# Applied Mathematics/Fourier Series

## IntroductionEdit

For the function $f(x)$, Taylor expansion is possible.

$f(x)=f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\cdots+\frac{f^{(n)}(a)}{n!}(x-a)^n+\cdots$

This is the Taylor expansion of $f(x)$. On the other hand, more generally speaking, $f(x)$ can be expanded by also Orthogonal f

## Fourier seriesEdit

For the function $f(x)$ which has $2\pi$ for its period, the series below is defined:

$\frac{a_0}{2}+\sum_{n=1}^\infty (a_n\cos nx+b_n\sin nx)\cdots(1)$

This series is referred to as Fourier series of $f(x)$. $a_n$ and $b_n$ are called Fourier coefficients.

$a_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx) dx$
$b_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin(nx) dx$

where $n$ is natural number. Especially when the Fourier series is equal to the $f(x)$, (1) is called Fourier series expansion of $f(x)$. Thus Fourier series expansion is defined as follows:

$f(x)=\frac{a_0}{2}+\sum_{n=1}^\infty (a_n\cos nx+b_n\sin nx)$