# §33.13 Complex Variable and Parameters

The functions $\mathop{F_{\ell}\/}\nolimits\!\left(\eta,\rho\right)$, $\mathop{G_{\ell}\/}\nolimits\!\left(\eta,\rho\right)$, and $\mathop{{H^{\pm}_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ may be extended to noninteger values of $\ell$ by generalizing $(2\ell+1)!=\mathop{\Gamma\/}\nolimits\!\left(2\ell+2\right)$, and supplementing (33.6.5) by a formula derived from (33.2.8) with $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ expanded via (13.2.42).

These functions may also be continued analytically to complex values of $\rho$, $\eta$, and $\ell$. The quantities $\mathop{C_{\ell}\/}\nolimits\!\left(\eta\right)$, $\mathop{{\sigma_{\ell}}\/}\nolimits\!\left(\eta\right)$, and $R_{\ell}$, given by (33.2.6), (33.2.10), and (33.4.1), respectively, must be defined consistently so that

 33.13.1 $\mathop{C_{\ell}\/}\nolimits\!\left(\eta\right)=2^{\ell}e^{\mathrm{i}\mathop{{% \sigma_{\ell}}\/}\nolimits\!\left(\eta\right)-(\pi\eta/2)}\mathop{\Gamma\/}% \nolimits\!\left(\ell+1-\mathrm{i}\eta\right)/\mathop{\Gamma\/}\nolimits\!% \left(2\ell+2\right),$

and

 33.13.2 $R_{\ell}=(2\ell+1)\mathop{C_{\ell}\/}\nolimits\!\left(\eta\right)/\mathop{C_{% \ell-1}\/}\nolimits\!\left(\eta\right).$

For further information see Dzieciol et al. (1999), Thompson and Barnett (1986), and Humblet (1984).