Editing Appendix/Ramblings/FourierSeries (section)

===Complex===
Here we make use of the ''exponential''/complex relation &#8212; also referred to as Euler's equation,
<div align="center">
<math>~e^{i\alpha} = \cos\alpha + i \sin\alpha \, ,</math>
&nbsp; &nbsp;&nbsp;<math>~\Rightarrow</math>&nbsp; &nbsp;&nbsp;
<math>~e^{-i\alpha} = \cos\alpha - i \sin\alpha \, ,</math>
</div>
in which case we may write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cos\alpha = \frac{1}{2} \biggl[ e^{i\alpha} + e^{-i\alpha}\biggr] \, ,</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\sin\alpha = \frac{1}{2i} \biggl[ e^{i\alpha} - e^{-i\alpha}\biggr]\, .</math>
  </td>
</tr>
</table>
</div>

Employing these definitions of the trigonometric relations <math>~\cos\alpha</math> and <math>~\sin\alpha</math>, the standard representation of the Fourier series may be rewritten as,
<div align="center" id="ComplexExpression">
<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center">''Complex'' Fourier Series Expression</th>
</tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f(x)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\sum_{n = -\infty}^{n = + \infty} d_n e^{i\omega_n x} \, ,
</math>
  </td>
</tr>
</table>
</td></tr></table>
</div>
where, for <math>~n = 0, \pm 1, \pm 2,  \pm 3, \dots~</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\omega_n</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{n\pi }{L} \, ,
</math>
  </td>
</tr>
</table>
</div>

and the ''complex'' coefficients,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d_n = a_{n} -i b_{n} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{L} \int_{-L}^{L} f(x) \cos\biggl( \frac{n\pi x}{L} \biggr) dx
- i\frac{1}{L} \int_{-L}^{L} f(x) \sin\biggl( \frac{n\pi x}{L} \biggr) dx
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{L} \int_{-L}^{L} f(x) \biggl[ \cos\biggl( \frac{n\pi x}{L} \biggr)
- i\sin\biggl( \frac{n\pi x}{L} \biggr)\biggr] dx
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{L} \int_{-L}^{L} f(x)e^{-i\omega_n x} dx \, .
</math>
  </td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8" width="70%">
<tr>
  <th align="center" bgcolor="yellow">
LaTeX mathematical expressions cut-and-pasted directly from
<br />
NIST's ''Digital Library of Mathematical Functions''
  </th>
</tr>
<tr>
  <td align="left">
As an additional primary point of reference, note that according to [https://dlmf.nist.gov/1.8 &sect;1.8(i) of NIST's ''Digital Library of Mathematical Functions''], this ''alternative form'' is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f(x)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sum^{\infty}_{n=-\infty}c_{n}e^{inx},</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~c_n</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2\pi}\int^{\pi}_{-\pi}f(x)e^{-inx}\mathrm{d}x.</math>
  </td>
</tr>
</table>

  </td>
</tr>
</table>


Let's demonstrate that this rewritten (complex) expression for <math>~f(x)</math> matches the standard Fourier series expression.  First, we will refer to the above ''standard'' definitions of <math>~a_n</math> and <math>~b_n</math> as, respectively, <math>~a_{|n|}</math> and <math>~b_{|n|}</math>, and recognize that, as the summation is extended to negative numbers, the following mapping is appropriate:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_n ~ \rightarrow ~ a_{|n|}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~b_n ~ \rightarrow ~ b_{|n|} \, ,</math>
  </td>
  <td align="right">&nbsp; &nbsp; for </td>
  <td align="left"><math>~n > 0 \, ;</math>
</tr>
<tr>
  <td align="right">
<math>~a_n ~ \rightarrow ~ a_{|n|}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~b_n ~ \rightarrow ~ - b_{|n|} \, ,</math>
  </td>
  <td align="right">&nbsp; &nbsp; for </td>
  <td align="left"><math>~n < 0 \, .</math>
</tr>
</table>
</div>

Hence, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~2f(x)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sum_{n = -\infty}^{n = + \infty} (a_n - ib_n)e^{i\omega_n x} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sum_{n = 1}^{n = + \infty} (a_{|n|} - ib_{|n|})e^{i\omega_{|n|} x} 
+ a_0
+ \sum_{n = \infty}^{n = 1} (a_{|n|} + ib_{|n|})e^{- i\omega_{|n|} x} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sum_{n = 1}^{n = + \infty} (a_{|n|} - ib_{|n|}) [ \cos (\omega_{|n|} x) + i\sin (\omega_{|n|} x)]
+ a_0
+ \sum_{n = 1}^{n = \infty} (a_{|n|} + ib_{|n|}) [ \cos (\omega_{|n|} x) - i\sin (\omega_{|n|} x)]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a_0 +
\sum_{n = 1}^{n = + \infty}\biggl\{  (a_{|n|} - ib_{|n|}) [ \cos (\omega_{|n|} x) + i\sin (\omega_{|n|} x)]
+ (a_{|n|} + ib_{|n|}) [ \cos (\omega_{|n|} x) - i\sin (\omega_{|n|} x)] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a_0 +
\sum_{n = 1}^{n = + \infty}\biggl\{2a_{|n|} \cos (\omega_{|n|} x)  + 2b_{|n|} \sin (\omega_{|n|} x)] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ f(x)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a_0}{2} +
\sum_{n = 1}^{n = + \infty}\biggl\{a_{|n|} \cos (\omega_{|n|} x)  + b_{|n|} \sin (\omega_{|n|} x)] \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>
Q.E.D.