Editing Appendix/Ramblings/FourierSeries (section)

===Example #1===
Here we are interested in examining a density distribution, <math>~\rho(\theta)</math>, that varies smoothly over the angular coordinate interval, <math>~0 < \theta \le 2\pi</math>, and whose distribution is strictly periodic over all other <math>~2\pi</math> intervals.  Suppose that we do not have a functional specification of this continuous density distribution but, instead, are given the value of the density, <math>~\rho_i(\theta_L)</math>, at <math>~L_\mathrm{max} = 8</math>, equally spaced discrete angular-coordinate locations, <math>~\theta_L = (2\pi L/L_\mathrm{max})</math>.  
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Suppose the density distribution is given by the expression,
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<math>~\rho_i(\theta)</math>
  </td>
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<math>~=</math>
  </td>
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<math>~
\bar\rho + \alpha_j\cos(j\theta+\zeta_{cj}) + \beta_j\sin(j\theta + \zeta_{sj}) 
+ \alpha_k\cos(k\theta + \zeta_{ck}) + \beta_k\sin(k\theta + \zeta_{sk}) \, .
</math>
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'''Example #1:''' Suppose <math>~\bar\rho = 3</math>; &nbsp; <math>~(j, k, L_\mathrm{max}) = (1, 2, 8)</math>;  &nbsp; and,
<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center"><math>~\alpha_1</math></td>
  <td align="center"><math>~\beta_1</math></td>
  <td align="center"><math>~\zeta_{c1}</math></td>
  <td align="center"><math>~\zeta_{s1}</math></td>
  <td align="center"><math>~\alpha_2</math></td>
  <td align="center"><math>~\beta_2</math></td>
  <td align="center"><math>~\zeta_{c2}</math></td>
  <td align="center"><math>~\zeta_{s2}</math></td>
</tr>
<tr>
  <td align="center">---</td>
  <td align="center"><math>~0.75</math></td>
  <td align="center">---</td>
  <td align="center"><math>~0.9 \pi</math></td>
  <td align="center"><math>~0.4</math></td>
  <td align="center">---</td>
  <td align="center"><math>~0.0 \pi</math></td>
  <td align="center">---</td>
</tr>
</table>
In this case, the specified density distribution is given by the expression,
<div align="center">
<math>~\rho_i = 
3 + \tfrac{3}{4}\sin(\theta + 0.9\pi) 
+ \tfrac{2}{5}\cos(2\theta)  \, .
</math>
</div>
</td>
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</table>
-->

<table border="0" align="center" cellpadding="5">
<tr>
  <th align="center">Figure 1a</th>
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  <td align="center">
[[File:Example1Data.png|550px|Modes]]
  </td>
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</table>

Specifically, suppose that the eight values of the the density and corresponding discrete angular-coordinate locations are those identified by the eight open circular markers in Figure 1a and listed, respectively, in columns 3 and 2 of Table 1.  Feeding these eight discrete values of <math>~\rho_i</math> and <math>~\theta_L</math> into the above Fourier transform equations, we have determined the <math>~a_m</math> and <math>~b_m</math> coefficient values and have listed them in columns 5 and 6, respectively, of Table 1.  

<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="9">Table 1:  &nbsp;Data Associated with Example #1</th>
</tr>
<tr>
  <th align="center" colspan="3">
Discrete Evaluation
  </th>
  <td align="center" rowspan="11">&nbsp;&nbsp;&nbsp;</td>
  <th align="center" colspan="3">
Fourier Amplitudes
  </th>
  <td align="center" rowspan="11">&nbsp;&nbsp;&nbsp;</td>
  <th align="center" colspan="3">
Reconstruction
  </th>
</tr>
<tr>
  <td align="center"><math>~L</math></td>
  <td align="center"><math>~\theta_L = \frac{2\pi L}{L_\mathrm{max}}</math></td>
  <td align="center"><math>~\rho_i(\theta_L)</math></td>
  <td align="center"><math>~m</math></td>
  <td align="center"><math>~a_m</math></td>
  <td align="center"><math>~b_m</math></td>
  <td align="center"><math>~\rho(\theta_L)</math></td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center"><math>~0</math></td>
  <td align="center" bgcolor="lightblue">6.0000</td>
  <td align="center">0.0000</td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~0.7854</math></td>
  <td align="center"><math>~2.6595</math></td>
  <td align="center"><math>~1</math></td>
  <td align="center" bgcolor="lightblue">0.2318</td>
  <td align="center" bgcolor="lightblue">-0.7133</td>
  <td align="center"><math>~2.6595</math></td>
</tr>
<tr>
  <td align="center"><math>~2</math></td>
  <td align="center"><math>~1.5708</math></td>
  <td align="center"><math>~1.8867</math></td>
  <td align="center"><math>~2</math></td>
  <td align="center" bgcolor="lightblue">0.4000</td>
  <td align="center" bgcolor="lightblue">0.0000</td>
  <td align="center"><math>~1.8867</math></td>
</tr>
<tr>
  <td align="center"><math>~3</math></td>
  <td align="center"><math>~2.3562</math></td>
  <td align="center"><math>~2.3317</math></td>
  <td align="center"><math>~3</math></td>
  <td align="center" bgcolor="lightblue">0.0000</td>
  <td align="center" bgcolor="lightblue">0.0000</td>
  <td align="center"><math>~2.3317</math></td>
</tr>
<tr>
  <td align="center"><math>~4</math></td>
  <td align="center"><math>~3.1416</math></td>
  <td align="center"><math>~3.1682</math></td>
  <td align="center"><math>~4</math></td>
  <td align="center" bgcolor="lightblue">0.0000</td>
  <td align="center">0.0000</td>
  <td align="center"><math>~3.1682</math></td>
</tr>
<tr>
  <td align="center"><math>~5</math></td>
  <td align="center"><math>~3.9270</math></td>
  <td align="center"><math>~3.3405</math></td>
  <td align="center"><math>~5</math></td>
  <td align="center">0.0000</td>
  <td align="center">0.0000</td>
  <td align="center"><math>~3.3405</math></td>
</tr>
<tr>
  <td align="center"><math>~6</math></td>
  <td align="center"><math>~4.7124</math></td>
  <td align="center"><math>~3.3133</math></td>
  <td align="center"><math>~6</math></td>
  <td align="center">0.4000</td>
  <td align="center">0.0000</td>
  <td align="center"><math>~3.3133</math></td>
</tr>
<tr>
  <td align="center"><math>~7</math></td>
  <td align="center"><math>~5.4978</math></td>
  <td align="center"><math>~3.6683</math></td>
  <td align="center"><math>~7</math></td>
  <td align="center">0.2318</td>
  <td align="center">+0.7133</td>
  <td align="center"><math>~3.6683</math></td>
</tr>
<tr>
  <td align="center"><math>~8</math></td>
  <td align="center"><math>~6.2832</math></td>
  <td align="center"><math>~3.6318</math></td>
  <td align="center"><math>~8</math></td>
  <td align="center">6.0000</td>
  <td align="center">0.0000</td>
  <td align="center"><math>~3.6318</math></td>
</tr>
</table>

Notice the following:
* Given that the initial discrete density distribution, <math>~\rho_i(\theta_L)</math>, is ''real'' (as opposed to ''complex'') and has been given only at <math>~L_\mathrm{max} = 8</math> angular locations over the coordinate range, <math>~0 < \theta_L \le 2\pi</math> &#8212; it repeats in a periodic fashion outside of this range &#8212; the Fourier series can have, at most, <math>~L_\mathrm{max} = 8</math> independent coefficient values.  In Table 1, the cells with light-blue backgrounds contain these eight independent values.  Elaboration:  For each Fourier mode over the range, <math>~1 \le m \le (\tfrac{1}{2}L_\mathrm{max} - 1)</math>, there are two relevant, independent coefficients, namely, <math>~a_m</math> and <math>~b_m</math>, giving, in our Example #1, six of the expected eight coefficient values.  The other two unique coefficient values arise from <math>~m = 0</math> and <math>~m = \tfrac{1}{2}L_\mathrm{max}</math>.  In both of these "edge" cases, only the <math>~a_m</math> coefficient provides relevant information; <math>~b_m</math> is irrelevant because, when <math>~m=0</math>, the argument of the sine function is always zero, and when <math>~m = \tfrac{1}{2}L_\mathrm{max}</math>, the argument of the sine function is <math>~m\theta_L = \tfrac{1}{2}L_\mathrm{max} \cdot 2\pi L/L_\mathrm{max} = \pi L</math>.

According to the [[#Standard_Setup|''standard setup'' presented above]], for this particular example analysis the density reconstruction is obtained via the expression,
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<math>~\rho(\theta_L)</math>
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<math>~=</math>
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  <td align="left">
<math>~\frac{a_0}{2} + a_1\cos(\theta_L) + b_1\sin(\theta_L)
+ a_2\cos(2\theta_L) + b_2\sin(2\theta_L)
+ a_3\cos(3\theta_L) + b_3\sin(3\theta_L)
+ a_4\cos(4\theta_L) 
</math>
  </td>
</tr>

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&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
3 + [0.2318\cos(\theta_L) - 0.7133\sin(\theta_L)] + 0.4\cos(2\theta_L)
\, .
</math>
  </td>
</tr>
</table>
</div>

The values of <math>~\rho(\theta_L)</math> listed in the last column (column 7) of Table 1 have been calculated from this expression and, indeed, across the board they match the originally specified, discrete values of <math>~\rho_i(\theta_L)</math>.  Notice, however, that this last expression not only can be used to generate correct values of the density at the originally identified discrete angular coordinate locations, it also offers an analytic prescription of a ''continuous'' function for the density distribution across the entire angular-coordinate domain.  In addition to redisplaying the initially specified discrete values of the density (open circular markers) from Figure 1a, Figure1b presents this continuous function's modal decomposition.  Specifically, the horizontal (black) dashed line is the <math>~m = 0</math> contribution; the green curve shows the sum of the two (sine and cosine) terms that represent the <math>~m=1</math> component; and the purple curve displays the <math>~m = 2</math> contribution.  

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  <th align="center">Figure 1b</th>
</tr>
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  <td align="center">
[[File:Example1Modes.png|550px|Modes]]
  </td>
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</table>

When added together, these three modal contributions give the continuous density distribution depicted by the dotted black curve in Figure 1b; as desired, this continuous curve runs through all eight initially specified discrete values of the density.