# §33.7 Integral Representations

 33.7.1 $F_{\ell}\left(\eta,\rho\right)=\frac{\rho^{\ell+1}2^{\ell}e^{\mathrm{i}\rho-(% \pi\eta/2)}}{|\Gamma\left(\ell+1+\mathrm{i}\eta\right)|}\int_{0}^{1}e^{-2% \mathrm{i}\rho t}t^{\ell+\mathrm{i}\eta}(1-t)^{\ell-\mathrm{i}\eta}\,\mathrm{d% }t,$
 33.7.2 ${H^{-}_{\ell}}\left(\eta,\rho\right)=\frac{e^{-\mathrm{i}\rho}\rho^{-\ell}}{(2% \ell+1)!C_{\ell}\left(\eta\right)}\int_{0}^{\infty}e^{-t}t^{\ell-\mathrm{i}% \eta}(t+2\mathrm{i}\rho)^{\ell+\mathrm{i}\eta}\,\mathrm{d}t,$
 33.7.3 ${H^{-}_{\ell}}\left(\eta,\rho\right)=\frac{-\mathrm{i}e^{-\pi\eta}\rho^{\ell+1% }}{(2\ell+1)!C_{\ell}\left(\eta\right)}\int_{0}^{\infty}\left(\frac{\exp\left(% -\mathrm{i}(\rho\tanh t-2\eta t)\right)}{(\cosh t)^{2\ell+2}}+\mathrm{i}(1+t^{% 2})^{\ell}\exp\left(-\rho t+2\eta\operatorname{arctan}t\right)\right)\,\mathrm% {d}t,$
 33.7.4 ${H^{+}_{\ell}}\left(\eta,\rho\right)=\frac{\mathrm{i}e^{-\pi\eta}\rho^{\ell+1}% }{(2\ell+1)!C_{\ell}\left(\eta\right)}\int_{-1}^{-\mathrm{i}\infty}e^{-\mathrm% {i}\rho t}(1-t)^{\ell-\mathrm{i}\eta}(1+t)^{\ell+\mathrm{i}\eta}\,\mathrm{d}t.$

Noninteger powers in (33.7.1)–(33.7.4) and the arctangent assume their principal values (§§4.2(i), 4.2(iv), 4.23(ii)).