Hankel function ($H_{\nu }^{(1)},H_{\nu }^{(2)}$)

The two Hankel functions $H_{\nu }^{(1)}=J_{\nu }+i{Y}_{\nu }$ and $H_{\nu }^{(2)}=J_{\nu }-i{Y}_{\nu }$ solve the Bessel differential equation. Their large-argument forms distinguish outgoing and incoming oscillations and have sector-dependent asymptotic expansions.

For fixed order, the standard Hankel function expansions in $e^{\pm iz}{z}^{-1/2}$ are valid in different sectors on a chosen argument sheet. Analytic continuation can mix exponentials; the sectors must be specified rather than treating a formal inverse-power series as global.