Fourier series type expansion
Schlömilch's series is a Fourier series type expansion of twice continuously differentiable function in the interval
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
has the following expansion:
![{\displaystyle f(x)=a_{0}+\sum _{n=1}^{\infty }a_{n}J_{0}(nx),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40c16473bd4ebc03116e129e8a9758075e50a05b)
where
![{\displaystyle {\begin{aligned}a_{0}&=f(0)+{\frac {1}{\pi }}\int _{0}^{\pi }\int _{0}^{\pi /2}uf'(u\sin \theta )\ d\theta \ du,\\a_{n}&={\frac {2}{\pi }}\int _{0}^{\pi }\int _{0}^{\pi /2}u\cos nu\ f'(u\sin \theta )\ d\theta \ du.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/290d966a63eb4f40c9632ef431be60a5e7c7cf39)
Examples
Some examples of Schlömilch's series are the following:
- Null functions in the interval
can be expressed by Schlömilch's Series,
, 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
; the series oscillates at
and diverges at
. This theorem is generalized so that
when
and
and also when
and
. These properties were identified by Niels Nielsen.[6] ![{\displaystyle x={\frac {\pi ^{2}}{4}}-2\sum _{n=1,3,...}^{\infty }{\frac {J_{0}(nx)}{n^{2}}},\quad 0<x<\pi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ee604aa3e25eb580e4cf0223951fc11fca9b1aa)
![{\displaystyle x^{2}={\frac {2\pi ^{2}}{3}}+8\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n^{2}}}J_{0}(nx),\quad -\pi <x<\pi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01cd6ea6289d0d0fa77926296941cf60555c322c)
![{\displaystyle {\frac {1}{x}}+\sum _{m=1}^{k}{\frac {2}{\sqrt {x^{2}-4m^{2}\pi ^{2}}}}={\frac {1}{2}}+\sum _{n=1}^{\infty }J_{0}(nx),\quad 2k\pi <x<2(k+1)\pi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1eb525f700d7d48dd5722700d0cccf427250fa72)
- If
are the cylindrical polar coordinates, then the series
is a solution of Laplace equation for
.
See also
References
- ^ Schlomilch, G. (1857). On Bessel's function. Zeitschrift fur Math, and Pkys., 2, 155-158.
- ^ Whittaker, E. T., & Watson, G. N. (1996). A Course of Modern Analysis. Cambridge university press.
- ^ Lord Rayleigh (1911). LXII. On a physical interpretation of Schlömilch's theorem in Bessel's functions. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 21(124), 567-571.
- ^ Watson, G. N. (1995). A treatise on the theory of Bessel functions. Cambridge university press.
- ^ Chapman, S. (1911). On the general theory of summability, with application to Fourier's and other series. Quarterly Journal, 43, 1-52.
- ^ Nielsen, N. (1904). Handbuch der theorie der cylinderfunktionen. BG Teubner.