33 Coulomb FunctionsVariables $r,ϵ$33.17 Recurrence Relations and Derivatives33.19 Power-Series Expansions in $r$

§33.18 Limiting Forms for Large $\mathrm{\ell }$

ⓘ

Keywords:

Coulomb functions: variables $r,ϵ$, limiting forms for large $\mathrm{\ell }$

Notes:

Combine (33.5.8) and (33.16.1), (33.16.2). For $f(ϵ,\mathrm{\ell };r)$ (33.19.1) can also be used.

Referenced by:

§33.23(iv)

Permalink:

http://dlmf.nist.gov/33.18

See also:

Annotations for Ch.33

As $\mathrm{\ell }\to \mathrm{\infty }$ with $ϵ$ and $r$ ($\ne 0$) fixed,

33.18.1		$f(ϵ,\mathrm{\ell };r)$	$\sim {\displaystyle \frac{{(2r)}^{\mathrm{\ell }+1}}{(2\mathrm{\ell }+1)!}},$
		$h(ϵ,\mathrm{\ell };r)$	$\sim {\displaystyle \frac{(2\mathrm{\ell })!}{\pi {(2r)}^{\mathrm{\ell }}}}.$
ⓘ Symbols: $f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter Permalink: http://dlmf.nist.gov/33.18.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.18 and Ch.33

33.17 Recurrence Relations and Derivatives33.19 Power-Series Expansions in $r$

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