\[{H^{-}_{\ell}}\left(\eta,\rho\right)=\frac{e^{-\mathrm{i}\rho}\rho^{-\ell}}{(2%
\ell+1)!C_{\ell}\left(\eta\right)}\int_{0}^{\infty}e^{-t}t^{\ell-\mathrm{i}%
\eta}(t+2\mathrm{i}\rho)^{\ell+\mathrm{i}\eta}\,\mathrm{d}t,\]