# Coefficients of Sine 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:

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

## Proof

 $\displaystyle f \left({x}\right)$ $=$ $\displaystyle \dfrac {a_0} 2 + \sum_{m \mathop = 1}^\infty \left({a_m \cos m x + b_m \sin m x}\right)$ $\displaystyle \leadsto \ \$ $\displaystyle \int_\alpha^{\alpha + 2 \pi} f \left({x}\right) \sin n x \, \mathrm d x$ $=$ $\displaystyle \int_\alpha^{\alpha + 2 \pi} \left({\dfrac {a_0} 2 + \sum_{m \mathop = 1}^\infty \left({a_m \cos m x + b_m \sin m x}\right)}\right) \sin n x \, \mathrm d x$ $\displaystyle$ $=$ $\displaystyle \int_\alpha^{\alpha + 2 \pi} \dfrac {a_0} 2 \sin n x \, \mathrm d x + \sum_{m \mathop = 1}^\infty \left({\int_\alpha^{\alpha + 2 \pi} \left({a_m \cos m x + b_m \sin m x}\right) \sin n x \, \mathrm d x}\right)$ $\displaystyle$ $=$ $\displaystyle 0 + \sum_{m \mathop = 1}^\infty \left({\int_\alpha^{\alpha + 2 \pi} \left({a_m \cos m x + b_m \sin m x}\right) \sin n x \, \mathrm d x}\right)$ $\displaystyle \int_\alpha^{\alpha + 2 \pi} \sin n x \, \mathrm d x = 0$ $\displaystyle$ $=$ $\displaystyle \sum_{m \mathop = 1}^\infty \left({\int_\alpha^{\alpha + 2 \pi} a_m \cos m x \sin n x \, \mathrm d x}\right) + \sum_{m \mathop = 1}^\infty \left({\int_\alpha^{\alpha + 2 \pi} b_m \sin m x \sin n x \, \mathrm d x}\right)$ $\displaystyle$ $=$ $\displaystyle \sum_{m \mathop = 1}^\infty \left({\int_\alpha^{\alpha + 2 \pi} b_m \sin m x \sin n x \, \mathrm d x}\right)$ $\displaystyle \int_\alpha^{\alpha + 2 \pi} \sin m x \cos n x \, \mathrm d x = 0$ $\displaystyle$ $=$ $\displaystyle \sum_{m \mathop = 1}^\infty b_m \pi \delta_{m n}$ $\displaystyle \int_\alpha^{\alpha + 2 \pi} \sin m x \sin n x \, \mathrm d x = \begin{cases} 0 & : m \ne n \\ \pi & : m = n \end{cases}$ $\displaystyle$ $=$ $\displaystyle b_n \pi$ Definition of Kronecker Delta

$\blacksquare$