Math — Fourier Series & Transform

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

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

\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)e^{-2\pi i x\xi}\,dx