\rho,\eta

\rho

33.13 Complex Variable and Parameters

§33.12 Asymptotic Expansions for Large $\eta$

ⓘ

Keywords:: Coulomb functions: variables $\rho,\eta$ , asymptotic expansions, large $\eta$
Permalink:: http://dlmf.nist.gov/33.12
See also:: Annotations for Ch.33

§33.12(i) Transition Region

ⓘ

Keywords:: Coulomb functions: variables $\rho,\eta$ , expansions in Airy functions, transition region
Referenced by:: §2.8(iii), §33.12(ii), Erratum (V1.0.11) for References
Permalink:: http://dlmf.nist.gov/33.12.i
Addition (effective with 1.0.11):: A reference to Temme (2015, Chapter 31) was added in the sentence following (33.12.7).
See also:: Annotations for §33.12 and Ch.33

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

33.12.1	$\displaystyle x$	$\displaystyle=(2\eta-\rho)/(2\eta)^{1/3},$
33.12.1	$\displaystyle\mu$	$\displaystyle=(2\eta)^{2/3}.$
ⓘ Defines: $x$ : variable (locally) and $\mu$ : variable (locally) Symbols: $\rho$ : nonnegative real variable and $\eta$ : real parameter Permalink: http://dlmf.nist.gov/33.12.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.12(i), §33.12 and Ch.33

Then as $\eta\to\infty$ ,

33.12.2		${F_{0}\left(\eta,\rho\right)\atop G_{0}\left(\eta,\rho\right)}\sim\pi^{1/2}(2% \eta)^{1/6}\left\{{\operatorname{Ai}\left(x\right)\atop\operatorname{Bi}\left(% x\right)}\left(1+\frac{B_{1}}{\mu}+\frac{B_{2}}{\mu^{2}}+\cdots\right)+{% \operatorname{Ai}'\left(x\right)\atop\operatorname{Bi}'\left(x\right)}\left(% \frac{A_{1}}{\mu}+\frac{A_{2}}{\mu^{2}}+\cdots\right)\right\},$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\rho$ : nonnegative real variable, $\eta$ : real parameter, $x$ : 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 See also: Annotations for §33.12(i), §33.12 and Ch.33

33.12.3		${F_{0}'\left(\eta,\rho\right)\atop G_{0}'\left(\eta,\rho\right)}\sim-\pi^{1/2}% (2\eta)^{-1/6}\left\{{\operatorname{Ai}\left(x\right)\atop\operatorname{Bi}% \left(x\right)}\left(\frac{B_{1}^{\prime}+xA_{1}}{\mu}+\frac{B_{2}^{\prime}+xA% _{2}}{\mu^{2}}+\cdots\right)+{\operatorname{Ai}'\left(x\right)\atop% \operatorname{Bi}'\left(x\right)}\left(\frac{B_{1}+A_{1}^{\prime}}{\mu}+\frac{% B_{2}+A_{2}^{\prime}}{\mu^{2}}+\cdots\right)\right\},$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\rho$ : nonnegative real variable, $\eta$ : real parameter, $x$ : 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 See also: Annotations for §33.12(i), §33.12 and Ch.33

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

33.12.4	$\displaystyle A_{1}$	$\displaystyle=\tfrac{1}{5}x^{2},$
	$\displaystyle A_{2}$	$\displaystyle=\tfrac{1}{35}(2x^{3}+6),$
	$\displaystyle A_{3}$	$\displaystyle=\tfrac{1}{15750}(21x^{7}+370x^{4}+580x),$
ⓘ Defines: $A_{j}$ : coefficients (locally) and $B_{j}$ : coefficients (locally) Symbols: $x$ : variable Permalink: http://dlmf.nist.gov/33.12.E4 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §33.12(i), §33.12 and Ch.33

33.12.5	$\displaystyle B_{1}$	$\displaystyle=-\tfrac{1}{5}x,$
	$\displaystyle B_{2}$	$\displaystyle=\tfrac{1}{350}(7x^{5}-30x^{2}),$
	$\displaystyle B_{3}$	$\displaystyle=\tfrac{1}{15750}(264x^{6}-290x^{3}-560).$
ⓘ Symbols: $x$ : variable and $B_{j}$ : coefficients Permalink: http://dlmf.nist.gov/33.12.E5 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §33.12(i), §33.12 and Ch.33

In particular,

33.12.6		${F_{0}\left(\eta,2\eta\right)\atop{3^{-\ifrac{1}{2}}G_{0}\left(\eta,2\eta% \right)}}\sim\frac{\Gamma\left(\frac{1}{3}\right)\omega^{1/2}}{2\sqrt{\pi}}% \left(1\mp\frac{2}{35}\frac{\Gamma\left(\frac{2}{3}\right)}{\Gamma\left(\frac{% 1}{3}\right)}\frac{1}{\omega^{4}}-\frac{8}{2025}\frac{1}{\omega^{6}}\mp\frac{5% 792}{46\,06875}\frac{\Gamma\left(\frac{2}{3}\right)}{\Gamma\left(\frac{1}{3}% \right)}\frac{1}{\omega^{10}}-\cdots\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\eta$ : real parameter and $\omega$ : factor Permalink: http://dlmf.nist.gov/33.12.E6 Encodings: TeX, pMML, png See also: Annotations for §33.12(i), §33.12 and Ch.33

33.12.7		${F_{0}'\left(\eta,2\eta\right)\atop{3^{-\ifrac{1}{2}}G_{0}'\left(\eta,2\eta% \right)}}\sim\frac{\Gamma\left(\frac{2}{3}\right)}{2\sqrt{\pi}\omega^{1/2}}% \left(\pm 1+\frac{1}{15}\frac{\Gamma\left(\frac{1}{3}\right)}{\Gamma\left(% \frac{2}{3}\right)}\frac{1}{\omega^{2}}\pm\frac{2}{14175}\frac{1}{\omega^{6}}+% \frac{1436}{23\,38875}\frac{\Gamma\left(\frac{1}{3}\right)}{\Gamma\left(\frac{% 2}{3}\right)}\frac{1}{\omega^{8}}\pm\cdots\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\eta$ : real parameter and $\omega$ : factor Referenced by: §33.12(i) Permalink: http://dlmf.nist.gov/33.12.E7 Encodings: TeX, pMML, png See also: Annotations for §33.12(i), §33.12 and Ch.33

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). For asymptotic expansions of $F_{\ell}\left(\eta,\rho\right)$ and $G_{\ell}\left(\eta,\rho\right)$ when $\eta\to\pm\infty$ see Temme (2015, Chapter 31).

§33.12(ii) Uniform Expansions

ⓘ

Keywords:: Coulomb functions: variables $\rho,\eta$ , asymptotic expansions, uniform expansions
Referenced by:: §2.8(iii), Erratum (V1.0.11) for References
Permalink:: http://dlmf.nist.gov/33.12.ii
Addition (effective with 1.0.11):: A reference to Temme (2015, §31.7) was added.
See also:: Annotations for §33.12 and Ch.33

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

33.12.8		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=\left(4\eta^{2}\left(\frac{1-z}{z}% \right)+\frac{\ell(\ell+1)}{z^{2}}\right)w.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\ell$ : nonnegative integer and $\eta$ : real parameter Permalink: http://dlmf.nist.gov/33.12.E8 Encodings: TeX, pMML, png See also: Annotations for §33.12(ii), §33.12 and Ch.33

Then, by application of the results given in §§2.8(iii) and 2.8(iv), two sets of asymptotic expansions can be constructed for $F_{\ell}\left(\eta,\rho\right)$ and $G_{\ell}\left(\eta,\rho\right)$ when $\eta\to\infty$ . See Temme (2015, §31.7).

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).

33.11 Asymptotic Expansions for Large

\rho

33.13 Complex Variable and Parameters

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§33.12 Asymptotic Expansions for Large $\eta$

Contents

§33.12(i) Transition Region

§33.12(ii) Uniform Expansions

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§33.12 Asymptotic Expansions for Large η

Contents

§33.12(i) Transition Region

§33.12(ii) Uniform Expansions

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§33.12 Asymptotic Expansions for Large $\eta$