Sonine (1880, pp. 51-52) evaluated the integral for all convergent parameter values, and Schafheitlin (1887) later investigated it in detail (Watson 1966, pp. 398-403).
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1474, 1980.Schafheitlin,
P. "Ueber die Darstellung der hypergeometrischen Reihe durch ein bestimmtes
Integral." Math. Ann.30, 157-178, 1887. https://eudml.org/doc/157311.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.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.
![R[mu+nu-lambda+1]>0](/images/equations/Sonine-SchafheitlinFormula/Inline1.svg)
,
![R[lambda]>-1](/images/equations/Sonine-SchafheitlinFormula/Inline2.svg)
,
and 
,
the Sonine-Schafheitlin formula is
![int_0^inftyJ_mu(at)J_nu(bt)t^(-lambda)dt=(a^muGamma[(mu+nu-lambda+1)/2])/(2^lambdab^(mu-lambda+1)Gamma[(-mu+nu+lambda+1)/2]Gamma(mu+1))_2F_1((mu+nu-lambda+1)/2,(mu-nu-lambda+1)/2;mu+1;a^2/b^2),](/images/equations/Sonine-SchafheitlinFormula/NumberedEquation1.svg)
is a Bessel function of the first kind,
is the gamma function, and 
is a hypergeometric
function. Equivalently, this is the 
specialization of the Weber-Schafheitlin
integral, since 
. The exponent parameter in that entry is 
.
