Definition:Fourier Series

Let $$f:\R \to \R \ $$ be a function such that $$\int_{-\pi}^\pi f(t)dt \ $$ converges absolutely. If we set

$$\pi a_n = \int_{-\pi}^\pi f(t) \cos(nt)dt, \qquad \pi b_n=\int_{-\pi}^\pi f(t)\sin(nt)dt \ $$,

then

$$\frac{a_0}{2} + \sum_{n=1}^\infty \left({ a_n \cos(nx) + b_n \sin(nx) }\right) \ $$

is called a Fourier Series for $$f \ $$.