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

Permalink:: http://dlmf.nist.gov/33.10

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

§33.10(i) Large $\rho$

Notes:: See Yost et al. (1936).
Keywords:: Coulomb functions: variables $\rho,\eta$
Permalink:: http://dlmf.nist.gov/33.10.i

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

33.10.1

$\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)=\mathop{\sin\/}\nolimits\!\left(\mathop{{\theta _{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)\right)+\mathop{o\/}\nolimits\!\left(1\right),$

$\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)=\mathop{\cos\/}\nolimits\!\left(\mathop{{\theta _{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)\right)+\mathop{o\/}\nolimits\!\left(1\right),$

Symbols:: $\mathop{{\theta _{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ : phase of Coulomb functions, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\mathop{\sin\/}\nolimits z$ : sine function, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter
Referenced by:: §33.21(i)
Permalink:: http://dlmf.nist.gov/33.10.E1
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33.10.2 $\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)\sim\mathop{\exp\/}\nolimits\!\left(\pm i\mathop{{\theta _{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)\right),$

Symbols:: $\mathop{{\theta _{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ : phase of Coulomb functions, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial functions, $\sim$ : asymptotic equality, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter
A&S Ref:: 14.6.5
Permalink:: http://dlmf.nist.gov/33.10.E2
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where $\mathop{{\theta _{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ is defined by (33.2.9).

§33.10(ii) Large Positive $\eta$

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).
Keywords:: Coulomb functions: variables $\rho,\eta$
Permalink:: http://dlmf.nist.gov/33.10.ii

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

33.10.3

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

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

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $!$ : $n!$ : factorial, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\sim$ : asymptotic equality, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter
A&S Ref:: 14.6.7
Permalink:: http://dlmf.nist.gov/33.10.E3
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In particular, for $\ell=0$ ,

33.10.4

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

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

Symbols:: $e$ : base of exponential function, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\sim$ : asymptotic equality, $\rho$ : nonnegative real variable and $\eta$ : real parameter
Permalink:: http://dlmf.nist.gov/33.10.E4
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33.10.5

${\mathop{F_{{0}}\/}\nolimits^{{\prime}}}\!\left(\eta,\rho\right)\sim e^{{-\pi\eta}}(2\pi\eta)^{{\ifrac{1}{2}}}\mathop{I_{{0}}\/}\nolimits\!\left((8\eta\rho)^{{\ifrac{1}{2}}}\right),$

${\mathop{G_{{0}}\/}\nolimits^{{\prime}}}\!\left(\eta,\rho\right)\sim-2e^{{\pi\eta}}\left(\ifrac{2\eta}{\pi}\right)^{{\ifrac{1}{2}}}\mathop{K_{{0}}\/}\nolimits\!\left((8\eta\rho)^{{\ifrac{1}{2}}}\right).$

Symbols:: $e$ : base of exponential function, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\sim$ : asymptotic equality, $\rho$ : nonnegative real variable and $\eta$ : real parameter
Permalink:: http://dlmf.nist.gov/33.10.E5
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Also,

33.10.6

$\mathop{{\sigma _{{0}}}\/}\nolimits\!\left(\eta\right)=\eta(\mathop{\ln\/}\nolimits\eta-1)+\tfrac{1}{4}\pi+\mathop{o\/}\nolimits\!\left(1\right),$

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

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $\mathop{{\sigma _{{\ell}}}\/}\nolimits\!\left(\eta\right)$ : Coulomb phase shift, $e$ : base of exponential function, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\sim$ : asymptotic equality and $\eta$ : real parameter
A&S Ref:: 14.6.16
Referenced by:: §33.10(ii)
Permalink:: http://dlmf.nist.gov/33.10.E6
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§33.10(iii) Large Negative $\eta$

Notes:: See Humblet (1985, Eqs. 4.7a,b). For (33.10.10) use (33.2.5), (33.2.10), and §5.11(i).
Keywords:: Coulomb functions: variables $\rho,\eta$
Permalink:: http://dlmf.nist.gov/33.10.iii

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

33.10.7

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

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

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $!$ : $n!$ : factorial, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : 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
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In particular, for $\ell=0$ ,

33.10.8

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

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

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\rho$ : nonnegative real variable and $\eta$ : real parameter
Permalink:: http://dlmf.nist.gov/33.10.E8
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33.10.9

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

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

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\rho$ : nonnegative real variable and $\eta$ : real parameter
Permalink:: http://dlmf.nist.gov/33.10.E9
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Also,

33.10.10

$\mathop{{\sigma _{{0}}}\/}\nolimits\!\left(\eta\right)=\eta(\mathop{\ln\/}\nolimits\!\left(-\eta\right)-1)-\tfrac{1}{4}\pi+\mathop{o\/}\nolimits\!\left(1\right),$

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

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $\mathop{{\sigma _{{\ell}}}\/}\nolimits\!\left(\eta\right)$ : Coulomb phase shift, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\sim$ : asymptotic equality and $\eta$ : real parameter
Referenced by:: §33.10(iii)
Permalink:: http://dlmf.nist.gov/33.10.E10
Encodings:: TeX, TeX, pMML, pMML, png, png

§33.10 Limiting Forms for Large or Large

Contents

§33.10(i) Large

§33.10(ii) Large Positive

§33.10(iii) Large Negative

§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 $\eta$