Editing Appendix/Ramblings/FourierSeries (section)

===Standard Setup===
A useful way of analyzing the growth and pattern speed of nonaxisymmetric structures is to Fourier transform the (discrete) density distribution, <math>~\rho(\theta_L)</math>, in angle space, <math>~\theta_L = L \delta\theta</math>, where, <math>~\delta\theta \equiv 2\pi/L_\mathrm{max}</math>.  On the discrete angular grid, the Fourier transform equations are
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_m</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2}{L_\mathrm{max}} \cdot \sum_{L=1}^{L_\mathrm{max}} \rho(\theta_L)  \cos(m\theta_L) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b_m</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2}{L_\mathrm{max}} \cdot \sum_{L=1}^{L_\mathrm{max}} \rho(\theta_L) \sin(m\theta_L) \, .
</math>
  </td>
</tr>
</table>
</div>
Notice that, <math>~a_0 = 2\bar\rho</math>, where <math>~\bar\rho</math> is the average density.  The density function can be reconstructed via the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho(\theta_L)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a_0}{2} + \sum_{m=1}^{L_\mathrm{max}/2} \biggl[ a_m \cos(m\theta_L) + b_m\sin(m\theta_L) \biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
so the <math>~a_m</math> and <math>~b_m</math> coefficients can, and should be, interpreted as amplitudes of various ''Fourier modes'' <math>~(m)</math>.