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

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§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 $\eta$