Applied Mathematics/Fourier Series

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Introduction[edit]

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[edit]

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)