Coefficients of Cosine 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:

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


Proof

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


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.

$\blacksquare$


Also see


Sources