Fourier Series/x squared over Minus Pi to Pi
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Theorem
For $x \in \openint {-\pi} \pi$:
- $\ds x^2 = \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \paren {\paren {-1}^n \frac 4 {n^2} \cos n x}$
Proof
From Even Power is Even Function, $x^2$ is an even function.
By Fourier Series for Even Function over Symmetric Range, we have:
- $\ds x^2 \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos n x$
where:
\(\ds a_n\) | \(=\) | \(\ds \frac 2 \pi \int_0^\pi x^2 \map \cos {n x} \rd x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac 1 2 a_0\) | \(=\) | \(\ds \frac 2 {2 \pi} \int_0^\pi x^2 \rd x\) | Cosine of Zero is One | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 \pi \cdot \frac {\pi^3} 3\) | Primitive of Power, Fundamental Theorem of Calculus | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\pi^2} 3\) |
Then:
\(\ds \frac 2 \pi \int_0^\pi x^2 \map \cos {n x} \rd x\) | \(=\) | \(\ds \frac 2 \pi \intlimits {\frac {2 x \cos n x} {n^2} + \paren {\frac {x^2} n - \frac 2 {n^3} } \sin n x} 0 \pi\) | Primitive of $x^2 \cos a x$, Fundamental Theorem of Calculus | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 \pi \intlimits {\frac {2 x \cos n x} {n^2} } 0 \pi\) | Sine of Multiple of Pi | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 \pi \cdot \frac {2 \pi \cos n \pi} {n^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {4 \cos n \pi} {n^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^n \frac 4 {n^2}\) | Cosine of Multiple of Pi |
Substituting for $a_n$ in $(1)$:
- $\ds x^2 = \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \paren {\paren {-1}^n \frac 4 {n^2} \cos n x}$
as required.
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 23$: Special Fourier Series and their Graphs: $23.14$