Content-Length: 122230 | pFad | https://dlmf.nist.gov/./././bib/../././././././././././././bib/.././././../././././././33.3

DLMF: §33.3 Graphics ‣ Variables 𝜌,𝜂 ‣ Chapter 33 Coulomb Functions

About the Project

33 Coulomb Functions Variables

\rho,\eta

33.2 Definitions and Basic Properties 33.4 Recurrence Relations and Derivatives

§33.3 Graphics

ⓘ

Keywords:: Coulomb functions: variables $\rho,\eta$ , graphics
Permalink:: http://dlmf.nist.gov/33.3
See also:: Annotations for Ch.33

Contents

§33.3(i) Line Graphs of the Coulomb Radial Functions $F_{\ell}\left(\eta,\rho\right)$ and $G_{\ell}\left(\eta,\rho\right)$

ⓘ

Notes:: These graphs were produced at NIST.
Permalink:: http://dlmf.nist.gov/33.3.i
See also:: Annotations for §33.3 and Ch.33

See accompanying text — Figure 33.3.1: $F_{\ell}\left(\eta,\rho\right)$ , $G_{\ell}\left(\eta,\rho\right)$ with $\ell=0$ , $\eta=-2$ . Magnify

See accompanying text — Figure 33.3.1: $F_{\ell}\left(\eta,\rho\right)$ , $G_{\ell}\left(\eta,\rho\right)$ with $\ell=0$ , $\eta=-2$ . Magnify

See accompanying text — Figure 33.3.3: $F_{\ell}\left(\eta,\rho\right)$ , $G_{\ell}\left(\eta,\rho\right)$ with $\ell=0$ , $\eta=2$ . The turning point is at $\rho_{\operatorname{tp}}\left(2,0\right)=4$ . Magnify

See accompanying text — Figure 33.3.3: $F_{\ell}\left(\eta,\rho\right)$ , $G_{\ell}\left(\eta,\rho\right)$ with $\ell=0$ , $\eta=2$ . The turning point is at $\rho_{\operatorname{tp}}\left(2,0\right)=4$ . Magnify

In Figures 33.3.5 and 33.3.6

33.3.1		$M_{\ell}\left(\eta,\rho\right)=({F_{\ell}}^{2}\left(\eta,\rho\right)+{G_{\ell}% }^{2}\left(\eta,\rho\right))^{1/2}=\left\|{H^{\pm}_{\ell}}\left(\eta,\rho\right% )\right\|.$
ⓘ Defines: $M_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : envelope of Coulomb functions Symbols: $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, ${H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial functions, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter Permalink: http://dlmf.nist.gov/33.3.E1 Encodings: TeX, pMML, png See also: Annotations for §33.3(i), §33.3 and Ch.33

See accompanying text — Figure 33.3.5: $F_{\ell}\left(\eta,\rho\right)$ , $G_{\ell}\left(\eta,\rho\right)$ , and $M_{\ell}\left(\eta,\rho\right)$ with $\ell=0$ , $\eta=\sqrt{15/2}$ . The turning point is at $\rho_{\operatorname{tp}}\left(\sqrt{15/2},0\right)=\sqrt{30}=5.47\dotsc$ . Magnify

See accompanying text — Figure 33.3.5: $F_{\ell}\left(\eta,\rho\right)$ , $G_{\ell}\left(\eta,\rho\right)$ , and $M_{\ell}\left(\eta,\rho\right)$ with $\ell=0$ , $\eta=\sqrt{15/2}$ . The turning point is at $\rho_{\operatorname{tp}}\left(\sqrt{15/2},0\right)=\sqrt{30}=5.47\dotsc$ . Magnify

§33.3(ii) Surfaces of the Coulomb Radial Functions $F_{0}\left(\eta,\rho\right)$ and $G_{0}\left(\eta,\rho\right)$

ⓘ

Keywords:: Coulomb functions: variables $\rho,\eta$ , graphics
Notes:: These surfaces were produced at NIST.
Permalink:: http://dlmf.nist.gov/33.3.ii
See also:: Annotations for §33.3 and Ch.33

See accompanying text — Figure 33.3.7: $F_{0}\left(\eta,\rho\right)$ , $-2\leq\eta\leq 2$ , $0\leq\rho\leq 5$ . Magnify 3D Help

See accompanying text — Figure 33.3.7: $F_{0}\left(\eta,\rho\right)$ , $-2\leq\eta\leq 2$ , $0\leq\rho\leq 5$ . Magnify 3D Help

33.2 Definitions and Basic Properties 33.4 Recurrence Relations and Derivatives

© 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