Coefficients of Cosine 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 $\openint \alpha {\alpha + 2 \pi}$:
- $\map f x = \dfrac {a_0} 2 + \ds \sum_{m \mathop = 1}^\infty \paren {a_m \cos m x + b_m \sin m x}$
Then:
- $\forall n \in \Z_{\ge 0}: a_n = \dfrac 1 \pi \ds \int_\alpha^{\alpha + 2 \pi} \map f x \cos n x \rd x$
Proof
\(\ds \map f x\) | \(=\) | \(\ds \dfrac {a_0} 2 + \sum_{m \mathop = 1}^\infty \paren {a_m \cos m x + b_m \sin m x}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int_\alpha^{\alpha + 2 \pi} \map f x \cos n x \rd x\) | \(=\) | \(\ds \int_\alpha^{\alpha + 2 \pi} \paren {\dfrac {a_0} 2 + \sum_{m \mathop = 1}^\infty \paren {a_m \cos m x + b_m \sin m x} } \cos n x \rd x\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_\alpha^{\alpha + 2 \pi} \dfrac {a_0} 2 \cos n x \rd x\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \sum_{m \mathop = 1}^\infty \paren {\int_\alpha^{\alpha + 2 \pi} \paren {a_m \cos m x + b_m \sin m x} \cos n x \rd x}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {a_0} 2 2 \pi \delta_{n 0} + \sum_{m \mathop = 1}^\infty \paren {\int_\alpha^{\alpha + 2 \pi} \paren {a_m \cos m x + b_m \sin m x} \cos n x \rd x}\) | $\ds \int_\alpha^{\alpha + 2 \pi} \cos n x \rd x = \begin {cases} 0 & : n \ne 0 \\ 2 \pi & : n = 0 \end {cases}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds a_0 \pi \delta_{n 0} + \sum_{m \mathop = 1}^\infty \paren {\int_\alpha^{\alpha + 2 \pi} a_m \cos m x \cos n x \rd x}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \sum_{m \mathop = 1}^\infty \paren {\int_\alpha^{\alpha + 2 \pi} b_m \sin m x \cos n x \rd x}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds a_0 \pi \delta_{n 0} + \sum_{m \mathop = 1}^\infty \paren {\int_\alpha^{\alpha + 2 \pi} a_m \cos m x \cos n x \rd x}\) | $\ds \int_\alpha^{\alpha + 2 \pi} \sin m x \cos n x \rd x = 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds a_0 \pi \delta_{n 0} + \sum_{m \mathop = 1}^\infty a_m \pi \delta_{m n}\) | $\ds \int_\alpha^{\alpha + 2 \pi} \cos m x \cos n x \rd x = \begin {cases} 0 & : m \ne n \\ \pi & : m = n \end {cases}$ |
Thus when $n = 0$ we have:
- $\ds \int_\alpha^{\alpha + 2 \pi} \map f x \rd x = \int_\alpha^{\alpha + 2 \pi} \map f x \cos 0 x \rd x = a_0 \pi$
and when $n \ne 0$ we have:
- $\ds \int_\alpha^{\alpha + 2 \pi} \map f x \cos n x \rd x = a_n \pi$
Hence the result.
$\blacksquare$
Also see
Sources
- 1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Chapter One: $\S 2$. Fourier Series: $(5 a)$, $(5 b)$