Arbitrary T period Fourier series

Definition of Fourier series

\(\begin{flalign*} f(t) = a_0 + \sum^\infty_{n=1} a_n \cos n\omega_0t + \sum^\infty_{n=1} b_n \sin n\omega_0t \end{flalign*}\)

Formula of Fourier coefficients

\(\begin{flalign*} & a_0 = \frac{1}{T}\int^T_0f(t)dt \\ \\ & a_n = \frac{2}{T}\int^T_0f(t)\cos n\omega_0tdt \\ \\ & b_n = \frac{2}{T}\int^T_0f(t)\sin n\omega_0tdt \end{flalign*}\)

Fourier coefficients of even function

\(\begin{flalign*} & a_0 = \frac{2}{T}\int^{T/2}_0f(t)dt \\ \\ & a_n = \frac{4}{T}\int^{T/2}_0f(t)\cos n\omega_0tdt \\ \\ & b_n = 0 \end{flalign*}\)

Fourier coefficients of odd function

\(\begin{flalign*} & a_0 = 0 \\ \\ & a_n = 0 \\ \\ & b_n = \frac{4}{T}\int^{T/2}_0f(t)\sin n\omega_0tdt \end{flalign*}\)