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