Coefficients of Sine Terms in Convergent Trigonometric Series

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Theorem

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

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


Then:

$\forall n \in \Z_{\ge 0}: b_n = \dfrac 1 \pi \ds \int_\alpha^{\alpha + 2 \pi} \map f x \sin n x \rd x$


Proof

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

$\blacksquare$


Also see


Sources

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