# Applied Mathematics/Fourier Series

For the function ${\displaystyle f(x)}$, Taylor expansion is possible.

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

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

## Fourier series

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

${\displaystyle {\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }(a_{n}\cos nx+b_{n}\sin nx)\cdots (1)}$

This series is referred to as Fourier series of ${\displaystyle f(x)}$ . ${\displaystyle a_{n}}$  and ${\displaystyle b_{n}}$  are called Fourier coefficients.

${\displaystyle a_{n}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)\cos(nx)dx}$
${\displaystyle b_{n}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)\sin(nx)dx}$

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

${\displaystyle f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }(a_{n}\cos nx+b_{n}\sin nx)}$