Content-Length: 35443 | pFad | https://dlmf.nist.gov/./../././bib/.././bib/.././bib/../././33.18

DLMF: §33.18 Limiting Forms for Large ℓ ‣ Variables 𝑟,ϵ ‣ Chapter 33 Coulomb Functions

About the Project

33 Coulomb Functions Variables

r,\epsilon

33.17 Recurrence Relations and Derivatives 33.19 Power-Series Expansions in

r

§33.18 Limiting Forms for Large $\ell$

ⓘ

Keywords:: Coulomb functions: variables $r,\epsilon$ , limiting forms for large $\ell$
Notes:: Combine (33.5.8) and (33.16.1), (33.16.2). For $f\left(\epsilon,\ell;r\right)$ (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 $\ell\to\infty$ with $\epsilon$ and $r$ ( $\neq 0$ ) fixed,

33.18.1		$\displaystyle f\left(\epsilon,\ell;r\right)$	$\displaystyle\sim\frac{(2r)^{\ell+1}}{(2\ell+1)!},$
33.18.1		$\displaystyle h\left(\epsilon,\ell;r\right)$	$\displaystyle\sim\frac{(2\ell)!}{\pi(2r)^{\ell}}.$
ⓘ Symbols: $f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\ell$ : nonnegative integer, $r$ : real variable and $\epsilon$ : 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 Derivatives 33.19 Power-Series Expansions in

r

© 2010–2024 NIST / Disclaimer / Feedback; Version 1.2.3; Release date 2024-12-15.

NIST

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov