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Bessel Function Neumann Series

For a function of a complex variable z, a Neumann series is a series expansion in terms of Bessel functions of the first kind given by

sum_(n=0)^inftya_nJ_(nu+n)(z),
(1)

where nu is a real.

Special cases include

z^nu=2^nuGamma(1/2nu+1)sum_(n=0)^infty((1/2z)^(nu/2+n))/(n!)J_(nu/2+n)(z),
(2)

where Gamma(z) is the gamma function, and

sum_(n=0)^inftyb_nz^(nu+n)=sum_(n=0)^inftya_n(1/2z)^((nu+n)/2)J_((nu+n)/2)(z),
(3)

where

a_n=sum_(m=0)^(|_n/2_|)(2^(nu+n-2m)Gamma(1/2nu+1/2n-m+1))/(m!)b_(n-2m),
(4)

and |_x_| is the floor function.

See also

Fourier-Bessel Series, Generalized Fourier Series, Kapteyn Series

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References

Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.

Referenced on Wolfram|Alpha

Bessel Function Neumann Series

Cite this as:

Weisstein, Eric W. "Bessel Function Neumann Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BesselFunctionNeumannSeries.html

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