Digital Library of Mathematical Functions

About the Project

33 Coulomb Functions Variables $r,\epsilon$ 33.17 Recurrence Relations and Derivatives 33.19 Power-Series Expansions in

§33.18 Limiting Forms for Large $\ell$

Notes:: Combine (33.5.8) and (33.16.1), (33.16.2). For $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ (33.19.1) can also be used.
Keywords:: Coulomb functions: variables $r,\epsilon$
Referenced by:: §33.23(iv)
Permalink:: http://dlmf.nist.gov/33.18

As $\ell\to\infty$ with $\epsilon$ and ( $\neq 0$ ) fixed,

33.18.1

$\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)\sim\frac{(2r)^{{\ell+1}}}{% (2\ell+1)!},$

$\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)\sim\frac{(2\ell)!}{\pi(2r)% ^{\ell}}.$

Symbols:: $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function, $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : irregular Coulomb function, : : factorial, $\sim$ : asymptotic equality, $\ell$ : nonnegative integer, : real variable and $\epsilon$ : real parameter
Permalink:: http://dlmf.nist.gov/33.18.E1
Encodings:: TeX, TeX, pMML, pMML, png, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 33.17 Recurrence Relations and Derivatives 33.19 Power-Series Expansions in