§33.12 Asymptotic Expansions for Large $\eta$

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

33.12.1		$x$	$=(2\eta -\rho )/{(2\eta )}^{1/3},$
		$\mu$	$={(2\eta )}^{2/3}.$
ⓘ 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 $\eta \to \mathrm{\infty }$,

33.12.2		$$\genfrac{}{}{0pt}{}{F_0(\eta ,\rho )}{G_0(\eta ,\rho )}\sim {\pi }^{1/2}{(2\eta )}^{1/6}\left\{\genfrac{}{}{0pt}{}{\mathrm{Ai}\left(x\right)}{\mathrm{Bi}\left(x\right)}\left(1+\frac{B_1}{\mu }+\frac{B_2}{{\mu }^2}+\mathrm{\cdots }\right)+\genfrac{}{}{0pt}{}{{\mathrm{Ai}}^{\prime }\left(x\right)}{{\mathrm{Bi}}^{\prime }\left(x\right)}\left(\frac{A_1}{\mu }+\frac{A_2}{{\mu }^2}+\mathrm{\cdots }\right)\right\},$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: 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		$$\genfrac{}{}{0pt}{}{F_0^{\prime }(\eta ,\rho )}{G_0^{\prime }(\eta ,\rho )}\sim -{\pi }^{1/2}{(2\eta )}^{-1/6}\{\begin{array}{l}\genfrac{}{}{0pt}{}{\mathrm{Ai}\left(x\right)}{\mathrm{Bi}\left(x\right)}\left(\frac{B_1^{\prime }+x{A}_1}{\mu }+\frac{B_2^{\prime }+x{A}_2}{{\mu }^2}+\mathrm{\cdots }\right)\\ +\genfrac{}{}{0pt}{}{{\mathrm{Ai}}^{\prime }\left(x\right)}{{\mathrm{Bi}}^{\prime }\left(x\right)}\left(\frac{B_1+A_1^{\prime }}{\mu }+\frac{B_2+A_2^{\prime }}{{\mu }^2}+\mathrm{\cdots }\right)\},\end{array}$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: 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 $\mathrm{Ai}$ and $\mathrm{Bi}$ are the Airy functions (§9.2), and

33.12.4		$A_1$	$=\frac{1}{5}{x}^2,$
		$A_2$	$=\frac{1}{35}(2x^3+6),$
		$A_3$	$=\frac{1}{15750}(21x^7+370x^4+580x),$
ⓘ 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		$B_1$	$=-\frac{1}{5}x,$
		$B_2$	$=\frac{1}{350}(7x^5-30x^2),$
		$B_3$	$=\frac{1}{15750}(264x^6-290x^3-560).$
ⓘ 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		$$\genfrac{}{}{0pt}{}{F_0(\eta ,2\eta )}{3^{-1/2}{G}_0(\eta ,2\eta )}\sim \frac{\mathrm{\Gamma }\left(\frac{1}{3}\right){\omega }^{1/2}}{2\sqrt{\pi }}\left(1\mp \frac{2}{35}\frac{\mathrm{\Gamma }\left(\frac{2}{3}\right)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\frac{1}{{\omega }^4}-\frac{8}{2025}\frac{1}{{\omega }^6}\mp \frac{5792}{\mathrm{46\hspace{0.17em}06875}}\frac{\mathrm{\Gamma }\left(\frac{2}{3}\right)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\frac{1}{{\omega }^{10}}-\mathrm{\cdots }\right),$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: 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		$$\genfrac{}{}{0pt}{}{F_0^{\prime }(\eta ,2\eta )}{3^{-1/2}{G}_0^{\prime }(\eta ,2\eta )}\sim \frac{\mathrm{\Gamma }\left(\frac{2}{3}\right)}{2\sqrt{\pi }{\omega }^{1/2}}\left(\pm 1+\frac{1}{15}\frac{\mathrm{\Gamma }\left(\frac{1}{3}\right)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\frac{1}{{\omega }^2}\pm \frac{2}{14175}\frac{1}{{\omega }^6}+\frac{1436}{\mathrm{23\hspace{0.17em}38875}}\frac{\mathrm{\Gamma }\left(\frac{1}{3}\right)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\frac{1}{{\omega }^8}\pm \mathrm{\cdots }\right),$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: 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 $\omega ={(\frac{2}{3}\eta )}^{1/3}$.

For derivations and additional terms in the expansions in this subsection see Abramowitz and Rabinowitz (1954) and Fröberg (1955). For asymptotic expansions of $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ when $\eta \to \pm \mathrm{\infty }$ see Temme (2015, Chapter 31).

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

33.12.8		$$\frac{d^{2}w}{{dz}^2}=\left(4{\eta }^2\left(\frac{1-z}{z}\right)+\frac{\mathrm{\ell }(\mathrm{\ell }+1)}{z^2}\right)w.$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{\ell }$: 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 $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ when $\eta \to \mathrm{\infty }$. See Temme (2015, §31.7).

The first set is in terms of Airy functions and the expansions are uniform for fixed $\mathrm{\ell }$ and , 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 $2\mathrm{\ell }+1$ and $2\mathrm{\ell }+2$, and they are uniform for fixed $\mathrm{\ell }$ and $0\le z\le 1-\delta$, where $\delta$ again denotes an arbitrary small positive constant.

Compare also §33.20(iv).