What is fourior Series

It is the method of expressing any given function in terms of sine and cosine series. The Fourier theorem’s can be expressed as “ any single valued function defined in the interval \([\pi,\pi]\) may be represented over this interval by trigonometric series.

\[f(x) = \frac{a_0}{2} + \sum_{i=1}^\infty[a_ncos(nx) + b_nsin(nx)]\]


\(a_0, a_n , b_n\) are called fourier coefficent, and \(a_0, a_n, b_n\)

are determined by the following formula,

\[1.\ a_0 = \frac{1}{\pi}\int_{-\pi}^\pi f(x) dx. \\ 2.\ a_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x)cos(nx) dx. \\ 3.\ b_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x)sin(nx) dx.\\\]

Where n = 1, 2, 3, …..

Fourier series expression for any arbitrary function in various interval

Thus far, the expansion interval has been restricted to \([\pi , \pi]\). To solve many physical problems, it is necessary to develop a fourier series that will be valid over a wide interval.

  • Fourire series in [a , b]

If f(x) is periodic in interval [a , b]

i;e \(f(x + a + b) = f(x)\)

the fourier series expansion of f(x) is,

\[f(x) = \frac{a_0}{2} + \sum_{i=1}^\infty[a_ncos\frac{2n.\pi.x}{b-a}+b_nsin\frac{2n.\pi.x}{b-a}] \\ where\\ a_0 = \frac{1}{b-a}\int_{a}^b f(x) dx.\\ a_n = \frac{1}{b-a}\int_{a}^b f(x)cos\frac{2n.\pi.x}{b-a} dx.\\ b_n = \frac{1}{b-a}\int_{a}^b f(x)sin\frac{2n.\pi.x}{b-a}. \\\]
  • Fourier series in [-l , l]

Similarly the function f(x) is periodic in the interval [-l , l]


\[f(x + 2l) = f(x)\]

the fourior series expansion of f(x) is,

\[f(x) = \frac{a_0}{2} + \sum_{i=1}^\infty[a_ncos\frac{2n.\pi.x}{l+l}+b_nsin\frac{2n.\pi.x}{l+l}] \\ where\\ a_0 = \frac{1}{l}\int_{-l}^l f(x) dx.\\ a_n = \frac{1}{l}\int_{-l}^l f(x)cos\frac{n.\pi.n}{l} dx.\\ b_n = \frac{1}{l}\int_{-l}^l f(x)sin\frac{n.\pi.n}{l} dx.\\\]