Half-Range Fourier Cosine Series/Cosine 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:
- $\map f x = \cos \lambda x$
Then its half-range Fourier cosine series can be expressed as:
\(\ds \map f x\) | \(\sim\) | \(\ds \frac {2 \lambda \sin \lambda \pi} \pi \paren {\frac 1 {2 \lambda^2} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {\cos n x} {\lambda^2 - n^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \lambda \sin \lambda \pi} \pi \paren {\frac 1 {2 \lambda^2} - \frac {\cos x} {\lambda^2 - 1} + \frac {\cos 2 x} {\lambda^2 - 4} - \frac {\cos 3 x} {\lambda^2 - 9} + \frac {\cos 4 x} {\lambda^2 - 16} - \dotsb}\) |
Proof
By definition of half-range Fourier cosine series:
- $\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos n x$
where for all $n \in \Z_{> 0}$:
- $a_n = \ds \frac 2 \pi \int_0^\pi \map f x \cos n x \rd x$
Thus by definition of $f$:
\(\ds a_0\) | \(=\) | \(\ds \frac 2 \pi \int_0^\pi \map f x \rd x\) | Cosine of Zero is One | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 \pi \int_0^\pi \cos \lambda x \rd x\) | Definition of $f$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 \pi \intlimits {\frac {\sin \lambda x} \lambda} 0 \pi\) | Primitive of $\cos a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 \pi \paren {\frac {\sin \lambda \pi} \lambda - \frac {\sin 0} \lambda}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \sin \lambda \pi} {\pi \lambda}\) | Sine of Zero is Zero |
$\Box$
Because $\lambda \notin \Z$ we have that $\lambda \ne n$ for all $n$.
Thus for $n > 0$:
\(\ds a_n\) | \(=\) | \(\ds \frac 2 \pi \int_0^\pi \map f x \cos n x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 \pi \int_0^\pi \cos \lambda x \cos n x \rd x\) | Definition of $f$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 \pi \intlimits {\frac {\sin \paren {\lambda - n} x} {2 \paren {\lambda - n} } + \frac {\sin \paren {\lambda + n} x} {2 \paren {\lambda + n} } } 0 \pi\) | Primitive of $\cos \lambda x \cos n x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 \pi \paren {\paren {\frac {\sin \paren {\lambda - n} \pi} {2 \paren {\lambda - n} } + \frac {\sin \paren {\lambda + n} \pi} {2 \paren {\lambda + n} } } - \paren {\frac {\sin 0} {2 \paren {\lambda - n} } + \frac {\sin 0} {2 \paren {\lambda + n} } } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 \pi \paren {\frac {\sin \paren {\lambda - n} \pi} {\lambda - n} + \frac {\sin \paren {\lambda + n} \pi} {\lambda + n} }\) | 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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^n \frac {2 \sin \lambda \pi} \pi \frac \lambda {\lambda^2 - n^2}\) | Difference of Two Squares |
$\Box$
Finally:
\(\ds \map f x\) | \(\sim\) | \(\ds \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos n x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \frac {2 \sin \lambda \pi} {\pi \lambda} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {2 \sin \lambda \pi} \pi \frac \lambda {\lambda^2 - n^2} \cos n x\) | substituting for $a_0$ and $a_n$ from above | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \sin \lambda \pi} \pi \paren {\frac 1 {2 \lambda} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac \lambda {\lambda^2 - n^2} \cos n x}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \lambda \sin \lambda \pi} \pi \paren {\frac 1 {2 \lambda^2} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {\cos n x} {\lambda^2 - n^2} }\) | further manipulation |
$\blacksquare$
Also presented as
Some sources present this result as:
\(\ds \cos \lambda x\) | \(\sim\) | \(\ds \frac {2 \lambda \sin \lambda \pi} \pi \paren {\frac 1 {2 \lambda^2} + \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {\cos n x} {n^2 - \lambda^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \lambda \sin \lambda \pi} \pi \paren {\frac 1 {2 \lambda^2} + \frac {\cos x} {1 - \lambda^2} - \frac {\cos 2 x} {2^2 - \lambda^2} + \frac {\cos 3 x} {3^2 - \lambda^2} - \frac {\cos 4 x} {4^2 - \lambda^2} + \dotsb}\) |
Sources
- 1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Exercises on Chapter $\text I$: $5$.
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 23$: Miscellanous Fourier Series: $23.20$