Applied Mathematics/Fourier Series

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 series

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:

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