Coefficients of Sine Terms in Convergent Trigonometric Series

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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

$\blacksquare$


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