Applied Mathematics/Fourier Series

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

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

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:

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

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

$a_n=\frac{1}{\pi }\underset{-\pi }{\overset{\pi }{\int }}f(x)\cos (nx)dx$

$b_n=\frac{1}{\pi }\underset{-\pi }{\overset{\pi }{\int }}f(x)\sin (nx)dx$

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:

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