33 Coulomb Functions Variables $\rho,\eta$ 33.12 Asymptotic Expansions for Large $\eta$ 33.14 Definitions and Basic Properties

§33.13 Complex Variable and Parameters

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Keywords:: Coulomb functions: variables $\rho,\eta$ , complex variable and parameters
Referenced by:: §33.22(vii)
Permalink:: http://dlmf.nist.gov/33.13
See also:: Annotations for Ch.33

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

These functions may also be continued analytically to complex values of $\rho$ , $\eta$ , and $\ell$ . The quantities $C_{\ell}\left(\eta\right)$ , ${\sigma_{\ell}}\left(\eta\right)$ , 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 $C_{\ell}\left(\eta\right)=2^{\ell}e^{\mathrm{i}{\sigma_{\ell}}\left(\eta\right% )-(\pi\eta/2)}\Gamma\left(\ell+1-\mathrm{i}\eta\right)/\Gamma\left(2\ell+2% \right),$

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