## Fourier coefficient integral for periodic function

Posted by peeterjoot on November 5, 2013

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In phy487 we’ve been using the fact that a periodic function

where

has a Fourier representation

Here is a vector in reciprocal space, say

where

Now let’s express the explicit form for the Fourier coefficient so that we can compute the Fourier representation for some periodic potentials for some numerical experimentation. In particular, let’s think about what it meant to integrate over a unit cell. Suppose we have a parameterization of the points in the unit cell

as sketched in fig. 1.1. Here . We can compute the values of for any vector in the cell by reciprocal projection

or

Let’s suppose that is period in the unit cell spanned by with , and integrate over the unit cube for that parameterization to compute

Let’s write

so that

Picking the integral of this integrand as representative, we have when

and when

This is just zero since is an integer, so we have

This gives us

This is our \textAndIndex{Fourier coefficient}. The \textAndIndex{Fourier series} written out in gory but explicit detail is

Also observe the unfortunate detail that we require integrability of the potential in the unit cell for the Fourier integrals to converge. This prohibits the use of the most obvious potential for numerical experimentation, the inverse radial .

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