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Kapteyn Series

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A Kapteyn series is a series of the form

sum_(n=0)^inftyalpha_nJ_(nu+n)[(nu+n)z],
(1)

where J_n(z) is a Bessel function of the first kind. Examples include Kapteyn's original series

1/(1-z)=1+2sum_(n=1)^inftyJ_n(nz)
(2)

and

(z^2)/(2(1-z^2))=sum_(n=1)^inftyJ_(2n)(2nz).
(3)

See also

Bessel Function Neumann Series, Bessel Function of the First Kind, Fourier-Bessel Series, Generalized Fourier Series, Lemon Surface, Schlömilch's Series

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References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1473, 1980.

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Kapteyn Series

Cite this as:

Weisstein, Eric W. "Kapteyn Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KapteynSeries.html

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