Coefficients of Cosine Terms in Convergent Trigonometric Series

Theorem
Let $S \left({x}\right)$ be a trigonometric series which converges to $f \left({x}\right)$ on the interval $\left[{\alpha \,.\,.\, \alpha + 2 \pi}\right]$:


 * $f \left({x}\right) = \dfrac {a_0} 2 + \displaystyle \sum_{m \mathop = 1}^\infty \left({a_m \cos m x + b_m \sin m x}\right)$

Then:
 * $\forall n \in \Z_{\ge 0}: a_n = \dfrac 1 \pi \displaystyle \int_\alpha^{\alpha + 2 \pi} f \left({x}\right) \cos n x \, \mathrm d x$

Proof
Thus when $n = 0$ we have:


 * $\displaystyle \int_\alpha^{\alpha + 2 \pi} f \left({x}\right) \, \mathrm d x = \int_\alpha^{\alpha + 2 \pi} f \left({x}\right) \cos 0 x \, \mathrm d x = a_0 \pi$

and when $n \ne 0$ we have:


 * $\displaystyle \int_\alpha^{\alpha + 2 \pi} f \left({x}\right) \cos n x \, \mathrm d x = a_n \pi$

Hence the result.

Also see

 * Coefficients of Sine Terms in Convergent Trigonometric Series