§33.13 Complex Variable and Parameters

Keywords:: Coulomb functions: variables $\rho,\eta$
Referenced by:: §33.22(vii)
Permalink:: http://dlmf.nist.gov/33.13

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

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $\mathop{{\sigma _{{\ell}}}\/}\nolimits\!\left(\eta\right)$ : Coulomb phase shift, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\ell$ : nonnegative integer and $\eta$ : real parameter
Permalink:: http://dlmf.nist.gov/33.13.E1
Encodings:: TeX, pMML, png

and

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

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $\ell$ : nonnegative integer, $\eta$ : real parameter and $R_{\ell}$ : factor
Permalink:: http://dlmf.nist.gov/33.13.E2
Encodings:: TeX, pMML, png

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