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Weber-Sonine Formula

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For R[mu+nu]>0, |argp|<pi/4, and a>0,

int_0^inftyJ_nu(at)e^(-p^2t^2)t^(mu-1)dt=(a/(2p))^nu(Gamma[1/2(nu+mu)])/(2p^muGamma(nu+1))_1F_1(1/2(nu+mu);nu+1;-(a^2)/(4p^2)),

where J_nu(z) is a Bessel function of the first kind, Gamma(z) is the gamma function, and _1F_1(a;b;z) is a confluent hypergeometric function of the first kind.

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References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1474, 1980.Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, p. 393, 1966.

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Weber-Sonine Formula

Cite this as:

Weisstein, Eric W. "Weber-Sonine Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Weber-SonineFormula.html

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