Weber's Formula

$${1\over 2p^2}e^{-(a^2+b^2)/(4p^2)}I_\nu\left({ab\over 2p^2}\right)=\int_0^\infty e^{-p^2t^2}J_\nu(at)J_\nu(bt)t\,dt,$$

where $\Re[\nu]>-1$, $\vert\arg p\vert<\pi/4$, and $a$, $b>0$, $J_\nu(z)$ is a Bessel Function of the First Kind, and $I_\nu(z)$ is a Modified Bessel Function of the First Kind.

See also Bessel Function of the First Kind, Modified Bessel Function of the First Kind

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1476, 1980.