/a_1 x + m_2 2\pi/a_2 y+m_3 2\pi/a_3 z)}$$

Where $c_k= c_{m_1 m_2 m_3}$ (some function of m1, m2, m3 that I have greatly simplified here, as the actual form is very long) and $\vec{\mathbf{k}}=[m_1 \pi/\ell_1, m_2 \pi/\ell_2, m_3 \pi/\ell_3]$.

How do I expand this notation into multiple summations? My ultimate goal is to implement the 3D Fourier expansion in code (Fortran) but at this point am I still trying to understand the notation.

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Since your function $e^{-\alpha(x^2+y^2+z^2)}$ is not periodic it cannot have a Fourier series. It should have a multidimensionsl Fourier transform, though: $$\iiint F(\vec k)e^{i(k_1 x+k_2 y+k_3 z)} d^3k $$ –  Henning Makholm Oct 28 '12 at 21:21
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A function doesn't need to be periodic to expand in a Fourier series. One only needs to $extend/define$ a function over an interval $[-L,L]$; in this case, the function would then have period $2L$ –  BobaFret Oct 28 '12 at 21:27
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1 Answer

First, I'm assuming a complex Fourier series (expanded in complex exponentials) is ok.

Here's the basics for the one-variable case:

$f(x)=e^{-\alpha x^2}$ defined on $-L_1 \leq x \leq L_1$ can be expanded in a Fourier series $$f(x)=\sum\limits_{m=- \infty}^{\infty}c_m e^{imx}$$

where

$$c_m = \frac{1}{2L_1} \int_{-L_1}^{L_1} f(x)e^{-imx}dx$$

The two- and three-variable case just generalizes this:

$\displaystyle f(x)=e^{-\alpha (x^2+y^2+z^2)}$ defined on $-L_1 \leq x \leq L_1, -L_2 \leq y \leq L_2, -L_3 \leq z \leq L_3$ can be expanded in a Fourier series (here the three summations are expressed as one sigma)

$$ f(x) = \sum\limits_{\ell, m, n=- \infty}^{\infty}c_{\ell mn} e^{i \ell x}e^{imy}e^{inz} = \sum\limits_{\ell =- \infty}^{\infty} \sum\limits_{m =- \infty}^{\infty} \sum\limits_{n =- \infty}^{\infty} c_{\ell mn} e^{i \ell x}e^{imy}e^{inz} $$

where

$$c_{\ell mn} = \frac{1}{8L_1 L_2 L_3} \iiint\limits_V f(x)e^{-i \ell x}e^{-imy}e^{-inz}dV$$

and $V=[-L_1,L_1]\times[-L_2,L_2]\times[-L_3,L_3]$.

We can write the Fourier series as a sum over a vector $\vec{k} =\langle \ell, m, n \rangle$ with integer components, where the complex exponential contains $\vec{k}$ "dotted" with $\vec{x}=\langle x,y,z \rangle$:

$$f(x)=\sum\limits_{\ell, m, n=- \infty}^{\infty}c_{\ell mn} e^{i \ell x}e^{imy}e^{inz}=\sum\limits_{\vec{k}} c_{\vec{k}} e^{\vec{k}\cdot\vec{x}}$$

I hope this helps.

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