Watson's Formula

Let $J_\nu(z)$ be a Bessel Function of the First Kind, $Y_\nu(z)$ a Bessel Function of the Second Kind, and $K_\nu(z)$ a Modified Bessel Function of the First Kind. Also let $\Re[z]>0$ and require $\Re[\mu-\nu]<1$. Then

$$J_\mu(z)Y_\nu(z)-J_\nu(z)Y_\mu(z) = {4\sin[(\mu-\nu)\pi]\over\pi^2}\int_0^\infty K_{\nu-\mu}(2z\sinh t)e^{-(\mu+\nu)t}\,dt.$$

The fourth edition of Gradshteyn and Ryzhik (1979), Iyanaga and Kawada (1980), and Ito (1987) erroneously give the exponential with a Plus Sign. A related integral is given by

$$J_\nu(z){\partial Y_\nu(z)\over\partial\nu}-Y_\nu(z){\partia... ...\nu} = -{4\over\pi}\int_0^\infty K_0(2z\sinh t)e^{-2\nu t}\,dt$$

for $\Re[z]>0$.

See also Dixon-Ferrar Formula, Nicholson's Formula

References

Gradshteyn, I. S. and Ryzhik, I. M. Eqns. 6.617.1 and 6.617.2 in Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 710, 1979.

Ito, K. (Ed.). Encyclopedic Dictionary of Mathematics, 2nd ed. Cambridge, MA: MIT Press, p. 1806, 1987.

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