Weber-Sonine Formula

For $\Re[\mu+nu]>0$, $\vert\arg p\vert<\pi/4$, and $a>0$,

$$\int_0^\infty J_\nu(at)e^{-p^2t^2}t^{\mu-1}\,dt = \left({a\o... ...extstyle{1\over 2}}(\nu+\mu); \nu+1; -{a^2\over 2p^2}}\right),$$

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.

References

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