# Β§33.12 Asymptotic Expansions for Large $\eta$

## Β§33.12(i) Transition Region

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 $\displaystyle x$ $\displaystyle=(2\eta-\rho)/(2\eta)^{1/3},$ $\displaystyle\mu$ $\displaystyle=(2\eta)^{2/3}.$ β 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 Β§33.12(i), Β§33.12 and Ch.33

Then as $\eta\to\infty$,

 33.12.2 ${F_{0}\left(\eta,\rho\right)\atop G_{0}\left(\eta,\rho\right)}\sim\pi^{1/2}(2% \eta)^{1/6}\left\{{\operatorname{Ai}\left(x\right)\atop\operatorname{Bi}\left(% x\right)}\left(1+\frac{B_{1}}{\mu}+\frac{B_{2}}{\mu^{2}}+\cdots\right)+{% \operatorname{Ai}'\left(x\right)\atop\operatorname{Bi}'\left(x\right)}\left(% \frac{A_{1}}{\mu}+\frac{A_{2}}{\mu^{2}}+\cdots\right)\right\},$
 33.12.3 ${F_{0}'\left(\eta,\rho\right)\atop G_{0}'\left(\eta,\rho\right)}\sim-\pi^{1/2}% (2\eta)^{-1/6}\left\{{\operatorname{Ai}\left(x\right)\atop\operatorname{Bi}% \left(x\right)}\left(\frac{B_{1}^{\prime}+xA_{1}}{\mu}+\frac{B_{2}^{\prime}+xA% _{2}}{\mu^{2}}+\cdots\right)+{\operatorname{Ai}'\left(x\right)\atop% \operatorname{Bi}'\left(x\right)}\left(\frac{B_{1}+A_{1}^{\prime}}{\mu}+\frac{% B_{2}+A_{2}^{\prime}}{\mu^{2}}+\cdots\right)\right\},$

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

 33.12.4 $\displaystyle A_{1}$ $\displaystyle=\tfrac{1}{5}x^{2},$ $\displaystyle A_{2}$ $\displaystyle=\tfrac{1}{35}(2x^{3}+6),$ $\displaystyle A_{3}$ $\displaystyle=\tfrac{1}{15750}(21x^{7}+370x^{4}+580x),$ β 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 Β§33.12(i), Β§33.12 and Ch.33
 33.12.5 $\displaystyle B_{1}$ $\displaystyle=-\tfrac{1}{5}x,$ $\displaystyle B_{2}$ $\displaystyle=\tfrac{1}{350}(7x^{5}-30x^{2}),$ $\displaystyle B_{3}$ $\displaystyle=\tfrac{1}{15750}(264x^{6}-290x^{3}-560).$ β 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 Β§33.12(i), Β§33.12 and Ch.33

In particular,

 33.12.6 ${F_{0}\left(\eta,2\eta\right)\atop{3^{-\ifrac{1}{2}}G_{0}\left(\eta,2\eta% \right)}}\sim\frac{\Gamma\left(\frac{1}{3}\right)\omega^{1/2}}{2\sqrt{\pi}}% \left(1\mp\frac{2}{35}\frac{\Gamma\left(\frac{2}{3}\right)}{\Gamma\left(\frac{% 1}{3}\right)}\frac{1}{\omega^{4}}-\frac{8}{2025}\frac{1}{\omega^{6}}\mp\frac{5% 792}{46\,06875}\frac{\Gamma\left(\frac{2}{3}\right)}{\Gamma\left(\frac{1}{3}% \right)}\frac{1}{\omega^{10}}-\cdots\right),$
 33.12.7 ${F_{0}'\left(\eta,2\eta\right)\atop{3^{-\ifrac{1}{2}}G_{0}'\left(\eta,2\eta% \right)}}\sim\frac{\Gamma\left(\frac{2}{3}\right)}{2\sqrt{\pi}\omega^{1/2}}% \left(\pm 1+\frac{1}{15}\frac{\Gamma\left(\frac{1}{3}\right)}{\Gamma\left(% \frac{2}{3}\right)}\frac{1}{\omega^{2}}\pm\frac{2}{14175}\frac{1}{\omega^{6}}+% \frac{1436}{23\,38875}\frac{\Gamma\left(\frac{1}{3}\right)}{\Gamma\left(\frac{% 2}{3}\right)}\frac{1}{\omega^{8}}\pm\cdots\right),$

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 FrΓΆberg (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).

## Β§33.12(ii) Uniform Expansions

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

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

Then, by application of the results given in Β§Β§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, Β§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 Β§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 Β§33.20(iv).