Bessel second-moment identity (${\sum }_{m\in \mathbb{Z}}{m}^2{J}_m(z{)}^2=z^2/2$)

For real $z$, the probabilities $J_m(z{)}^2$ over integer angular-momentum modes satisfy ${\sum }_m{m}^2{J}_m(z{)}^2=z^2/2$. Differentiate the unit-modulus function $e^{iz\cos \theta }$ and use Parseval to obtain the identity; its zero first moment follows from the symmetry of opposite orders.