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:
where
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]
- If are the cylindrical polar coordinates, then the series is a solution of Laplace equation for .
See also
References
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