Fourier Cosine Series (even extension)

$\displaystyle f(t) = \frac{a_0}{2}+\sum\limits_{n=1}^\infty a_n\cos\left(\frac{n\pi t}{L}\right)$

$\displaystyle a_0 = \frac{2}{L}\int_{0}^Lf(t) dt$

$\displaystyle a_n = \frac{2}{L}\int_{0}^Lf(t)\cos\left(\frac{n\pi t}{L}\right) dt$

Fourier Sine Series (odd extension)

$\displaystyle f(t) = \sum\limits_{n=1}^\infty b_n\sin\left(\frac{n\pi t}{L}\right)$

$\displaystyle b_n = \frac{2}{L}\int_{0}^L f(t)\sin\left(\frac{n\pi t}{L}\right) dt$

Fourier Series on [0,L] (identity reflection)

$\displaystyle f(t) = \frac{a_0}{2}+\sum\limits_{n=1}^\infty\left(a_n\cos\left(\frac{2 n\pi t}{L}\right)+b_n\sin\left(\frac{2 n\pi t}{L}\right)\right)$

$\displaystyle a_0 = \frac{2}{L}\int_{0}^L f(t) dt$

$\displaystyle a_n = \frac{2}{L}\int_{0}^L f(t)\cos\left(\frac{2 n\pi t}{L}\right) dt$

$\displaystyle b_n = \frac{2}{L}\int_{0}^L f(t)\sin\left(\frac{2 n\pi t}{L}\right) dt$