Dixon-Ferrar 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 $\vert\Re[z]\vert<1/2$. Then

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

See also Nicholson's Formula, Watson's Formula

References

Gradshteyn, I. S. and Ryzhik, I. M. Eqn. 6.518 in Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 671, 1979.

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