33 Coulomb Functions Variables $\rho,\eta$ 33.11 Asymptotic Expansions for Large $\rho$ 33.13 Complex Variable and Parameters

§33.12 Asymptotic Expansions for Large $\eta$

Keywords:: Coulomb functions: variables $\rho,\eta$
Permalink:: http://dlmf.nist.gov/33.12

§33.12(i) Transition Region
§33.12(ii) Uniform Expansions

§33.12(i) Transition Region

Keywords:: Coulomb functions: variables $\rho,\eta$
Referenced by:: §2.8(iii), §33.12(ii)
Permalink:: http://dlmf.nist.gov/33.12.i

When $\ell=0$ and $\eta>0$ , the outer turning point is given by $\mathop{\rho_{{\mathrm{tp}}}\/}\nolimits\!\left(\eta,0\right)=2\eta$ ; compare (33.2.2). Define

33.12.1

$x=(2\eta-\rho)/(2\eta)^{{1/3}},$

$\mu=(2\eta)^{{2/3}}.$

Symbols:: $\rho$ : nonnegative real variable, $\eta$ : real parameter, : variable and $\mu$ : variable
Permalink:: http://dlmf.nist.gov/33.12.E1
Encodings:: TeX, TeX, pMML, pMML, png, png

Then as $\eta\to\infty$ ,

33.12.2 ${\mathop{F_{{0}}\/}\nolimits\!\left(\eta,\rho\right)\atop\mathop{G_{{0}}\/}% \nolimits\!\left(\eta,\rho\right)}\sim\pi^{{1/2}}(2\eta)^{{1/6}}\left\{{% \mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)\atop\mathop{\mathrm{Bi}\/}% \nolimits\!\left(x\right)}\left(1+\frac{B_{1}}{\mu}+\frac{B_{2}}{\mu^{2}}+% \cdots\right)+{{\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(x\right)% \atop{\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(x\right)}\left(\frac{% A_{1}}{\mu}+\frac{A_{2}}{\mu^{2}}+\cdots\right)\right\},$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\{\ldots\}$ : sequence, asymptotic sequence (or scale), or enumerable set, $\sim$ : asymptotic equality, $\rho$ : nonnegative real variable, $\eta$ : real parameter, : 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:: TeX, pMML, png

33.12.3 ${{\mathop{F_{{0}}\/}\nolimits^{{\prime}}}\!\left(\eta,\rho\right)\atop{\mathop% {G_{{0}}\/}\nolimits^{{\prime}}}\!\left(\eta,\rho\right)}\sim-\pi^{{1/2}}(2% \eta)^{{-1/6}}\left\{{\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)\atop% \mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)}\left(\frac{B_{1}^{{\prime}}+% xA_{1}}{\mu}+\frac{B_{2}^{{\prime}}+xA_{2}}{\mu^{2}}+\cdots\right)+{{\mathop{% \mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(x\right)\atop{\mathop{\mathrm{Bi}\/% }\nolimits^{{\prime}}}\!\left(x\right)}\left(\frac{B_{1}+A_{1}^{{\prime}}}{\mu% }+\frac{B_{2}+A_{2}^{{\prime}}}{\mu^{2}}+\cdots\right)\right\},$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\{\ldots\}$ : sequence, asymptotic sequence (or scale), or enumerable set, $\sim$ : asymptotic equality, $\rho$ : nonnegative real variable, $\eta$ : real parameter, : 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:: TeX, pMML, png

uniformly for bounded values of $\left|(\rho-2\eta)/\eta^{{1/3}}\right|$ . Here $\mathop{\mathrm{Ai}\/}\nolimits$ and $\mathop{\mathrm{Bi}\/}\nolimits$ are the Airy functions (§9.2), and

33.12.4

$A_{1}=\tfrac{1}{5}x^{2},$

$A_{2}=\tfrac{1}{35}(2x^{3}+6),$

$A_{3}=\tfrac{1}{15750}(21x^{7}+370x^{4}+580x),$

Symbols:: : variable, $A_{{j}}$ : coefficients and $B_{{j}}$ : coefficients
Permalink:: http://dlmf.nist.gov/33.12.E4
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

33.12.5

$B_{1}=-\tfrac{1}{5}x,$

$B_{2}=\tfrac{1}{350}(7x^{5}-30x^{2}),$

$B_{3}=\tfrac{1}{15750}(264x^{6}-290x^{3}-560).$

Symbols:: : variable and $B_{{j}}$ : coefficients
Permalink:: http://dlmf.nist.gov/33.12.E5
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

In particular,

33.12.6 ${\mathop{F_{{0}}\/}\nolimits\!\left(\eta,2\eta\right)\atop{3^{{-\ifrac{1}{2}}}% \mathop{G_{{0}}\/}\nolimits\!\left(\eta,2\eta\right)}}\sim\frac{\mathop{\Gamma% \/}\nolimits\!\left(\frac{1}{3}\right)\omega^{{1/2}}}{2\sqrt{\pi}}\left(1\mp% \frac{2}{35}\frac{\mathop{\Gamma\/}\nolimits\!\left(\frac{2}{3}\right)}{% \mathop{\Gamma\/}\nolimits\!\left(\frac{1}{3}\right)}\frac{1}{\omega^{4}}-% \frac{8}{2025}\frac{1}{\omega^{6}}\mp\frac{5792}{46\,06875}\frac{\mathop{% \Gamma\/}\nolimits\!\left(\frac{2}{3}\right)}{\mathop{\Gamma\/}\nolimits\!% \left(\frac{1}{3}\right)}\frac{1}{\omega^{{10}}}-\cdots\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\sim$ : asymptotic equality, $\eta$ : real parameter and $\omega$ : factor
Permalink:: http://dlmf.nist.gov/33.12.E6
Encodings:: TeX, pMML, png

33.12.7 ${{\mathop{F_{{0}}\/}\nolimits^{{\prime}}}\!\left(\eta,2\eta\right)\atop{3^{{-% \ifrac{1}{2}}}{\mathop{G_{{0}}\/}\nolimits^{{\prime}}}\!\left(\eta,2\eta\right% )}}\sim\frac{\mathop{\Gamma\/}\nolimits\!\left(\frac{2}{3}\right)}{2\sqrt{\pi}% \omega^{{1/2}}}\left(\pm 1+\frac{1}{15}\frac{\mathop{\Gamma\/}\nolimits\!\left% (\frac{1}{3}\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{2}{3}\right)}% \frac{1}{\omega^{2}}\pm\frac{2}{14175}\frac{1}{\omega^{6}}+\frac{1436}{23\,388% 75}\frac{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{3}\right)}{\mathop{\Gamma% \/}\nolimits\!\left(\frac{2}{3}\right)}\frac{1}{\omega^{{8}}}\pm\cdots\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\sim$ : asymptotic equality, $\eta$ : real parameter and $\omega$ : factor
Permalink:: http://dlmf.nist.gov/33.12.E7
Encodings:: TeX, pMML, png

where $\omega=(\tfrac{2}{3}\eta)^{{1/3}}$ .

For derivations and additional terms in the expansions in this subsection see Abramowitz and Rabinowitz (1954) and Fröberg (1955).

§33.12(ii) Uniform Expansions

Keywords:: Coulomb functions: variables $\rho,\eta$
Referenced by:: §2.8(iii)
Permalink:: http://dlmf.nist.gov/33.12.ii

With the substitution $\rho=2\eta z$ , Equation (33.2.1) becomes

33.12.8 $\frac{{d}^{2}w}{{dz}^{2}}=\left(4\eta^{2}\left(\frac{1-z}{z}\right)+\frac{\ell% (\ell+1)}{z^{2}}\right)w.$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , $\ell$ : nonnegative integer and $\eta$ : real parameter
Permalink:: http://dlmf.nist.gov/33.12.E8
Encodings:: TeX, pMML, png

Then, by application of the results given in §§2.8(iii) and 2.8(iv), two sets of asymptotic expansions can be constructed for $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ and $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ when $\eta\to\infty$ .

The first set is in terms of Airy functions and the expansions are uniform for fixed $\ell$ and $\delta\leq z<\infty$ , where $\delta$ is an arbitrary small positive constant. They would include the results of §33.12(i) as a special case.

The second set is in terms of Bessel functions of orders $2\ell+1$ and $2\ell+2$ , and they are uniform for fixed $\ell$ and $0\leq z\leq 1-\delta$ , where $\delta$ again denotes an arbitrary small positive constant.

Compare also §33.20(iv).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 33.11 Asymptotic Expansions for Large $\rho$ 33.13 Complex Variable and Parameters

§33.12 Asymptotic Expansions for Large

Contents

§33.12(i) Transition Region

§33.12(ii) Uniform Expansions

§33.12 Asymptotic Expansions for Large $\eta$