Coefficients of Cosine Terms in Convergent Trigonometric Series

Theorem
Let $\map S x$ be a trigonometric series which converges to $\map f x$ on the interval $\closedint \alpha {\alpha + 2 \pi}$:


 * $\map f x = \dfrac {a_0} 2 + \displaystyle \sum_{m \mathop = 1}^\infty \paren {a_m \cos m x + b_m \sin m x}$

Then:
 * $\forall n \in \Z_{\ge 0}: a_n = \dfrac 1 \pi \displaystyle \int_\alpha^{\alpha + 2 \pi} \map f x \cos n x \rd x$

Proof
Thus when $n = 0$ we have:


 * $\displaystyle \int_\alpha^{\alpha + 2 \pi} \map f x \rd x = \int_\alpha^{\alpha + 2 \pi} \map f x \cos 0 x \rd x = a_0 \pi$

and when $n \ne 0$ we have:


 * $\displaystyle \int_\alpha^{\alpha + 2 \pi} \map f x \cos n x \rd x = a_n \pi$

Hence the result.

Also see

 * Coefficients of Sine Terms in Convergent Trigonometric Series