\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 $\rho$

ⓘ

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	$\displaystyle F_{\ell}\left(\eta,\rho\right)$	$\displaystyle=\sin\left({\theta_{\ell}}\left(\eta,\rho\right)\right)+o\left(1% \right),$
33.10.1	$\displaystyle G_{\ell}\left(\eta,\rho\right)$	$\displaystyle=\cos\left({\theta_{\ell}}\left(\eta,\rho\right)\right)+o\left(1% \right),$
ⓘ Symbols: ${\theta_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : phase of Coulomb functions, $\cos\NVar{z}$ : cosine function, $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : 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, $\sin\NVar{z}$ : sine function, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter Referenced by: item 1 Permalink: http://dlmf.nist.gov/33.10.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.10(i), §33.10 and Ch.33

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	$\displaystyle F_{\ell}\left(\eta,\rho\right)$	$\displaystyle\sim\dfrac{(2\ell+1)!C_{\ell}\left(\eta\right)}{(2\eta)^{\ell+1}}% (2\eta\rho)^{\ifrac{1}{2}}I_{2\ell+1}\left((8\eta\rho)^{\ifrac{1}{2}}\right),$
33.10.3	$\displaystyle G_{\ell}\left(\eta,\rho\right)$	$\displaystyle\sim\dfrac{2(2\eta)^{\ell}}{(2\ell+1)!C_{\ell}\left(\eta\right)}(% 2\eta\rho)^{\ifrac{1}{2}}K_{2\ell+1}\left((8\eta\rho)^{\ifrac{1}{2}}\right).$
ⓘ Symbols: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $\sim$ : asymptotic equality, $!$ : factorial (as in $n!$ ), $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : 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, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter A&S Ref: 14.6.7 Permalink: http://dlmf.nist.gov/33.10.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.10(ii), §33.10 and Ch.33

In particular, for $\ell=0$ ,

33.10.4	$\displaystyle F_{0}\left(\eta,\rho\right)$	$\displaystyle\sim e^{-\pi\eta}(\pi\rho)^{\ifrac{1}{2}}I_{1}\left((8\eta\rho)^{% \ifrac{1}{2}}\right),$
33.10.4	$\displaystyle G_{0}\left(\eta,\rho\right)$	$\displaystyle\sim 2e^{\pi\eta}\left(\ifrac{\rho}{\pi}\right)^{\ifrac{1}{2}}K_{% 1}\left((8\eta\rho)^{\ifrac{1}{2}}\right),$
ⓘ 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	$\displaystyle F_{0}'\left(\eta,\rho\right)$	$\displaystyle\sim e^{-\pi\eta}(2\pi\eta)^{\ifrac{1}{2}}I_{0}\left((8\eta\rho)^% {\ifrac{1}{2}}\right),$
33.10.5	$\displaystyle G_{0}'\left(\eta,\rho\right)$	$\displaystyle\sim-2e^{\pi\eta}\left(\ifrac{2\eta}{\pi}\right)^{\ifrac{1}{2}}K_% {0}\left((8\eta\rho)^{\ifrac{1}{2}}\right).$
ⓘ 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	$\displaystyle{\sigma_{0}}\left(\eta\right)$	$\displaystyle=\eta(\ln\eta-1)+\tfrac{1}{4}\pi+o\left(1\right),$
33.10.6	$\displaystyle C_{0}\left(\eta\right)$	$\displaystyle\sim(2\pi\eta)^{1/2}e^{-\pi\eta}.$
ⓘ 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	$\displaystyle F_{\ell}\left(\eta,\rho\right)$	$\displaystyle=\dfrac{(2\ell+1)!C_{\ell}\left(\eta\right)}{(-2\eta)^{\ell+1}}% \left((-2\eta\rho)^{\ifrac{1}{2}}\*J_{2\ell+1}\left((-8\eta\rho)^{\ifrac{1}{2}% }\right)+o\left({\left\|\eta\right\|}^{\ifrac{1}{4}}\right)\right),$
33.10.7	$\displaystyle G_{\ell}\left(\eta,\rho\right)$	$\displaystyle=-\dfrac{\pi(-2\eta)^{\ell}}{(2\ell+1)!C_{\ell}\left(\eta\right)}% \left((-2\eta\rho)^{\ifrac{1}{2}}\*Y_{2\ell+1}\left((-8\eta\rho)^{\ifrac{1}{2}% }\right)+o\left({\left\|\eta\right\|}^{\ifrac{1}{4}}\right)\right).$
ⓘ 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, $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : 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, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter A&S Ref: 14.6.7 Permalink: http://dlmf.nist.gov/33.10.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.10(iii), §33.10 and Ch.33

In particular, for $\ell=0$ ,

33.10.8	$\displaystyle F_{0}\left(\eta,\rho\right)$	$\displaystyle=(\pi\rho)^{\ifrac{1}{2}}J_{1}\left((-8\eta\rho)^{\ifrac{1}{2}}% \right)+o\left({\left\|\eta\right\|}^{-\ifrac{1}{4}}\right),$
33.10.8	$\displaystyle G_{0}\left(\eta,\rho\right)$	$\displaystyle=-(\pi\rho)^{\ifrac{1}{2}}Y_{1}\left((-8\eta\rho)^{\ifrac{1}{2}}% \right)+o\left({\left\|\eta\right\|}^{-\ifrac{1}{4}}\right).$
ⓘ 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	$\displaystyle F_{0}'\left(\eta,\rho\right)$	$\displaystyle=(-2\pi\eta)^{\ifrac{1}{2}}J_{0}\left((-8\eta\rho)^{\ifrac{1}{2}}% \right)+o\left({\left\|\eta\right\|}^{\ifrac{1}{4}}\right),$
33.10.9	$\displaystyle G_{0}'\left(\eta,\rho\right)$	$\displaystyle=-(-2\pi\eta)^{\ifrac{1}{2}}Y_{0}\left((-8\eta\rho)^{\ifrac{1}{2}% }\right)+o\left({\left\|\eta\right\|}^{\ifrac{1}{4}}\right).$
ⓘ 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.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.10(iii), §33.10 and Ch.33

Also,

33.10.10	$\displaystyle{\sigma_{0}}\left(\eta\right)$	$\displaystyle=\eta(\ln\left(-\eta\right)-1)-\tfrac{1}{4}\pi+o\left(1\right),$
33.10.10	$\displaystyle C_{0}\left(\eta\right)$	$\displaystyle\sim(-2\pi\eta)^{1/2}.$
ⓘ 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, $o\left(\NVar{x}\right)$ : order less than, $\ln\NVar{z}$ : principal branch of logarithm function and $\eta$ : real parameter Referenced by: §33.10(iii) Permalink: http://dlmf.nist.gov/33.10.E10 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.10(iii), §33.10 and Ch.33

33.9 Expansions in Series of Bessel Functions 33.11 Asymptotic Expansions for Large

\rho

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§33.10 Limiting Forms for Large ρ or Large |η|

Contents

§33.10(i) Large ρ

§33.10(ii) Large Positive η

§33.10(iii) Large Negative η

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

§33.10(i) Large $\rho$

§33.10(ii) Large Positive $\eta$

§33.10(iii) Large Negative $\eta$