33 Coulomb Functions Variables $\rho,\eta$ 33.9 Expansions in Series of Bessel Functions 33.11 Asymptotic Expansions for Large $\rho$

§33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$

ⓘ

Permalink:: http://dlmf.nist.gov/33.10
See also:: Annotations for Ch.33

§33.10(i) Large $\rho$
§33.10(ii) Large Positive $\eta$
§33.10(iii) Large Negative $\eta$

§33.10(i) Large $\rho$

ⓘ

Keywords:: Coulomb functions: variables $\rho,\eta$ , large $\rho$ , limiting forms
Notes:: See Yost et al. (1936).
Permalink:: http://dlmf.nist.gov/33.10.i
See also:: Annotations for §33.10 and Ch.33

As $\rho\to\infty$ with $\eta$ fixed,

33.10.1

$F_{\ell}\left(\eta,\rho\right)=\sin\left({\theta_{\ell}}\left(\eta,\rho\right)% \right)+o\left(1\right),$

$G_{\ell}\left(\eta,\rho\right)=\cos\left({\theta_{\ell}}\left(\eta,\rho\right)% \right)+o\left(1\right),$

33.10.2 ${H^{\pm}_{\ell}}\left(\eta,\rho\right)\sim\exp\left(\pm\mathrm{i}{\theta_{\ell% }}\left(\eta,\rho\right)\right),$

ⓘ

Symbols:: ${\theta_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : phase of Coulomb functions, $\sim$ : asymptotic equality, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, ${H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial functions, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter
A&S Ref:: 14.6.5
Permalink:: http://dlmf.nist.gov/33.10.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §33.10(i), §33.10 and Ch.33

where ${\theta_{\ell}}\left(\eta,\rho\right)$ is defined by (33.2.9).

§33.10(ii) Large Positive $\eta$

ⓘ

Keywords:: Coulomb functions: variables $\rho,\eta$ , large $\left|\eta\right|$ , limiting forms
Notes:: See Fröberg (1955), Humblet (1984), Humblet (1985, Eqs. 2.10a,b). For (33.10.6) use (33.2.5), (33.2.10), and §5.11(i).
Referenced by:: Erratum (V1.0.19) for Subsections 33.10(ii), 33.10(iii)
Permalink:: http://dlmf.nist.gov/33.10.ii
Errata (effective with 1.0.19):: In the first sentence, originally it was stated that $\rho$ was fixed. This has been corrected to state that $\eta\rho$ is fixed.
Reported 2018-05-15 by Ian Thompson
See also:: Annotations for §33.10 and Ch.33

As $\eta\to\infty$ with $\eta\rho$ fixed,

33.10.3

$F_{\ell}\left(\eta,\rho\right)\sim\dfrac{(2\ell+1)!C_{\ell}\left(\eta\right)}{% (2\eta)^{\ell+1}}(2\eta\rho)^{\ifrac{1}{2}}I_{2\ell+1}\left((8\eta\rho)^{% \ifrac{1}{2}}\right),$

$G_{\ell}\left(\eta,\rho\right)\sim\dfrac{2(2\eta)^{\ell}}{(2\ell+1)!C_{\ell}% \left(\eta\right)}(2\eta\rho)^{\ifrac{1}{2}}K_{2\ell+1}\left((8\eta\rho)^{% \ifrac{1}{2}}\right).$

In particular, for $\ell=0$ ,

33.10.4

$F_{0}\left(\eta,\rho\right)\sim e^{-\pi\eta}(\pi\rho)^{\ifrac{1}{2}}I_{1}\left% ((8\eta\rho)^{\ifrac{1}{2}}\right),$

$G_{0}\left(\eta,\rho\right)\sim 2e^{\pi\eta}\left(\ifrac{\rho}{\pi}\right)^{% \ifrac{1}{2}}K_{1}\left((8\eta\rho)^{\ifrac{1}{2}}\right),$

ⓘ

Symbols:: $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\rho$ : nonnegative real variable and $\eta$ : real parameter
Permalink:: http://dlmf.nist.gov/33.10.E4
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §33.10(ii), §33.10 and Ch.33

33.10.5

$F_{0}'\left(\eta,\rho\right)\sim e^{-\pi\eta}(2\pi\eta)^{\ifrac{1}{2}}I_{0}% \left((8\eta\rho)^{\ifrac{1}{2}}\right),$

$G_{0}'\left(\eta,\rho\right)\sim-2e^{\pi\eta}\left(\ifrac{2\eta}{\pi}\right)^{% \ifrac{1}{2}}K_{0}\left((8\eta\rho)^{\ifrac{1}{2}}\right).$

ⓘ

Symbols:: $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\rho$ : nonnegative real variable and $\eta$ : real parameter
Permalink:: http://dlmf.nist.gov/33.10.E5
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §33.10(ii), §33.10 and Ch.33

Also,

33.10.6

${\sigma_{0}}\left(\eta\right)=\eta(\ln\eta-1)+\tfrac{1}{4}\pi+o\left(1\right),$

$C_{0}\left(\eta\right)\sim(2\pi\eta)^{1/2}e^{-\pi\eta}.$

ⓘ

Symbols:: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, ${\sigma_{\NVar{\ell}}}\left(\NVar{\eta}\right)$ : Coulomb phase shift, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $o\left(\NVar{x}\right)$ : order less than, $\ln\NVar{z}$ : principal branch of logarithm function and $\eta$ : real parameter
A&S Ref:: 14.6.16
Referenced by:: §33.10(ii)
Permalink:: http://dlmf.nist.gov/33.10.E6
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §33.10(ii), §33.10 and Ch.33

§33.10(iii) Large Negative $\eta$

ⓘ

Keywords:: Coulomb functions: variables $\rho,\eta$ , large $\left|\eta\right|$ , limiting forms
Notes:: See Humblet (1985, Eqs. 4.7a,b). For (33.10.10) use (33.2.5), (33.2.10), and §5.11(i).
Referenced by:: Erratum (V1.0.19) for Subsections 33.10(ii), 33.10(iii)
Permalink:: http://dlmf.nist.gov/33.10.iii
Errata (effective with 1.0.19):: In the first sentence, originally it was stated that $\rho$ was fixed. This has been corrected to state that $\eta\rho$ is fixed.
Reported 2018-05-15 by Ian Thompson
See also:: Annotations for §33.10 and Ch.33

As $\eta\to-\infty$ with $\eta\rho$ fixed,

33.10.7

$F_{\ell}\left(\eta,\rho\right)=\dfrac{(2\ell+1)!C_{\ell}\left(\eta\right)}{(-2% \eta)^{\ell+1}}\left((-2\eta\rho)^{\ifrac{1}{2}}\*J_{2\ell+1}\left((-8\eta\rho% )^{\ifrac{1}{2}}\right)+o\left({\left|\eta\right|}^{\ifrac{1}{4}}\right)\right),$

$G_{\ell}\left(\eta,\rho\right)=-\dfrac{\pi(-2\eta)^{\ell}}{(2\ell+1)!C_{\ell}% \left(\eta\right)}\left((-2\eta\rho)^{\ifrac{1}{2}}\*Y_{2\ell+1}\left((-8\eta% \rho)^{\ifrac{1}{2}}\right)+o\left({\left|\eta\right|}^{\ifrac{1}{4}}\right)% \right).$

In particular, for $\ell=0$ ,

33.10.8

$F_{0}\left(\eta,\rho\right)=(\pi\rho)^{\ifrac{1}{2}}J_{1}\left((-8\eta\rho)^{% \ifrac{1}{2}}\right)+o\left({\left|\eta\right|}^{-\ifrac{1}{4}}\right),$

$G_{0}\left(\eta,\rho\right)=-(\pi\rho)^{\ifrac{1}{2}}Y_{1}\left((-8\eta\rho)^{% \ifrac{1}{2}}\right)+o\left({\left|\eta\right|}^{-\ifrac{1}{4}}\right).$

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, $o\left(\NVar{x}\right)$ : order less than, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\rho$ : nonnegative real variable and $\eta$ : real parameter
Permalink:: http://dlmf.nist.gov/33.10.E8
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §33.10(iii), §33.10 and Ch.33

33.10.9

$F_{0}'\left(\eta,\rho\right)=(-2\pi\eta)^{\ifrac{1}{2}}J_{0}\left((-8\eta\rho)% ^{\ifrac{1}{2}}\right)+o\left({\left|\eta\right|}^{\ifrac{1}{4}}\right),$

$G_{0}'\left(\eta,\rho\right)=-(-2\pi\eta)^{\ifrac{1}{2}}Y_{0}\left((-8\eta\rho% )^{\ifrac{1}{2}}\right)+o\left({\left|\eta\right|}^{\ifrac{1}{4}}\right).$