Editing Appendix/Ramblings/FourierSeries (section)

===Example #2===

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  <td align="center">Periodic Square Wave</td>
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[[File:SquareWaveCRCduplicate01.png|350px|right|Taken from p. 464 of Edition 19 of the CRC Standard Mathematical Tables (1971)]]
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The figure/equation pair displayed here, on the right, is an example of a "Fourier expansion for a basic periodic function" that has been presented on p. 464 of the [[Appendix/References#CRC|CRC Standard Mathematical Tables (19th Edition)]].  The following derivation, analysis, and discussion has been motivated by this published CRC example.

Referring back to the [[#Standard|''standard representation'' presented above]], suppose that <math>f(x)</math> is a square wave (unity amplitude) that extends over the range <math>~\pm c</math>, where, <math>c < L</math>.  (The dashed-black curve in [[#Figure2|Figure 2]], below, displays one cycle of this function for the specific case of c/L = 1/3.)  Replacing <math>~x</math> by the angle,
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<math>\theta \equiv \frac{n\pi x}{L} \, ,</math>
</div>
the two coefficients are (for all ''n'' > 0), 

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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>a_n</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{L} \int_{-c}^{c} \cos\biggl( \frac{n\pi x}{L} \biggr) dx
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
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<math>
\biggl( \frac{1}{n\pi} \biggr) \int_{-(n\pi c/L)}^{(n\pi c/L)} \cos ( \theta ) d\theta
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
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<math>
\biggl( \frac{1}{n\pi} \biggr) \biggl[\sin ( \theta ) \biggr]_{-(n\pi c/L)}^{(n\pi c/L)}
</math>
  </td>
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  <td align="right">
&nbsp;
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  <td align="center">
<math>=</math>
  </td>
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<math>
\biggl( \frac{2}{n\pi} \biggr) \sin \biggl( \frac{n\pi c}{L} \biggr) \, ,
</math>
  </td>
</tr>

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  <td align="right">
<math>b_n</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{L} \int_{-c}^{c} \sin\biggl( \frac{n\pi x}{L} \biggr) dx
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{1}{n\pi} \biggr) \int_{-(n\pi c/L)}^{(n\pi c/L)} \sin ( \theta ) d\theta
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\biggl( \frac{1}{n\pi} \biggr) \biggl[\cos ( \theta ) \biggr]_{-(n\pi c/L)}^{(n\pi c/L)}
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
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<math>
0 \, .
</math>
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For the special case of <math>~n = 0</math>, we have,
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<math>a_0 = \frac{1}{L}\int_{-c}^c dx = \frac{2c}{L} \, ,</math>
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&nbsp; &nbsp; and &nbsp; &nbsp;
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<math>b_0 = 0 \, .</math>
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</table>
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Hence, the Fourier series expression for the square-wave function, itself, is,

<div align="center" id="StandardExpression">
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  <td align="right">
<math>f(x)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{a_0}{2} + \sum_{n=1}^{\infty} \biggl[ a_n\cos \biggl(\frac{n\pi x}{L}\biggr) + b_n\sin  \biggl(\frac{n\pi x}{L}\biggr) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{c}{L} + \sum_{n=1}^{\infty} \biggl( \frac{2}{n\pi} \biggr) \sin \biggl( \frac{n\pi c}{L} \biggr) \cos \biggl(\frac{n\pi x}{L}\biggr) \, .
</math>
  </td>
</tr>
</table>
</div>

<div align="center" id="Figure2">
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Figure 2: &nbsp; Approximate Series Expressions for a Square-Wave<br />
<br />
<math>\frac{c}{L} = \frac{1}{3}</math>
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  <td align="center">[[File:SquareWave02.gif|Square Wave Fourier Series Expression]]</td>
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The dotted-red curve displays the behavior of the (approximate) function <math>f(x)</math>,  as the series expression is truncated for various values of the index, <math>n</math>, as indicated by the (time-varying) integer legend.  The dashed-black "square wave" shows the input function, which should be the behavior of <math>f(x)</math> when the series expression includes an infinite number of terms.
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Notice the following:
<ol start="1">
  <li>If we define,
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<math>\alpha_n \equiv \frac{n\pi c}{L} \, ,</math>
</div>
then, as a function of <math>~\alpha</math>, the Fourier mode amplitude is,
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>a_n(\alpha_n)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{2c}{L} \biggr) \frac{\sin \alpha_n}{\alpha_n} = \biggl( \frac{2c}{L} \biggr) \mathrm{sinc}(\alpha_n)  \, .</math>
  </td>
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</table>
</div>
</li>
<li>
The inverse transform may also then be rewritten as,
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  <td align="right">
<math>f(x)</math>
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  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2c}{L}\biggl[ \frac{1}{2} +\sum_{n=1}^{\infty} \mathrm{sinc}(\alpha_n) \cos \biggl(\frac{n\pi x}{L}\biggr) \biggr] \, .
</math>
  </td>
</tr>
</table>
</li>
<li>
The ''sinc'' function goes to zero each time its argument, <math>\alpha_n</math>, is an integer multiple of &pi;.  This means that the first zero arises at <math>n = L/c</math>, if <math>L</math> is an integer-multiple of <math>c</math>.  This also means that the discrete array that marks the Fourier mode amplitude, <math>a_n</math>, will resolve the central amplitude peak only if <math>c/L \ll 1</math>.
</li>
</ol>