33 Coulomb Functions Variables $\rho,\eta$ 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$ 33.12 Asymptotic Expansions for Large $\eta$

§33.11 Asymptotic Expansions for Large $\rho$

Notes:: See Fröberg (1955).
Keywords:: Coulomb functions: variables $\rho,\eta$
Referenced by:: §33.23(ii), §33.23(iii), §33.23(iii)
Permalink:: http://dlmf.nist.gov/33.11

For large $\rho$ , with $\ell$ and $\eta$ fixed,

33.11.1 $\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)=e^{{\pm i% \mathop{{\theta_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)}}\sum_{{k=0}}^{% \infty}\frac{\left(a\right)_{{k}}\left(b\right)_{{k}}}{k!(\mp 2i\rho)^{k}},$

Symbols:: $\mathop{{\theta_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ : phase of Coulomb functions, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : base of exponential function, : : factorial, $\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial functions, : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter, : coefficient and : coefficient
A&S Ref:: 14.5.9
Permalink:: http://dlmf.nist.gov/33.11.E1
Encodings:: TeX, pMML, png

where $\mathop{{\theta_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ is defined by (33.2.9), and and are defined by (33.8.3).

With arguments $(\eta,\rho)$ suppressed, an equivalent formulation is given by

33.11.2

$\mathop{F_{{\ell}}\/}\nolimits=g\mathop{\cos\/}\nolimits\mathop{{\theta_{{\ell% }}}\/}\nolimits+f\mathop{\sin\/}\nolimits\mathop{{\theta_{{\ell}}}\/}\nolimits,$

$\mathop{G_{{\ell}}\/}\nolimits=f\mathop{\cos\/}\nolimits\mathop{{\theta_{{\ell% }}}\/}\nolimits-g\mathop{\sin\/}\nolimits\mathop{{\theta_{{\ell}}}\/}\nolimits,$

Symbols:: $\mathop{{\theta_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ : phase of Coulomb functions, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\mathop{\sin\/}\nolimits z$ : sine function, $\ell$ : nonnegative integer, : real part and : imaginary part
A&S Ref:: 14.5.1 14.5.2
Permalink:: http://dlmf.nist.gov/33.11.E2
Encodings:: TeX, TeX, pMML, pMML, png, png

33.11.3

${\mathop{F_{{\ell}}\/}\nolimits^{{\prime}}}=\widehat{g}\mathop{\cos\/}% \nolimits\mathop{{\theta_{{\ell}}}\/}\nolimits+\widehat{f}\mathop{\sin\/}% \nolimits\mathop{{\theta_{{\ell}}}\/}\nolimits,$

${\mathop{G_{{\ell}}\/}\nolimits^{{\prime}}}=\widehat{f}\mathop{\cos\/}% \nolimits\mathop{{\theta_{{\ell}}}\/}\nolimits-\widehat{g}\mathop{\sin\/}% \nolimits\mathop{{\theta_{{\ell}}}\/}\nolimits,$