# 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.

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

Where

$$a_0, a_n , b_n$$ 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]

i;e

$f(x + 2l) = f(x)$

the fourior series expansion of f(x) is,

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

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