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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: $$\iiint F(\vec k)e^{i(k_1 x+k_2 y+k_3 z)} d^3k $$2012-10-28
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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$2012-10-28

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