Half-Range Fourier Sine Series/Sine of Non-Integer Multiple of x over 0 to Pi
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Theorem
Let $\lambda \in \R \setminus \Z$ be a real number which is not an integer.
Let $\map f x$ be the real function defined on $\openint 0 \pi$ as:
- $\map f x = \sin \lambda x$
Then its half-range Fourier sine series can be expressed as:
\(\ds \map f x\) | \(\sim\) | \(\ds \frac {2 \sin \lambda \pi} \pi \paren {\sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {n \sin n x} {\lambda^2 - n^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \sin \lambda \pi} \pi \paren {-\frac {\sin x} {\lambda^2 - 1} + \frac {2 \sin 2 x} {\lambda^2 - 4} - \frac {3 \sin 3 x} {\lambda^2 - 9} + \frac {4 \sin 4 x} {\lambda^2 - 16} - \dotsb}\) |
Proof
By definition of half-range Fourier sine series:
- $\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \sin n x$
where for all $n \in \Z_{> 0}$:
- $a_n = \ds \frac 2 \pi \int_0^\pi \map f x \sin n x \rd x$
Because $\lambda \notin \Z$ we have that $\lambda \ne n$ for all $n$.
Thus for $n > 0$:
\(\ds a_n\) | \(=\) | \(\ds \frac 2 \pi \int_0^\pi \map f x \sin n x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 \pi \int_0^\pi \sin \lambda x \sin n x \rd x\) | Definition of $f$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 \pi \intlimits {\frac {\map \sin {\lambda - n} x} {2 \paren {\lambda - n} } - \frac {\map \sin {\lambda + n} x} {2 \paren {\lambda + n} } } 0 \pi\) | Primitive of $\sin \lambda x \sin n x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 \pi \paren {\paren {\frac {\sin \paren {\lambda - n} \pi} {2 \paren {\lambda - n} } - \frac {\sin \paren {\lambda + n} \pi} {2 \paren {\lambda + n} } } - \paren {\frac {\sin 0} {2 \paren {\lambda - n} } - \frac {\sin 0} {2 \paren {\lambda + n} } } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 \pi \paren {\frac {\sin \paren {\lambda - n} \pi} {\lambda - n} - \frac {\sin \paren {\lambda + n} \pi} {\lambda + n} }\) | Sine of Multiple of Pi and simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 \pi \paren {\frac {\sin \lambda \pi \cos n \pi - \cos \lambda \pi \sin n \pi} {\lambda - n} - \frac {\sin \lambda \pi \cos n \pi + \cos \lambda \pi \sin n \pi} {\lambda + n} }\) | Sine of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sin \lambda \pi \cos n \pi} \pi \paren {\frac 1 {\lambda - n} - \frac 1 {\lambda + n} }\) | Sine of Multiple of Pi and simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {-1}^n \sin \lambda \pi} \pi \frac {\lambda + n - \lambda + n} {\paren {\lambda - n} \paren {\lambda + n} }\) | Cosine of Multiple of Pi and manipulation | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^n \frac {2 \sin \lambda \pi} \pi \frac n {\lambda^2 - n^2}\) | Difference of Two Squares |
$\Box$
Finally:
\(\ds \map f x\) | \(\sim\) | \(\ds \sum_{n \mathop = 1}^\infty b_n \sin n x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {2 \sin \pi} \pi \frac n {\lambda^2 - n^2} \sin n x\) | substituting for $b_n$ from above | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \sin \lambda \pi} \pi \paren {\sum_{n \mathop = 1}^\infty \paren {-1}^n \frac n {\lambda^2 - n^2} \sin n x}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \sin \lambda \pi} \pi \paren {\sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {n \sin n x} {\lambda^2 - n^2} }\) | further manipulation |
$\blacksquare$
Also presented as
Some sources present this result as:
\(\ds \sin \lambda x\) | \(\sim\) | \(\ds \frac {2 \sin \lambda \pi} \pi \paren {\sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {n \sin n x} {n^2 - \lambda^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \sin \lambda \pi} \pi \paren {\frac {\sin x} {1 - \lambda^2} - \frac {2 \sin 2 x} {2^2 - \lambda^2} + \frac {3 \sin 3 x} {3^2 - \lambda^2} - \frac {4 \sin 4 x} {4^2 - \lambda^2} + \dotsb}\) |
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 23$: Miscellanous Fourier Series: $23.19$