Sonine-Schafheitlin Formula

$\int_0^\infty J_\mu(at)J_\nu(bt)t^{-\lambda}\,dt = {a^\mu\Gamma[(\mu+\nu-\lambd... ...]\over 2^\lambda b^{\mu-\lambda+1} \Gamma[(-\mu+\nu+\lambda+1)/2]\Gamma(\mu+1)}$
$\times\, {}_2F_1((\mu+\nu-\lambda+1)/2, (\mu-\nu-\lambda+1)/2; \mu+1; a^2/b^2),$

where $\Re[\mu+\nu-\lambda+1]>0$, $\Re[\lambda]>-1$, $0<a<b$, $J_\nu(x)$ is a Bessel Function of the First Kind, $\Gamma(x)$ is the Gamma Function, and ${}_2F_1(a,b;c;x)$ is a Hypergeometric Function.

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1474, 1980.