33 Coulomb FunctionsVariables $\rho ,\eta$33.2 Definitions and Basic Properties33.4 Recurrence Relations and Derivatives

§33.3 Graphics

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Keywords:

Coulomb functions: variables $\rho ,\eta$, graphics

Permalink:

http://dlmf.nist.gov/33.3

See also:

Annotations for Ch.33

Contents

1. §33.3(i) Line Graphs of the Coulomb Radial Functions $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$
2. §33.3(ii) Surfaces of the Coulomb Radial Functions $F_0(\eta ,\rho )$ and $G_0(\eta ,\rho )$

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

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Notes:

These graphs were produced at NIST.

Permalink:

http://dlmf.nist.gov/33.3.i

See also:

Annotations for §33.3 and Ch.33

ⓘ

Symbols:

$G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Permalink:

http://dlmf.nist.gov/33.3.F1

Encodings:

pdf, png

See also:

Annotations for §33.3(i), §33.3 and Ch.33

ⓘ

Symbols:

$G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Permalink:

http://dlmf.nist.gov/33.3.F2

Encodings:

pdf, png

See also:

Annotations for §33.3(i), §33.3 and Ch.33

ⓘ

Symbols:

${\rho }_{\mathrm{tp}}(\eta ,\mathrm{\ell })$: outer turning point for Coulomb radial functions, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Permalink:

http://dlmf.nist.gov/33.3.F3

Encodings:

pdf, png

See also:

Annotations for §33.3(i), §33.3 and Ch.33

ⓘ

Symbols:

${\rho }_{\mathrm{tp}}(\eta ,\mathrm{\ell })$: outer turning point for Coulomb radial functions, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Permalink:

http://dlmf.nist.gov/33.3.F4

Encodings:

pdf, png

See also:

Annotations for §33.3(i), §33.3 and Ch.33

In Figures 33.3.5 and 33.3.6

33.3.1		$$M_{\mathrm{\ell }}(\eta ,\rho )={(F_{\mathrm{\ell }}^2(\eta ,\rho )+G_{\mathrm{\ell }}^2(\eta ,\rho ))}^{1/2}=\left|H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )\right|.$$
ⓘ Defines: $M_{\mathrm{\ell }}(\eta ,\rho )$: envelope of Coulomb functions Symbols: $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: 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

ⓘ

Symbols:

${\rho }_{\mathrm{tp}}(\eta ,\mathrm{\ell })$: outer turning point for Coulomb radial functions, $M_{\mathrm{\ell }}(\eta ,\rho )$: envelope of Coulomb functions, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Referenced by:

Figure 33.3.6, Figure 33.3.6, §33.3(i)

Permalink:

http://dlmf.nist.gov/33.3.F5

Encodings:

pdf, png

See also:

Annotations for §33.3(i), §33.3 and Ch.33

ⓘ

Symbols:

${\rho }_{\mathrm{tp}}(\eta ,\mathrm{\ell })$: outer turning point for Coulomb radial functions, $M_{\mathrm{\ell }}(\eta ,\rho )$: envelope of Coulomb functions, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Referenced by:

§33.3(i)

Permalink:

http://dlmf.nist.gov/33.3.F6

Encodings:

pdf, png

See also:

Annotations for §33.3(i), §33.3 and Ch.33

§33.3(ii) Surfaces of the Coulomb Radial Functions $F_0(\eta ,\rho )$ and $G_0(\eta ,\rho )$

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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

ⓘ

Symbols:

$F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\rho$: nonnegative real variable and $\eta$: real parameter

Permalink:

http://dlmf.nist.gov/33.3.F7

Encodings:

Vizualization, pdf, png

See also:

Annotations for §33.3(ii), §33.3 and Ch.33

ⓘ

Symbols:

$G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $\rho$: nonnegative real variable and $\eta$: real parameter

Permalink:

http://dlmf.nist.gov/33.3.F8

Encodings:

Vizualization, pdf, png

See also:

Annotations for §33.3(ii), §33.3 and Ch.33

33.2 Definitions and Basic Properties33.4 Recurrence Relations and Derivatives

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