Fourier Series/Absolute Value of x over Minus Pi to Pi/Proof 1
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Theorem
For $x \in \openint {-\pi} \pi$:
- $\ds \size x = \frac \pi 2 - \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\map \cos {2 n - 1} x} {\paren {2 n - 1}^2}$
Proof
By definition, the absolute value function is an even function:
- $\size {-x} = x = \size x$
Thus by Fourier Series for Even Function over Symmetric Range, $\size x$ can be expressed as:
- $\ds \size x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos n x$
where for all $n \in \Z_{\ge 0}$:
- $a_n = \ds \frac 2 \pi \int_0^\pi \size x \cos n x \rd x$
On the real interval $\openint 0 \pi$:
- $\size x = x$
and so for all $n \in \Z_{\ge 0}$:
- $a_n = \ds \frac 2 \pi \int_0^\pi x \cos n x \rd x$
Thus Half-Range Fourier Cosine Series for Identity Function over $\openint 0 \pi$ can be applied directly.
So for $x \in \openint {-\pi} \pi$:
- $\ds \size x = \frac \pi 2 - \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\map \cos {2 n - 1} x} {\paren {2 n - 1}^2}$
$\blacksquare$