# §33.7 Integral Representations

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

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)).