Fourier Series/Pi minus x over 0 to 2 Pi

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Theorem

For $x \in \openint 0 {2 \pi}$:

$\ds \pi - x = 2 \sum_{n \mathop = 1}^\infty \frac {\sin n x} n$


Proof

By definition of Fourier series:

$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$

where:

\(\ds a_n\) \(=\) \(\ds \dfrac 1 \pi \int_0^{2 \pi} \map f x \cos n x \rd x\)
\(\ds b_n\) \(=\) \(\ds \dfrac 1 \pi \int_0^{2 \pi} \map f x \sin n x \rd x\)

for all $n \in \Z_{>0}$.


Thus:

\(\ds a_0\) \(=\) \(\ds \frac 1 \pi \int_0^{2 \pi} \map f x \rd x\) Cosine of Zero is One
\(\ds \) \(=\) \(\ds \frac 1 \pi \int_0^{2 \pi} \pi - x \rd x\) Definition of $f$
\(\ds \) \(=\) \(\ds \frac 1 \pi \intlimits {\pi x - \frac 1 2 x^2} 0 {2 \pi}\) Primitive of Power
\(\ds \) \(=\) \(\ds 2 \pi^2 - \frac 4 2 \pi^2\)
\(\ds \) \(=\) \(\ds 0\)

$\Box$


For $n > 0$:

\(\ds a_n\) \(=\) \(\ds \frac 1 \pi \int_0^{2 \pi} \map f x \cos n x \rd x\)
\(\ds \) \(=\) \(\ds \frac 1 \pi \int_0^{2 \pi} \paren {\pi - x} \cos n x \rd x\) Definition of $f$
\(\ds \) \(=\) \(\ds \pi \int_0^{2 \pi} \cos n x \rd x - \frac 1 \pi \int_0^{2 \pi} x \cos n x \rd x\)
\(\ds \) \(=\) \(\ds - \frac 1 \pi \int_0^{2 \pi} x \cos n x \rd x\) Integral over $2 \pi$ of $\cos n x$
\(\ds \) \(=\) \(\ds - \frac 1 \pi \intlimits {\frac {\cos n x} {n^2} + \frac {x \sin n x} n} 0 {2 \pi}\) Primitive of $x \cos n x$
\(\ds \) \(=\) \(\ds - \frac 1 \pi \paren {\cos 0 - \cos 2 \pi}\) Sine of Integer Multiple of Pi
\(\ds \) \(=\) \(\ds 0\) Sine and Cosine are Periodic on Reals

$\Box$


Now for the $\sin n x$ terms:

\(\ds b_n\) \(=\) \(\ds \frac 1 \pi \int_0^{2 \pi} \map f x \sin n x \rd x\)
\(\ds \) \(=\) \(\ds \frac 1 \pi \int_0^{2 \pi} \paren {\pi - x} \sin n x \rd x\) Definition of $f$
\(\ds \) \(=\) \(\ds \int_0^{2 \pi} \sin n x \rd x - \frac 1 \pi \int_0^{2 \pi} x \sin n x \rd x\)
\(\ds \) \(=\) \(\ds - \frac 1 \pi \int_0^{2 \pi} x \sin n x \rd x\) Integral over $2 \pi$ of $\sin n x$
\(\ds \) \(=\) \(\ds - \frac 1 \pi \intlimits {\frac {\sin n x} {n^2} - \frac {x \cos n x} n} 0 {2 \pi}\) Primitive of $x \sin n x$
\(\ds \) \(=\) \(\ds \frac 1 \pi \frac {2 \pi \cos 2 n \pi} n\) Sine of Multiple of Pi
\(\ds \) \(=\) \(\ds \frac 2 n\) Cosine of Multiple of Pi


Finally:

\(\ds \map f x\) \(\sim\) \(\ds \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac 2 n \sin n x\) substituting for $a_0$, $a_n$ and $b_n$ from above
\(\ds \) \(=\) \(\ds 2 \sum_{n \mathop = 1}^\infty \frac {\sin n x} n\) rearranging

$\blacksquare$


Sources


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