Fourier Series
Overview
A Fourier series expresses a periodic function as the sum of
sines
and cosines.
Fourier Series Expansion
A fourier series expansion expresses a function as the weighted sum of sine and cosine functions.
{% f(x) \approx \frac{a_0}{2} + \sum_{i=1} ^n (a_n cos \frac{n \pi x}{c} + b_n sin \frac{n \pi x}{c})%}
the coefficients can be calculated by the following formulas
{% a_n = \frac{1}{\pi} \int_{- c}^{c} \; f(x)\; cos (\frac{n \pi x}{c}) \; dx %}
{% b_n = \frac{1}{\pi} \int_{- c}^{c} \; f(x)\; sin(\frac{n \pi x}{c}) \; dx %}
Note that c is generally taken to be equal to {% \pi %}, however, it can taken to be any size.
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