Sommerfeld's Formula

There are (at least) two equations known as Sommerfeld's formula. The first is

$$J_\nu(z)={1\over 2\pi}\int_{-\eta+i\infty}^{2\pi-\eta+i\infty} e^{iz\cos t}e^{i\nu(t-\pi/2)}\,dt,$$

where $J_\nu(z)$ is a Bessel Function of the First Kind. The second states that under appropriate restrictions,

$$\int_0^\infty J_0(\tau r)e^{-\vert x\vert\sqrt{\tau^2-k^2}} ... ...{\tau^2-k^2}} = {e^{ik\sqrt{\tau^2+k^2}}\over\sqrt{r^2+x^2}}.$$

See also Weyrich's Formula

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1472 and 1474, 1980.