Schlömilch's series explained

Schlömilch's series is a Fourier series type expansion of twice continuously differentiable function in the interval

( 0 ,\pi) in terms of the Bessel function of the first kind, named after the German mathematician Oskar Schlömilch, who derived the series in 1857.^{[1] [2] [3] [4] [5]} The real-valued function f(x) has the following expansion:f(x) = a _{0} + \sum ^{inf ty} _{n =1} a _{n} J _{0(nx),}

where

\begin{align} a _{0} & = f( 0 ) + (1)/(\pi) \int ^{\pi} _{0} \int ^{\pi/ 2} _{0} u f'(u\sin\theta) d\theta du, \\ a _{n} & = (2)/(\pi) \int ^{\pi} _{0} \int ^{\pi/ 2} _{0} u\cos nu f'(u\sin\theta) d\theta du. \end{align}

Examples

Some examples of Schlömilch's series are the following:

• Null functions in the interval

( 0 ,\pi) can be expressed by Schlömilch's Series, 0 = (1)/(2) + \sum ^{inf ty} _{n =1} ( -1 ) ^{n} J _{0(nx)} , which cannot be obtained by Fourier Series. This is particularly interesting because the null function is represented by a series expansion in which not all the coefficients are zero. The series converges only when 0 <x<\pi ; the series oscillates at x =0 and diverges at x = \pi . This theorem is generalized so that 0 = (1)/(2 \Gamma(\nu +1 )) + \sum ^{inf ty} _{n =1} ( -1 ) ^{n} J ^{\nu} _{0 (nx)/(nx/ 2 )} when -1 / 2 <\nu\leq 1 / 2 and 0 <x<\pi and also when \nu> 1 / 2 and 0 <x\leq \pi . These properties were identified by Niels Nielsen.^[6] x = (\pi ^{2})/(4) -2 \sum ^{inf ty} _{n =1 , 3 ,...} (J _{0(nx)})/(n ^{2}) , 0 <x<\pi. x ^{2} = (2 \pi ^{2})/(3) + 8 \sum ^{inf ty} _{n =1} (( -1 ) ^{n})/(n ^{2}) J _{0(nx),} - \pi<x<\pi. (1)/(x) + \sum ^{k (2)/(\sqrt{x ^{2-4m} ^{2\pi} ^{2})} _{m =1} } = \frac + \sum_^\infty J_0(nx), \quad 2k\pi

• If

(r,z) are the cylindrical polar coordinates, then the series 1+ \sum ^{inf ty} _{n =1} e ^{-nz} J _{0(nr)} is a solution of Laplace equation for z> 0 .

See also

• Kapteyn series

Notes and References

1. Schlomilch, G. (1857). On Bessel's function. Zeitschrift fur Math, and Phys., 2, 155-158.
2. [E. T. Whittaker|Whittaker, E. T.]
3. [John William Strutt, 3rd Baron Rayleigh|Lord Rayleigh]
4. [G. N. Watson|Watson, G. N.]
5. Chapman, S. (1911). On the general theory of summability, with application to Fourier's and other series. Quarterly Journal, 43, 1-52.
6. Nielsen, N. (1904). Handbuch der theorie der cylinderfunktionen. BG Teubner.

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