Nicholson'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 $\Re[z]>0$. Then

$$J_\nu^2(z)+Y_\nu^2(z)={8\over\pi^2}\int_0^\infty K_0(2z\sinh t)\cos(2\nu t)\,dt.$$

See also Dixon-Ferrar Formula, Watson's Formula

References

Gradshteyn, I. S. and Ryzhik, I. M. Eqn. 6.664.4 in Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 727, 1979.

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