Editing Fourier series (section)

=== Synthesis ===
A Fourier series can be written in several equivalent forms, shown here as the <math>N^\text{th}</math> [[Series_(mathematics)#Partial_sum_of_a_series|partial sums]] <math>s_N(x)</math> of the Fourier series of <math>s(x)</math>:<ref>
{{Citation
| last = Strang      
| first = Gilbert 
| author-link = Gilbert Strang
| title = Fourier Series And Integrals
| publisher = Wellesley-Cambridge Press
| year = 2008
| edition = 2
| chapter = 4.1
| chapter-url = https://math.mit.edu/~gs/cse/websections/cse41.pdf
| page = 323 (eq 19)
| url = https://math.mit.edu/~gs/cse/
}}
</ref>
[[File:Fourier_series_illustration.svg|right|thumb|400x400px|Fig 1. The top graph shows a non-periodic function <math>s(x)</math> in blue defined only over the red interval from 0 to ''P''. The function can be analyzed over this interval to produce the Fourier series in the bottom graph. The Fourier series is always a periodic function, even if original function <math>s(x)</math> is not.]]

{{Equation box 1 |title=Sine-cosine form
|indent=:|border|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6
|equation={{NumBlk|
|<math>s_N(x) = a_0 + \sum_{n=1}^N \left( 
a_n \cos \left(2 \pi \tfrac{n}{P} x \right) + 
b_n \sin \left(2 \pi \tfrac{n}{P} x  \right) \right)</math> &nbsp; &nbsp;
|{{EquationRef|Eq.1}}}}}}<br/>

{{Equation box 1 |title=Exponential form
|indent=:|border|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6
|equation={{NumBlk|
|<math>s_N(x) = \sum_{n=-N}^N c_n\ e^{i 2\pi \tfrac{n}{P}x}</math> &nbsp; &nbsp;
|{{EquationRef|Eq.2}}}}}}
The harmonics are indexed by an integer, <math>n,</math> which is also the number of cycles the corresponding sinusoids make in interval <math> P</math>. Therefore, the sinusoids have''':'''

* a [[wavelength]] equal to <math>\tfrac{P}{n}</math> in the same units as <math>x</math>.
* a [[frequency]] equal to <math>\tfrac{n}{P}</math> in the reciprocal units of <math>x</math>.

These series can represent functions that are just a sum of one or more frequencies in the [[harmonic spectrum]].  In the limit <math>N\to\infty</math>, a trigonometric series can also represent the intermediate frequencies or non-sinusoidal functions because of the infinite number of terms.