§33.3 Graphics

Keywords:: Coulomb functions: variables $\rho,\eta$
Permalink:: http://dlmf.nist.gov/33.3

§33.3(i) Line Graphs of the Coulomb Radial Functions $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ and $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$
§33.3(ii) Surfaces of the Coulomb Radial Functions $\mathop{F_{{0}}\/}\nolimits\!\left(\eta,\rho\right)$ and $\mathop{G_{{0}}\/}\nolimits\!\left(\eta,\rho\right)$

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

Notes:: These graphs were produced at NIST.
Permalink:: http://dlmf.nist.gov/33.3.i

Figure 33.3.1: $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ , $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ with $\ell=0$ , $\eta=-2$ .

Symbols:: $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter
Permalink:: http://dlmf.nist.gov/33.3.F1
Encodings:: pdf, png

Figure 33.3.2: $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ , $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ with $\ell=0$ , $\eta=0$ .

Symbols:: $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter
Permalink:: http://dlmf.nist.gov/33.3.F2
Encodings:: pdf, png

Figure 33.3.3: $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ , $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ with $\ell=0$ , $\eta=2$ . The turning point is at $\mathop{\rho _{{\mathrm{tp}}}\/}\nolimits\!\left(2,0\right)=4$ .

Symbols:: $\mathop{\rho _{{\mathrm{tp}}}\/}\nolimits\!\left(\eta,\ell\right)$ : outer turning point for Coulomb radial functions, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter
Permalink:: http://dlmf.nist.gov/33.3.F3
Encodings:: pdf, png

Figure 33.3.4: $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ , $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ with $\ell=0$ , $\eta=10$ . The turning point is at $\mathop{\rho _{{\mathrm{tp}}}\/}\nolimits\!\left(10,0\right)=20$ .

Symbols:: $\mathop{\rho _{{\mathrm{tp}}}\/}\nolimits\!\left(\eta,\ell\right)$ : outer turning point for Coulomb radial functions, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter
Permalink:: http://dlmf.nist.gov/33.3.F4
Encodings:: pdf, png

In Figures 33.3.5 and 33.3.6

33.3.1 $\mathop{M_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)=({\mathop{F_{{\ell}}\/}\nolimits^{{2}}}\!\left(\eta,\rho\right)+{\mathop{G_{{\ell}}\/}\nolimits^{{2}}}\!\left(\eta,\rho\right))^{{1/2}}=\left|\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)\right|.$

Defines:: $\mathop{M_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : envelope of Coulomb functions
Symbols:: $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial functions, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : 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

Figure 33.3.5: $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ , $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ , and $\mathop{M_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ with $\ell=0$ , $\eta=\sqrt{15/2}$ . The turning point is at $\mathop{\rho _{{\mathrm{tp}}}\/}\nolimits\!\left(\sqrt{15/2},0\right)=\sqrt{30}=5.47\ldots$ .

Symbols:: $\mathop{\rho _{{\mathrm{tp}}}\/}\nolimits\!\left(\eta,\ell\right)$ : outer turning point for Coulomb radial functions, $\mathop{M_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : envelope of Coulomb functions, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\ell$ : 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

Figure 33.3.6: $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ , $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ , and $\mathop{M_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ with $\ell=5$ , $\eta=0$ . The turning point is at $\mathop{\rho _{{\mathrm{tp}}}\/}\nolimits\!\left(0,5\right)=\sqrt{30}$ (as in Figure 33.3.5).

Symbols:: $\mathop{\rho _{{\mathrm{tp}}}\/}\nolimits\!\left(\eta,\ell\right)$ : outer turning point for Coulomb radial functions, $\mathop{M_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : envelope of Coulomb functions, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\ell$ : 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

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

Notes:: These surfaces were produced at NIST.
Keywords:: Coulomb functions: variables $\rho,\eta$
Permalink:: http://dlmf.nist.gov/33.3.ii

Visualization Help

Figure 33.3.7: $\mathop{F_{{0}}\/}\nolimits\!\left(\eta,\rho\right)$ , $-2\leq\eta\leq 2$ , $0\leq\rho\leq 5$ .

Symbols:: $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\rho$ : nonnegative real variable and $\eta$ : real parameter
Permalink:: http://dlmf.nist.gov/33.3.F7
Encodings:: VRML, X3D, pdf, png

Visualization Help

Figure 33.3.8: $\mathop{G_{{0}}\/}\nolimits\!\left(\eta,\rho\right)$ , $-2\leq\eta\leq 2$ , $0<\rho\leq 5$ .

Symbols:: $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\rho$ : nonnegative real variable and $\eta$ : real parameter
Permalink:: http://dlmf.nist.gov/33.3.F8
Encodings:: VRML, X3D, pdf, png

§33.3 Graphics

Contents

§33.3(i) Line Graphs of the Coulomb Radial Functions and

§33.3(ii) Surfaces of the Coulomb Radial Functions and

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

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