# §33.7 Integral Representations

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

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)).