Coulomb radial functions

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11: 33.12 Asymptotic Expansions for Large $\eta$

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33.12.2 $$\genfrac{}{}{0pt}{}{F_0(\eta ,\rho )}{G_0(\eta ,\rho )}\sim {\pi }^{1/2}{(2\eta )}^{1/6}\left\{\genfrac{}{}{0pt}{}{\mathrm{Ai}\left(x\right)}{\mathrm{Bi}\left(x\right)}\left(1+\frac{B_1}{\mu }+\frac{B_2}{{\mu }^2}+\mathrm{\cdots }\right)+\genfrac{}{}{0pt}{}{{\mathrm{Ai}}^{\prime }\left(x\right)}{{\mathrm{Bi}}^{\prime }\left(x\right)}\left(\frac{A_1}{\mu }+\frac{A_2}{{\mu }^2}+\mathrm{\cdots }\right)\right\},$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\rho$: nonnegative real variable, $\eta$: real parameter, $x$: variable, $\mu$: variable, $A_j$: coefficients and $B_j$: coefficients

A&S Ref:

14.4.9

Permalink:

http://dlmf.nist.gov/33.12.E2

Encodings:

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See also:

Annotations for §33.12(i), §33.12 and Ch.33

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33.12.3 $$\genfrac{}{}{0pt}{}{F_0^{\prime }(\eta ,\rho )}{G_0^{\prime }(\eta ,\rho )}\sim -{\pi }^{1/2}{(2\eta )}^{-1/6}\left\{\genfrac{}{}{0pt}{}{\mathrm{Ai}\left(x\right)}{\mathrm{Bi}\left(x\right)}\left(\frac{B_1^{\prime }+x{A}_1}{\mu }+\frac{B_2^{\prime }+x{A}_2}{{\mu }^2}+\mathrm{\cdots }\right)+\genfrac{}{}{0pt}{}{{\mathrm{Ai}}^{\prime }\left(x\right)}{{\mathrm{Bi}}^{\prime }\left(x\right)}\left(\frac{B_1+A_1^{\prime }}{\mu }+\frac{B_2+A_2^{\prime }}{{\mu }^2}+\mathrm{\cdots }\right)\right\},$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\rho$: nonnegative real variable, $\eta$: real parameter, $x$: variable, $\mu$: variable, $A_j$: coefficients and $B_j$: coefficients

A&S Ref:

14.4.10

Permalink:

http://dlmf.nist.gov/33.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.12(i), §33.12 and Ch.33

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33.12.6 $$\genfrac{}{}{0pt}{}{F_0(\eta ,2\eta )}{3^{-1/2}{G}_0(\eta ,2\eta )}\sim \frac{\mathrm{\Gamma }\left(\frac{1}{3}\right){\omega }^{1/2}}{2\sqrt{\pi }}\left(1\mp \frac{2}{35}\frac{\mathrm{\Gamma }\left(\frac{2}{3}\right)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\frac{1}{{\omega }^4}-\frac{8}{2025}\frac{1}{{\omega }^6}\mp \frac{5792}{\mathrm{46\hspace{0.17em}06875}}\frac{\mathrm{\Gamma }\left(\frac{2}{3}\right)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\frac{1}{{\omega }^{10}}-\mathrm{\cdots }\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\eta$: real parameter and $\omega$: factor

Permalink:

http://dlmf.nist.gov/33.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.12(i), §33.12 and Ch.33

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33.12.7 $$\genfrac{}{}{0pt}{}{F_0^{\prime }(\eta ,2\eta )}{3^{-1/2}{G}_0^{\prime }(\eta ,2\eta )}\sim \frac{\mathrm{\Gamma }\left(\frac{2}{3}\right)}{2\sqrt{\pi }{\omega }^{1/2}}\left(\pm 1+\frac{1}{15}\frac{\mathrm{\Gamma }\left(\frac{1}{3}\right)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\frac{1}{{\omega }^2}\pm \frac{2}{14175}\frac{1}{{\omega }^6}+\frac{1436}{\mathrm{23\hspace{0.17em}38875}}\frac{\mathrm{\Gamma }\left(\frac{1}{3}\right)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\frac{1}{{\omega }^8}\pm \mathrm{\cdots }\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\eta$: real parameter and $\omega$: factor

Referenced by:

§33.12(i)

Permalink:

http://dlmf.nist.gov/33.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §33.12(i), §33.12 and Ch.33

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12: 33.9 Expansions in Series of Bessel Functions

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33.9.1 $$F_{\mathrm{\ell }}(\eta ,\rho )=\rho \sum _{k=0}^{\mathrm{\infty }}{a}_k{𝗃}_{\mathrm{\ell }+k}\left(\rho \right),$$

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Symbols:

$F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $a_k$: coefficients

A&S Ref:

14.4.5

Referenced by:

§33.9(i), §33.9(i)

Permalink:

http://dlmf.nist.gov/33.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.9(i), §33.9 and Ch.33

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33.9.3 $$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\frac{(2\mathrm{\ell }+1)!}{{(2\eta )}^{2\mathrm{\ell }+1}}{\rho }^{-\mathrm{\ell }}\sum _{k=2\mathrm{\ell }+1}^{\mathrm{\infty }}{b}_k{t}^{k/2}{I}_k\left(2\sqrt{t}\right),$$ $\eta >0$,

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $t$: parameter and $b_k$: coefficient

A&S Ref:

14.4.1

Referenced by:

§33.9(ii), §33.9(ii)

Permalink:

http://dlmf.nist.gov/33.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.9(ii), §33.9 and Ch.33

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33.9.4 $$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\frac{(2\mathrm{\ell }+1)!}{{(2\left|\eta \right|)}^{2\mathrm{\ell }+1}}{\rho }^{-\mathrm{\ell }}\sum _{k=2\mathrm{\ell }+1}^{\mathrm{\infty }}{b}_k{t}^{k/2}{J}_k\left(2\sqrt{t}\right),$$ .

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $!$: factorial (as in $n!$), $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $t$: parameter and $b_k$: coefficient

Referenced by:

§33.9(ii), §33.9(ii)

Permalink:

http://dlmf.nist.gov/33.9.E4

Encodings:

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See also:

Annotations for §33.9(ii), §33.9 and Ch.33

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33.9.6 $$G_{\mathrm{\ell }}(\eta ,\rho )\sim \frac{{\rho }^{-\mathrm{\ell }}}{(\mathrm{\ell }+\frac{1}{2}){\lambda }_{\mathrm{\ell }}(\eta )C_{\mathrm{\ell }}\left(\eta \right)}\sum _{k=2\mathrm{\ell }+1}^{\mathrm{\infty }}{(-1)}^k{b}_k{t}^{k/2}{K}_k\left(2\sqrt{t}\right),$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\sim$: asymptotic equality, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $t$: parameter, $b_k$: coefficient and ${\lambda }_{\mathrm{\ell }}(\eta )$: coefficient

A&S Ref:

14.4.2

Referenced by:

§33.9(ii)

Permalink:

http://dlmf.nist.gov/33.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.9(ii), §33.9 and Ch.33

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13: 33.16 Connection Formulas

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33.16.1 $$F_{\mathrm{\ell }}(\eta ,\rho )=\frac{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}{{(-2\eta )}^{\mathrm{\ell }+1}}f(1/{\eta }^2,\mathrm{\ell };-\eta \rho ),$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $!$: factorial (as in $n!$), $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Referenced by:

§33.18

Permalink:

http://dlmf.nist.gov/33.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.16(i), §33.16 and Ch.33

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33.16.2 $$G_{\mathrm{\ell }}(\eta ,\rho )=\frac{\pi {(-2\eta )}^{\mathrm{\ell }}}{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}h(1/{\eta }^2,\mathrm{\ell };-\eta \rho ),$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Referenced by:

§33.18

Permalink:

http://dlmf.nist.gov/33.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.16(i), §33.16 and Ch.33

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33.16.4 $$f(ϵ,\mathrm{\ell };r)={\left(\frac{2}{\pi \tau }\frac{1-{\mathrm{e}}^{-2\pi /\tau }}{A(ϵ,\mathrm{\ell })}\right)}^{1/2}{F}_{\mathrm{\ell }}(-1/\tau ,\tau r),$$

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Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function and $\tau$: parameter

Referenced by:

§33.16(iv), item 1

Permalink:

http://dlmf.nist.gov/33.16.E4

Encodings:

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See also:

Annotations for §33.16(ii), §33.16 and Ch.33

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33.16.5 $$h(ϵ,\mathrm{\ell };r)={\left(\frac{2}{\pi \tau }\frac{A(ϵ,\mathrm{\ell })}{1-{\mathrm{e}}^{-2\pi /\tau }}\right)}^{1/2}{G}_{\mathrm{\ell }}(-1/\tau ,\tau r).$$

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Symbols:

$h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function and $\tau$: parameter

Permalink:

http://dlmf.nist.gov/33.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §33.16(ii), §33.16 and Ch.33

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33.16.6 $$f(ϵ,\mathrm{\ell };r)={(-1)}^{\mathrm{\ell }+1}{\left(\frac{2}{\pi \tau }\frac{{\mathrm{e}}^{2\pi /\tau }-1}{A(ϵ,\mathrm{\ell })}\right)}^{1/2}{F}_{\mathrm{\ell }}(1/\tau ,-\tau r),$$

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Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function and $\tau$: parameter

Permalink:

http://dlmf.nist.gov/33.16.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.16(ii), §33.16 and Ch.33

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14: Bibliography S

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The accuracy of iterated JWBK approximations for Coulomb radial functions. Comput. Phys. Comm. 32 (2), pp. 115–119.

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External Links:

Document

Referenced by:

§33.23(vii).

Encodings:

BibTeX

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FGH, a code for the calculation of Coulomb radial wave functions from series expansions. Comput. Phys. Comm. 146 (2), pp. 250–253.

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Notes:

Single-precision Fortran code.

External Links:

Document, Code, ZentralBlatt

Referenced by:

10th item.

Encodings:

BibTeX

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15: 18.39 Applications in the Physical Sciences

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a) Spherical Radial Coulomb Wave Functions Expressed in terms of Laguerre OP’s

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c) Spherical Radial Coulomb Wave Functions

… ►The radial Coulomb wave functions $R_{n,l}(r)$, solutions of … ►

d) Radial Coulomb Wave Functions Expressed in Terms of the Associated Coulomb–Laguerre OP’s

… ►The bound state $L^2$ eigenfunctions of the radial Coulomb Schrödinger operator are discussed in §§18.39(i) and 18.39(ii), and the $\delta$-function normalized (non-$L^2$) in Chapter 33, where the solutions appear as Whittaker functions. …

16: 33.23 Methods of Computation

§33.23 Methods of Computation

… ►The methods used for computing the Coulomb functions described below are similar to those in §13.29. … ►Inside the turning points, that is, when , there can be a loss of precision by a factor of approximately ${|G_{\mathrm{\ell }}|}^2$. … ►WKBJ approximations (§2.7(iii)) for $\rho >{\rho }_{\mathrm{tp}}(\eta ,\mathrm{\ell })$ are presented in Hull and Breit (1959) and Seaton and Peach (1962: in Eq. … ►Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for $F_0$ and $G_0$ in the region inside the turning point: .

17: Bibliography B

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Methods of calculation of radial wave functions and new tables of Coulomb functions. Physical Rev. (2) 80, pp. 553–560.

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External Links:

Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§33.7.

Encodings:

BibTeX

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18: Bibliography Y

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Tables of Airy Functions and Their Derivatives. Izdat. Nauka, Moscow (Russian).

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Referenced by:

4th item.

Encodings:

BibTeX

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$J$-matrix method: Extensions to arbitrary angular momentum and to Coulomb scattering. J. Math. Phys. 16, pp. 410–420.

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Referenced by:

§18.39(iv).

Encodings:

BibTeX

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$L$-squared discretizations of the continuum: Radial kinetic energy and the Coulomb Hamiltonian. Phys. Rev. A 11 (4), pp. 1144–1156.

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Referenced by:

§18.39(iv), §18.39(iv), §18.39(iv), §18.40(ii).

Encodings:

BibTeX

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Coulomb wave functions in repulsive fields. Phys. Rev. 49 (2), pp. 174–189.

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External Links:

Document, ZentralBlatt

Referenced by:

§33.10(i), §33.2(ii), §33.2(iii), §33.2(iv), §33.5(i), §33.5(iv), §33.9(ii).

Encodings:

BibTeX

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Bessel Functions. Part IV: Kelvin Functions. Royal Society Mathematical Tables, Volume 10, Cambridge University Press, Cambridge-New York.

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External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

§10.65(ii), §10.68(iii), §10.68(iv), §10.71(i), 1st item.

Encodings:

BibTeX

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19: 33.22 Particle Scattering and Atomic and Molecular Spectra

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§33.22(i) Schrödinger Equation

… ►At positive energies $E>0$, $\rho \ge 0$, and: … ►Positive-energy functions correspond to processes such as Rutherford scattering and Coulomb excitation of nuclei (Alder et al. (1956)), and atomic photo-ionization and electron-ion collisions (Bethe and Salpeter (1977)). … ►

§33.22(iv) Klein–Gordon and Dirac Equations

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20: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Functions $f,g\in L^2(X,d\alpha )$ for which $⟨f-g,f-g⟩=0$ are identified with each other. … ►Eigenfunctions corresponding to the continuous spectrum are non-$L^2$ functions. … ►

Example 2: Radial 3D Schrödinger operators, including the Coulomb potential

… ►In unusual cases $N=\mathrm{\infty }$, even for all $\mathrm{\ell }$, such as in the case of the Schrödinger–Coulomb problem ($V=-r^{-1}$) discussed in §18.39 and §33.14, where the point spectrum actually accumulates at the onset of the continuum at $\lambda =0$, implying an essential singularity, as well as a branch point, in matrix elements of the resolvent, (1.18.66). … …