\[F_{\ell}\left(\eta,\rho\right)=\frac{\rho^{\ell+1}2^{\ell}e^{\mathrm{i}\rho-(%
\pi\eta/2)}}{|\Gamma\left(\ell+1+\mathrm{i}\eta\right)|}\int_{0}^{1}e^{-2%
\mathrm{i}\rho t}t^{\ell+\mathrm{i}\eta}(1-t)^{\ell-\mathrm{i}\eta}\,\mathrm{d%
}t,\]