# §12.10 Uniform Asymptotic Expansions for Large Parameter

## §12.10(i) Introduction

In this section we give asymptotic expansions of PCFs for large values of the parameter $a$ that are uniform with respect to the variable $z$, when both $a$ and $z$ $(=x)$ are real. These expansions follow from Olver (1959), where detailed information is also given for complex variables.

With the transformations

 12.10.1 $\displaystyle a$ $\displaystyle=\pm\tfrac{1}{2}\mu^{2},$ $\displaystyle x$ $\displaystyle=\mu t\sqrt{2},$ Symbols: $x$: real variable and $a$: real or complex parameter Permalink: http://dlmf.nist.gov/12.10.E1 Encodings: TeX, TeX, pMML, pMML, png, png

(12.2.2) becomes

 12.10.2 $\frac{{d}^{2}w}{{dt}^{2}}=\mu^{4}(t^{2}\pm 1)w.$ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$ Referenced by: §12.10(i) Permalink: http://dlmf.nist.gov/12.10.E2 Encodings: TeX, pMML, png

With the upper sign in (12.10.2), expansions can be constructed for large $\mu$ in terms of elementary functions that are uniform for $t\in(-\infty,\infty)$2.8(ii)). With the lower sign there are turning points at $t=\pm 1$, which need to be excluded from the regions of validity. These cases are treated in §§12.10(ii)12.10(vi).

The turning points can be included if expansions in terms of Airy functions are used instead of elementary functions (§2.8(iii)). These cases are treated in §§12.10(vii)12.10(viii).

Throughout this section the symbol $\delta$ again denotes an arbitrary small positive constant.

## §12.10(ii) Negative $a$, $2\sqrt{-a}

As $a\to-\infty$

 12.10.3 $\mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)\sim% \frac{g(\mu)e^{-\mu^{2}\xi}}{(t^{2}-1)^{\frac{1}{4}}}\sum_{s=0}^{\infty}\frac{% {\cal A}_{s}(t)}{\mu^{2s}},$
 12.10.4 $\mathop{U\/}\nolimits'\!\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)\sim-% \frac{\mu}{\sqrt{2}}g(\mu)(t^{2}-1)^{\frac{1}{4}}\*e^{-\mu^{2}\xi}\sum_{s=0}^{% \infty}\frac{{\cal B}_{s}(t)}{\mu^{2s}},$
 12.10.5 $\mathop{V\/}\nolimits\!\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)\sim% \frac{2g(\mu)}{\mathop{\Gamma\/}\nolimits(\frac{1}{2}+\frac{1}{2}\mu^{2})}% \frac{e^{\mu^{2}\xi}}{(t^{2}-1)^{\frac{1}{4}}}\*\sum_{s=0}^{\infty}(-1)^{s}% \frac{{\cal A}_{s}(t)}{\mu^{2s}},$
 12.10.6 $\mathop{V\/}\nolimits'\!\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)\sim% \frac{\sqrt{2}\mu g(\mu)}{\mathop{\Gamma\/}\nolimits(\frac{1}{2}+\frac{1}{2}% \mu^{2})}(t^{2}-1)^{\frac{1}{4}}\*e^{\mu^{2}\xi}\sum_{s=0}^{\infty}(-1)^{s}% \frac{{\cal B}_{s}(t)}{\mu^{2s}},$

uniformly for $t\in[1+\delta,\infty)$, where

 12.10.7 $\xi=\tfrac{1}{2}t\sqrt{t^{2}-1}-\tfrac{1}{2}\mathop{\ln\/}\nolimits\!\left(t+% \sqrt{t^{2}-1}\right).$ Defines: $\xi$ (locally) Symbols: $\mathop{\ln\/}\nolimits z$: principal branch of logarithm function Referenced by: §12.10(vi), §12.10(vii), §12.14(ix) Permalink: http://dlmf.nist.gov/12.10.E7 Encodings: TeX, pMML, png

The coefficients are given by

 12.10.8 ${\cal A}_{s}(t)=\frac{u_{s}(t)}{(t^{2}-1)^{\frac{3}{2}s}},~{}{\cal B}_{s}(t)=% \frac{v_{s}(t)}{(t^{2}-1)^{\frac{3}{2}s}},$ Defines: $\mathcal{A}_{s}(t)$: coefficients (locally) and $\mathcal{B}_{s}(t)$: coefficients (locally) Symbols: $s$: nonnegative integer, $u_{s}(t)$: solution and $v_{s}(t)$: solution Referenced by: §12.10(iv) Permalink: http://dlmf.nist.gov/12.10.E8 Encodings: TeX, pMML, png

where $u_{s}(t)$ and $v_{s}(t)$ are polynomials in $t$ of degree $3s$, ($s$ odd), $3s-2$ ($s$ even, $s\geq 2$). For $s=0,1,2$,

 12.10.9 $\displaystyle u_{0}(t)$ $\displaystyle=1,$ $\displaystyle u_{1}(t)$ $\displaystyle=\frac{t(t^{2}-6)}{24},$ $\displaystyle u_{2}(t)$ $\displaystyle=\frac{-9t^{4}+249t^{2}+145}{1152},$ Defines: $u_{s}(t)$: solution (locally) Symbols: $s$: nonnegative integer Permalink: http://dlmf.nist.gov/12.10.E9 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
 12.10.10 $\displaystyle v_{0}(t)$ $\displaystyle=1,$ $\displaystyle v_{1}(t)$ $\displaystyle=\frac{t(t^{2}+6)}{24},$ $\displaystyle v_{2}(t)$ $\displaystyle=\frac{15t^{4}-327t^{2}-143}{1152}.$ Defines: $v_{s}(t)$: solution (locally) Symbols: $s$: nonnegative integer Permalink: http://dlmf.nist.gov/12.10.E10 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

Higher polynomials $u_{s}(t)$ can be calculated from the recurrence relation

 12.10.11 $(t^{2}-1)u^{\prime}_{s}(t)-3stu_{s}(t)=r_{s-1}(t),$

where

 12.10.12 $8r_{s}(t)=(3t^{2}+2)u_{s}(t)-12(s+1)tr_{s-1}(t)+4(t^{2}-1)r^{\prime}_{s-1}(t),$ Defines: $r_{s}(t)$: polynomial (locally) Symbols: $s$: nonnegative integer and $u_{s}(t)$: solution Permalink: http://dlmf.nist.gov/12.10.E12 Encodings: TeX, pMML, png

and the $v_{s}(t)$ then follow from

 12.10.13 $v_{s}(t)=u_{s}(t)+\tfrac{1}{2}tu_{s-1}(t)-r_{s-2}(t).$

Lastly, the function $g(\mu)$ in (12.10.3) and (12.10.4) has the asymptotic expansion:

 12.10.14 $g(\mu)\sim h(\mu)\left(1+\frac{1}{2}\sum_{s=1}^{\infty}\frac{\gamma_{s}}{(% \frac{1}{2}\mu^{2})^{s}}\right),$ Defines: $g(\mu)$: expansion (locally) Symbols: $\sim$: asymptotic equality, $s$: nonnegative integer, $h(\mu)$: expansion and $\gamma_{s}$: coefficients Referenced by: §12.10(vii) Permalink: http://dlmf.nist.gov/12.10.E14 Encodings: TeX, pMML, png

where

 12.10.15 $h(\mu)=2^{-\frac{1}{4}\mu^{2}-\frac{1}{4}}e^{-\frac{1}{4}\mu^{2}}\mu^{\frac{1}% {2}\mu^{2}-\frac{1}{2}},$ Defines: $h(\mu)$: expansion (locally) Symbols: $e$: base of exponential function Referenced by: §12.10(vi) Permalink: http://dlmf.nist.gov/12.10.E15 Encodings: TeX, pMML, png

and the coefficients $\gamma_{s}$ are defined by

 12.10.16 $\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}+z\right)\sim\sqrt{2\pi}e^{-z}z^% {z}\sum_{s=0}^{\infty}\frac{\gamma_{s}}{z^{s}};$ Defines: $\gamma_{s}$: coefficients (locally) Symbols: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$: gamma function, $e$: base of exponential function, $\sim$: asymptotic equality, $z$: complex variable and $s$: nonnegative integer Permalink: http://dlmf.nist.gov/12.10.E16 Encodings: TeX, pMML, png

compare (5.11.8). For $s\leq 4$

 12.10.17 $\displaystyle\gamma_{0}$ $\displaystyle=1,$ $\displaystyle\gamma_{1}$ $\displaystyle=-\tfrac{1}{24},$ $\displaystyle\gamma_{2}$ $\displaystyle=\tfrac{1}{1152},$ $\displaystyle\gamma_{3}$ $\displaystyle=\tfrac{1003}{4\;14720},$ $\displaystyle\gamma_{4}$ $\displaystyle=-\tfrac{4027}{398\;13120}.$ Symbols: $\gamma_{s}$: coefficients Permalink: http://dlmf.nist.gov/12.10.E17 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png

## §12.10(iii) Negative $a$, $-\infty

When $\mu\to\infty$, asymptotic expansions for the functions $\mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right)$ and $\mathop{V\/}\nolimits\!\left(-\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right)$ that are uniform for $t\in[1+\delta,\infty)$ are obtainable by substitution into (12.2.15) and (12.2.16) by means of (12.10.3) and (12.10.5). Similarly for $\mathop{U\/}\nolimits'\!\left(-\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right)$ and $\mathop{V\/}\nolimits'\!\left(-\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right)$.

## §12.10(iv) Negative $a$, $-2\sqrt{-a}

As $a\to-\infty$

 12.10.18 $\displaystyle\mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)$ $\displaystyle\sim\frac{2g(\mu)}{(1-t^{2})^{\frac{1}{4}}}\*\left(\mathop{\cos\/% }\nolimits\kappa\sum_{s=0}^{\infty}(-1)^{s}\frac{\widetilde{\cal{A}}_{2s}(t)}{% \mu^{4s}}-\mathop{\sin\/}\nolimits\kappa\sum_{s=0}^{\infty}(-1)^{s}\frac{% \widetilde{\cal{A}}_{2s+1}(t)}{\mu^{4s+2}}\right),$ 12.10.19 $\displaystyle\mathop{U\/}\nolimits'\!\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)$ $\displaystyle\sim\mu\sqrt{2}g(\mu)(1-t^{2})^{\frac{1}{4}}\*\left(\mathop{\sin% \/}\nolimits\kappa\sum_{s=0}^{\infty}(-1)^{s}\frac{\widetilde{\cal{B}}_{2s}(t)% }{\mu^{4s}}+\mathop{\cos\/}\nolimits\kappa\sum_{s=0}^{\infty}(-1)^{s}\frac{% \widetilde{\cal{B}}_{2s+1}(t)}{\mu^{4s+2}}\right),$ 12.10.20 $\displaystyle\mathop{V\/}\nolimits\!\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)$ $\displaystyle\sim\frac{2g(\mu)}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}% +\tfrac{1}{2}\mu^{2}\right)(1-t^{2})^{\frac{1}{4}}}\*\left(\mathop{\cos\/}% \nolimits\chi\sum_{s=0}^{\infty}(-1)^{s}\frac{\widetilde{\cal{A}}_{2s}(t)}{\mu% ^{4s}}-\mathop{\sin\/}\nolimits\chi\sum_{s=0}^{\infty}(-1)^{s}\frac{\widetilde% {\cal{A}}_{2s+1}(t)}{\mu^{4s+2}}\right),$ 12.10.21 $\displaystyle\mathop{V\/}\nolimits'\!\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)$ $\displaystyle\sim\frac{\mu\sqrt{2}g(\mu)(1-t^{2})^{\frac{1}{4}}}{\mathop{% \Gamma\/}\nolimits\!\left(\tfrac{1}{2}+\tfrac{1}{2}\mu^{2}\right)}\*\left(% \mathop{\sin\/}\nolimits\chi\sum_{s=0}^{\infty}(-1)^{s}\frac{\widetilde{\cal{B% }}_{2s}(t)}{\mu^{4s}}+\mathop{\cos\/}\nolimits\chi\sum_{s=0}^{\infty}(-1)^{s}% \frac{\widetilde{\cal{B}}_{2s+1}(t)}{\mu^{4s+2}}\right),$

uniformly for $t\in[-1+\delta,1-\delta]$. The quantities $\kappa$ and $\chi$ are defined by

 12.10.22 $\displaystyle\kappa$ $\displaystyle=\mu^{2}\eta-\tfrac{1}{4}\pi,$ $\displaystyle\chi$ $\displaystyle=\mu^{2}\eta+\tfrac{1}{4}\pi,$ Defines: $\kappa$ (locally) and $\chi$ (locally) Symbols: $\eta$ Permalink: http://dlmf.nist.gov/12.10.E22 Encodings: TeX, TeX, pMML, pMML, png, png

where

 12.10.23 $\eta=\tfrac{1}{2}\mathop{\mathrm{arccos}\/}\nolimits t-\tfrac{1}{2}t\sqrt{1-t^% {2}},$ Defines: $\eta$ (locally) Symbols: $\mathop{\mathrm{arccos}\/}\nolimits z$: arccosine function Referenced by: §12.10(vii), §12.14(ix) Permalink: http://dlmf.nist.gov/12.10.E23 Encodings: TeX, pMML, png

and the coefficients $\widetilde{\cal{A}}_{s}(t)$ and $\widetilde{\cal{B}}_{s}(t)$ are given by

 12.10.24 $\displaystyle\widetilde{\cal{A}}_{s}(t)$ $\displaystyle=\frac{u_{s}(t)}{(1-t^{2})^{\frac{3}{2}s}},$ $\displaystyle\widetilde{\cal{B}}_{s}(t)$ $\displaystyle=\frac{v_{s}(t)}{(1-t^{2})^{\frac{3}{2}s}};$ Symbols: $s$: nonnegative integer, $u_{s}(t)$: solution and $v_{s}(t)$: solution Referenced by: §12.14(ix) Permalink: http://dlmf.nist.gov/12.10.E24 Encodings: TeX, TeX, pMML, pMML, png, png

compare (12.10.8).

## §12.10(v) Positive $a$, $-\infty

As $a\to\infty$

 12.10.25 $\mathop{U\/}\nolimits\!\left(\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)\sim\frac% {\overline{g}(\mu)e^{-\mu^{2}{\overline{\xi}}}}{(t^{2}+1)^{\frac{1}{4}}}\sum_{% s=0}^{\infty}\frac{\overline{u}_{s}(t)}{(t^{2}+1)^{\frac{3}{2}s}}\frac{1}{\mu^% {2s}},$

uniformly for $t\in\Real$. Here bars do not denote complex conjugates; instead

 12.10.26 $\overline{\xi}=\tfrac{1}{2}t\sqrt{t^{2}+1}+\tfrac{1}{2}\mathop{\ln\/}\nolimits% \!\left(t+\sqrt{t^{2}+1}\right),$ Symbols: $\mathop{\ln\/}\nolimits z$: principal branch of logarithm function Permalink: http://dlmf.nist.gov/12.10.E26 Encodings: TeX, pMML, png
 12.10.27 $\overline{u}_{s}(t)=i^{s}u_{s}(-it),$ Symbols: $s$: nonnegative integer and $u_{s}(t)$: solution Permalink: http://dlmf.nist.gov/12.10.E27 Encodings: TeX, pMML, png

and the function $\overline{g}(\mu)$ has the asymptotic expansion

 12.10.28 $\overline{g}(\mu)\sim\frac{1}{\mu\sqrt{2}h(\mu)}\left(1+\frac{1}{2}\sum_{s=1}^% {\infty}(-1)^{s}\frac{\gamma_{s}}{(\frac{1}{2}\mu^{2})^{s}}\right),$

where $h(\mu)$ and $\gamma_{s}$ are as in §12.10(ii).

With the same conditions

 12.10.29 $\mathop{U\/}\nolimits'\!\left(\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)\sim{-% \frac{\mu}{\sqrt{2}}\overline{g}(\mu)(t^{2}+1)^{\frac{1}{4}}e^{-\mu^{2}{% \overline{\xi}}}\sum_{s=0}^{\infty}\frac{\overline{v}_{s}(t)}{(t^{2}+1)^{\frac% {3}{2}s}}\frac{1}{\mu^{2s}}},$

where

 12.10.30 $\overline{v}_{s}(t)=i^{s}v_{s}(-it).$ Symbols: $s$: nonnegative integer and $v_{s}(t)$: solution Permalink: http://dlmf.nist.gov/12.10.E30 Encodings: TeX, pMML, png

## §12.10(vi) Modifications of Expansions in Elementary Functions

In Temme (2000) modifications are given of Olver’s expansions. An example is the following modification of (12.10.3)

 12.10.31 $\mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)\sim% \frac{h(\mu)e^{-\mu^{2}\xi}}{(t^{2}-1)^{\frac{1}{4}}}\sum_{s=0}^{\infty}\frac{% \mathsf{A}_{s}(\tau)}{\mu^{2s}},$

where $\xi$ and $h(\mu)$ are as in (12.10.7) and (12.10.15) ,

 12.10.32 $\tau=\frac{1}{2}\left(\frac{t}{\sqrt{t^{2}-1}}-1\right),$ Defines: $\tau$ (locally) Permalink: http://dlmf.nist.gov/12.10.E32 Encodings: TeX, pMML, png

and the coefficients $\mathsf{A}_{s}(\tau)$ are the product of $\tau^{s}$ and a polynomial in $\tau$ of degree $2s$. They satisfy the recursion

 12.10.33 $\mathsf{A}_{s+1}(\tau)=-4\tau^{2}(\tau+1)^{2}\frac{d}{d\tau}\mathsf{A}_{s}(% \tau)-\frac{1}{4}\int_{0}^{\tau}\left(20u^{2}+20u+3\right)\mathsf{A}_{s}(u)du,$ $s=0,1,2,\dots$, Defines: $\mathsf{A}_{s}(\tau)$: coefficients (locally) Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $dx$: differential of $x$, $\int$: integral, $s$: nonnegative integer and $\tau$ Permalink: http://dlmf.nist.gov/12.10.E33 Encodings: TeX, pMML, png

starting with $\mathsf{A}_{o}(\tau)=1$. Explicitly,

 12.10.34 $\displaystyle\mathsf{A}_{1}(\tau)$ $\displaystyle=-\tfrac{1}{12}\tau(20\tau^{2}+30\tau+9),$ $\displaystyle\mathsf{A}_{2}(\tau)$ $\displaystyle=\tfrac{1}{288}\tau^{2}(6160\tau^{4}+18480\tau^{3}+19404\tau^{2}+% 8028\tau+945).$ Symbols: $\tau$ and $\mathsf{A}_{s}(\tau)$: coefficients Permalink: http://dlmf.nist.gov/12.10.E34 Encodings: TeX, TeX, pMML, pMML, png, png

The modified expansion (12.10.31) shares the property of (12.10.3) that it applies when $\mu\to\infty$ uniformly with respect to $t\in[1+\delta,\infty)$. In addition, it enjoys a double asymptotic property: it holds if either or both $\mu$ and $t$ tend to infinity. Observe that if $t\to\infty$, then $\mathsf{A}_{s}(\tau)=\mathop{O\/}\nolimits\!\left(t^{-2s}\right)$, whereas ${\cal A}_{s}(t)=\mathop{O\/}\nolimits(1)$ or $\mathop{O\/}\nolimits\!\left(t^{-2}\right)$ according as $s$ is even or odd. The proof of the double asymptotic property then follows with the aid of error bounds; compare §10.41(iv).

## §12.10(vii) Negative $a$, $-2\sqrt{-a}. Expansions in Terms of Airy Functions

The following expansions hold for large positive real values of $\mu$, uniformly for $t\in[-1+\delta,\infty)$. (For complex values of $\mu$ and $t$ see Olver (1959).)

 12.10.35 $\displaystyle\mathop{U\/}\nolimits\!\left(-\frac{1}{2}\mu^{2},\mu t\sqrt{2}\right)$ $\displaystyle\sim 2\pi^{\frac{1}{2}}\mu^{\frac{1}{3}}g(\mu)\phi(\zeta)\*\left(% \mathop{\mathrm{Ai}\/}\nolimits\!\left(\mu^{\frac{4}{3}}\zeta\right)\sum_{s=0}% ^{\infty}\frac{A_{s}(\zeta)}{\mu^{4s}}+\frac{\mathop{\mathrm{Ai}\/}\nolimits'% \!\left(\mu^{\frac{4}{3}}\zeta\right)}{\mu^{\frac{8}{3}}}\sum_{s=0}^{\infty}% \frac{B_{s}(\zeta)}{\mu^{4s}}\right),$ 12.10.36 $\displaystyle\mathop{U\/}\nolimits'\!\left(-\frac{1}{2}\mu^{2},\mu t\sqrt{2}\right)$ $\displaystyle\sim\frac{(2\pi)^{\frac{1}{2}}\mu^{\frac{2}{3}}g(\mu)}{\phi(\zeta% )}\*\left(\frac{\mathop{\mathrm{Ai}\/}\nolimits\!\left(\mu^{\frac{4}{3}}\zeta% \right)}{\mu^{\frac{4}{3}}}\sum_{s=0}^{\infty}\frac{C_{s}(\zeta)}{\mu^{4s}}+% \mathop{\mathrm{Ai}\/}\nolimits'\!\left(\mu^{\frac{4}{3}}\zeta\right)\sum_{s=0% }^{\infty}\frac{D_{s}(\zeta)}{\mu^{4s}}\right),$ 12.10.37 $\displaystyle\mathop{V\/}\nolimits\!\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)$ $\displaystyle\sim\frac{2\pi^{\frac{1}{2}}\mu^{\frac{1}{3}}g(\mu)\phi(\zeta)}{% \mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}+\tfrac{1}{2}\mu^{2}\right)}\*% \left(\mathop{\mathrm{Bi}\/}\nolimits\!\left(\mu^{\frac{4}{3}}\zeta\right)\sum% _{s=0}^{\infty}\frac{A_{s}(\zeta)}{\mu^{4s}}+\frac{\mathop{\mathrm{Bi}\/}% \nolimits'\!\left(\mu^{\frac{4}{3}}\zeta\right)}{\mu^{\frac{8}{3}}}\sum_{s=0}^% {\infty}\frac{B_{s}(\zeta)}{\mu^{4s}}\right),$ 12.10.38 $\displaystyle\mathop{V\/}\nolimits'\!\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)$ $\displaystyle\sim\frac{(2\pi)^{\frac{1}{2}}\mu^{\frac{2}{3}}g(\mu)}{\phi(\zeta% )\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}+\tfrac{1}{2}\mu^{2}\right)}\*% \left(\frac{\mathop{\mathrm{Bi}\/}\nolimits\!\left(\mu^{\frac{4}{3}}\zeta% \right)}{\mu^{\frac{4}{3}}}\sum_{s=0}^{\infty}\frac{C_{s}(\zeta)}{\mu^{4s}}+% \mathop{\mathrm{Bi}\/}\nolimits'\!\left(\mu^{\frac{4}{3}}\zeta\right)\sum_{s=0% }^{\infty}\frac{D_{s}(\zeta)}{\mu^{4s}}\right).$

The variable $\zeta$ is defined by

 12.10.39 $\displaystyle\tfrac{2}{3}\zeta^{\frac{3}{2}}$ $\displaystyle=\xi,\quad\text{1\leq t},\text{(\zeta\geq 0)};$ $\displaystyle\tfrac{2}{3}(-\zeta)^{\frac{3}{2}}$ $\displaystyle=\eta,\quad\text{-1 Defines: $\zeta$: change of variable (locally) Symbols: $\xi$ and $\eta$ Referenced by: §12.11(iii) Permalink: http://dlmf.nist.gov/12.10.E39 Encodings: TeX, TeX, pMML, pMML, png, png

where $\xi,\eta$ are given by (12.10.7), (12.10.23), respectively, and

 12.10.40 $\phi(\zeta)=\left(\frac{\zeta}{t^{2}-1}\right)^{\frac{1}{4}}.$ Defines: $\phi(\zeta)$: function (locally) Symbols: $\zeta$: change of variable Referenced by: §12.10(vii) Permalink: http://dlmf.nist.gov/12.10.E40 Encodings: TeX, pMML, png

The function $\zeta=\zeta(t)$ is real for $t>-1$ and analytic at $t=1$. Inversely, with $w=2^{-\frac{1}{3}}\zeta$,

 12.10.41 $t=1+w-\tfrac{1}{10}w^{2}+\tfrac{11}{350}w^{3}-\tfrac{823}{63000}w^{4}+\tfrac{1% \;50653}{242\;55000}w^{5}+\cdots,$ $|\zeta|<\left(\tfrac{3}{4}\pi\right)^{\frac{2}{3}}.$ Symbols: $\zeta$: change of variable Referenced by: §12.11(iii), §12.11(iii) Permalink: http://dlmf.nist.gov/12.10.E41 Encodings: TeX, pMML, png

For $g(\mu)$ see (12.10.14). The coefficients $A_{s}(\zeta)$ and $B_{s}(\zeta)$ are given by

 12.10.42 $\displaystyle A_{s}(\zeta)$ $\displaystyle=\zeta^{-3s}\sum_{m=0}^{2s}\beta_{m}(\phi(\zeta))^{6(2s-m)}u_{2s-% m}(t),$ $\displaystyle\zeta^{2}B_{s}(\zeta)$ $\displaystyle=-\zeta^{-3s}\sum_{m=0}^{2s+1}\alpha_{m}(\phi(\zeta))^{6(2s-m+1)}% u_{2s-m+1}(t),$

where $\phi(\zeta)$ is as in (12.10.40), $u_{k}(t)$ is as in §12.10(ii), $\alpha_{0}=1$, and

 12.10.43 $\displaystyle\alpha_{m}$ $\displaystyle=\frac{(2m+1)(2m+3)\cdots(6m-1)}{m!(144)^{m}},$ $\displaystyle\beta_{m}$ $\displaystyle=-\frac{6m+1}{6m-1}\alpha_{m}.$ Defines: $\alpha_{m}$: coefficients (locally) and $\beta_{m}$: coefficients (locally) Symbols: $!$: factorial (as in $n!$) Permalink: http://dlmf.nist.gov/12.10.E43 Encodings: TeX, TeX, pMML, pMML, png, png

The coefficients $C_{s}(\zeta)$ and $D_{s}(\zeta)$ in (12.10.36) and (12.10.38) are given by

 12.10.44 $\displaystyle C_{s}(\zeta)$ $\displaystyle=\chi(\zeta)A_{s}(\zeta)+A^{\prime}_{s}(\zeta)+\zeta B_{s}(\zeta),$ $\displaystyle D_{s}(\zeta)$ $\displaystyle=A_{s}(\zeta)+\chi(\zeta)B_{s-1}(\zeta)+B^{\prime}_{s-1}(\zeta),$

where

 12.10.45 $\chi(\zeta)=\frac{\phi^{\prime}(\zeta)}{\phi(\zeta)}=\frac{1-2t(\phi(\zeta))^{% 6}}{4\zeta}.$ Defines: $\chi(\zeta)$ (locally) Symbols: $\zeta$: change of variable and $\phi(\zeta)$: function Permalink: http://dlmf.nist.gov/12.10.E45 Encodings: TeX, pMML, png

Explicitly,

 12.10.46 $\displaystyle\zeta C_{s}(\zeta)$ $\displaystyle=-\zeta^{-3s}\sum_{m=0}^{2s+1}\beta_{m}(\phi(\zeta))^{6(2s-m+1)}v% _{2s-m+1}(t),$ $\displaystyle D_{s}(\zeta)$ $\displaystyle=\zeta^{-3s}\sum_{m=0}^{2s}\alpha_{m}(\phi(\zeta))^{6(2s-m)}v_{2s% -m}(t),$

where $v_{k}(t)$ is as in §12.10(ii).

### Modified Expansions

The expansions (12.10.35)–(12.10.38) can be modified, again see Temme (2000), and the new expansions hold if either or both $\mu$ and $t$ tend to infinity. This is provable by the methods used in §10.41(v).

## §12.10(viii) Negative $a$, $-\infty. Expansions in Terms of Airy Functions

When $\mu\to\infty$, asymptotic expansions for $\mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right)$ and $\mathop{V\/}\nolimits\!\left(-\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right)$ that are uniform for $t\in[-1+\delta,\infty)$ are obtained by substitution into (12.2.15) and (12.2.16) by means of (12.10.35) and (12.10.37). Similarly for $\mathop{U\/}\nolimits'\!\left(-\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right)$ and $\mathop{V\/}\nolimits'\!\left(-\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right)$.