Definition:Fourier Series

Also defined as
The form given here is more general than that usually given.

The usual form is one of the cases where $\alpha = 0$ or $\alpha = -\lambda$, thus giving a range of integration of either $\openint 0 {2 \lambda}$ or $\openint {-\lambda} \lambda$.

The actual range may often be chosen for convenience of analysis.

Also see

 * Coefficients of Cosine Terms in Convergent Trigonometric Series
 * Coefficients of Sine Terms in Convergent Trigonometric Series