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Weber-Schafheitlin Integral

The Weber-Schafheitlin integral is the n=2 case of the integral

I_n=int_0^inftyt^(mu-1)product_(j=1)^nJ_(nu_j)(x_jt)dt,
(1)

where J_nu(z) is a Bessel function of the first kind. Let

M=mu+nu_1+...+nu_n.
(2)

For an integer n>=2 and positive real numbers x_1,...,x_n satisfying x_n>x_1+...+x_(n-1), a generalization due to Srivastava and Exton (1979) is

I_n=(2^(mu-1)x_1^(nu_1)...x_(n-1)^(nu_(n-1))x_n^(nu_n-M)Gamma(1/2M))/(Gamma(nu_1+1)...Gamma(nu_(n-1)+1)Gamma(nu_n-1/2M+1))×F_C^((n-1))(1/2M,1/2M-nu_n;nu_1+1,...,nu_(n-1)+1;(x_1^2)/(x_n^2),...,(x_(n-1)^2)/(x_n^2)),
(3)

provided that

R(1+nu_1+...+nu_n)>R(1-mu)>-n/2.
(4)

Here F_C^((n-1)) is a Lauricella function. When n=2, F_C^((1)) is the Gauss hypergeometric function, and the formula reduces to the Sonine-Schafheitlin formula. When n=3, F_C^((2)) is the Appell function F_4. A simultaneous permutation of the pairs (nu_j,x_j) gives analogous forms when another x_j is greater than the sum of the remaining arguments.

Weber (1873, pp. 75-80) treated several special cases of the n=2 integral, Sonine (1880, pp. 51-52) evaluated it for all convergent parameter values, and Schafheitlin (1887) later investigated it extensively (Watson 1966, pp. 398-403).

See also

Appell Hypergeometric Function, Bessel Function of the First Kind, Lauricella Functions, Sonine-Schafheitlin Formula, Weber-Sonine Formula

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References

Schafheitlin, P. "Ueber die Darstellung der hypergeometrischen Reihe durch ein bestimmtes Integral." Math. Ann. 30, 157-178, 1887. https://eudml.org/doc/157311.Schafheitlin, P. "Ueber die Gausssche und Besselsche Differentialgleichung und eine neue Integralform der letzteren." J. Reine Angew. Math. 114, 31-44, 1895. https://eudml.org/doc/148937.Sonine, N. J. "Recherches sur les fonctions cylindriques et le développement des fonctions continues en séries." Math. Ann. 16, 1-80, 1880. https://eudml.org/doc/156875.Srivastava, H. M. and Exton, H. "A Generalization of the Weber-Schafheitlin Integral." J. Reine Angew. Math. 309, 1-6, 1979. https://doi.org/10.1515/crll.1979.309.1.Watson, G. N. "The Discontinuous Integral of Weber and Schafheitlin." §13.4 in A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, pp. 398-403, 1966.Weber, H. "Ueber die Besselschen Functionen und ihre Anwendung auf die Theorie der elektrischen Ströme." J. reine angew. Math. 75, 75-105, 1873. https://eudml.org/doc/148193.

Cite this as:

Weisstein, Eric W. "Weber-Schafheitlin Integral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Weber-SchafheitlinIntegral.html

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