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