§12.10 Uniform Asymptotic Expansions for Large Parameter

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		$a$	$=\pm \frac{1}{2}{\mu }^2,$
		$x$	$=\mu t\sqrt{2},$
ⓘ 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 See also: Annotations for §12.10(i), §12.10 and Ch.12

(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 See also: Annotations for §12.10(i), §12.10 and Ch.12

With the upper sign in (12.10.2), expansions can be constructed for large $\mu$ in terms of elementary functions that are uniform for $t\in (-\mathrm{\infty },\mathrm{\infty })$ (§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.

As $a\to -\mathrm{\infty }$

12.10.3		$$U(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim \frac{g(\mu ){\mathrm{e}}^{-{\mu }^2\xi }}{{(t^2-1)}^{\frac{1}{4}}}\sum _{s=0}^{\mathrm{\infty }}\frac{{𝒜}_s(t)}{{\mu }^{2s}},$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $s$: nonnegative integer, $\xi$, ${𝒜}_s(t)$: coefficients and $g(\mu )$: expansion Referenced by: §12.10(ii), §12.10(iii), §12.10(vi), §12.10(vi) Permalink: http://dlmf.nist.gov/12.10.E3 Encodings: TeX, pMML, png See also: Annotations for §12.10(ii), §12.10 and Ch.12

12.10.4		$$U^{\prime }(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim -\frac{\mu }{\sqrt{2}}g(\mu ){(t^2-1)}^{\frac{1}{4}}{\mathrm{e}}^{-{\mu }^2\xi }\sum _{s=0}^{\mathrm{\infty }}\frac{{ℬ}_s(t)}{{\mu }^{2s}},$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $s$: nonnegative integer, $\xi$, ${ℬ}_s(t)$: coefficients and $g(\mu )$: expansion Referenced by: §12.10(ii) Permalink: http://dlmf.nist.gov/12.10.E4 Encodings: TeX, pMML, png See also: Annotations for §12.10(ii), §12.10 and Ch.12

12.10.5		$$V(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim \frac{2g(\mu )}{\mathrm{\Gamma }\left(\frac{1}{2}+\frac{1}{2}{\mu }^2\right)}\frac{{\mathrm{e}}^{{\mu }^2\xi }}{{(t^2-1)}^{\frac{1}{4}}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{𝒜}_s(t)}{{\mu }^{2s}},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $V(a,z)$: parabolic cylinder function, $s$: nonnegative integer, $\xi$, ${𝒜}_s(t)$: coefficients and $g(\mu )$: expansion Referenced by: §12.10(iii) Permalink: http://dlmf.nist.gov/12.10.E5 Encodings: TeX, pMML, png See also: Annotations for §12.10(ii), §12.10 and Ch.12

12.10.6		$$V^{\prime }(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim \frac{\sqrt{2}\mu g(\mu )}{\mathrm{\Gamma }\left(\frac{1}{2}+\frac{1}{2}{\mu }^2\right)}{(t^2-1)}^{\frac{1}{4}}{\mathrm{e}}^{{\mu }^2\xi }\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{ℬ}_s(t)}{{\mu }^{2s}},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $V(a,z)$: parabolic cylinder function, $s$: nonnegative integer, $\xi$, ${ℬ}_s(t)$: coefficients and $g(\mu )$: expansion Permalink: http://dlmf.nist.gov/12.10.E6 Encodings: TeX, pMML, png See also: Annotations for §12.10(ii), §12.10 and Ch.12

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

12.10.7		$$\xi =\frac{1}{2}t\sqrt{t^2-1}-\frac{1}{2}\ln \left(t+\sqrt{t^2-1}\right).$$
ⓘ Defines: $\xi$ (locally) Symbols: $\ln z$: 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 See also: Annotations for §12.10(ii), §12.10 and Ch.12

The coefficients are given by

12.10.8		$${𝒜}_s(t)=\frac{u_s(t)}{{(t^2-1)}^{\frac{3}{2}s}},{ℬ}_s(t)=\frac{v_s(t)}{{(t^2-1)}^{\frac{3}{2}s}},$$
ⓘ Defines: ${𝒜}_s(t)$: coefficients (locally) and ${ℬ}_s(t)$: 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 See also: Annotations for §12.10(ii), §12.10 and Ch.12

where $u_s(t)$ and $v_s(t)$ are polynomials in $t$ of degree $3s$, ($s$ odd), $3s-2$ ($s$ even, $s\ge 2$). For $s=0,1,2$,

12.10.9		$u_0(t)$	$=1,$
		$u_1(t)$	$={\displaystyle \frac{t(t^2-6)}{24}},$
		$u_2(t)$	$={\displaystyle \frac{-9t^4+249t^2+145}{1152}},$
ⓘ 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 See also: Annotations for §12.10(ii), §12.10 and Ch.12

12.10.10		$v_0(t)$	$=1,$
		$v_1(t)$	$={\displaystyle \frac{t(t^2+6)}{24}},$
		$v_2(t)$	$={\displaystyle \frac{15t^4-327t^2-143}{1152}}.$
ⓘ 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 See also: Annotations for §12.10(ii), §12.10 and Ch.12

Higher polynomials $u_s(t)$ can be calculated from the recurrence relation

12.10.11		$$(t^2-1)u_s^{\prime }(t)-3st{u}_s(t)=r_{s-1}(t),$$
ⓘ Symbols: $s$: nonnegative integer, $u_s(t)$: solution and $r_s(t)$: polynomial Permalink: http://dlmf.nist.gov/12.10.E11 Encodings: TeX, pMML, png See also: Annotations for §12.10(ii), §12.10 and Ch.12

where

12.10.12		$$8r_s(t)=(3t^2+2)u_s(t)-12(s+1)t{r}_{s-1}(t)+4(t^2-1)r_{s-1}^{\prime }(t),$$
ⓘ 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 See also: Annotations for §12.10(ii), §12.10 and Ch.12

and the $v_s(t)$ then follow from

12.10.13		$$v_s(t)=u_s(t)+\frac{1}{2}t{u}_{s-1}(t)-r_{s-2}(t).$$
ⓘ Symbols: $s$: nonnegative integer, $u_s(t)$: solution, $v_s(t)$: solution and $r_s(t)$: polynomial Permalink: http://dlmf.nist.gov/12.10.E13 Encodings: TeX, pMML, png See also: Annotations for §12.10(ii), §12.10 and Ch.12

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

12.10.14		$$g(\mu )\sim h(\mu )\left(1+\frac{1}{2}\sum _{s=1}^{\mathrm{\infty }}\frac{{\gamma }_s}{{(\frac{1}{2}{\mu }^2)}^s}\right),$$
ⓘ Defines: $g(\mu )$: expansion (locally) Symbols: $\sim$: Poincaré asymptotic expansion, $s$: 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 See also: Annotations for §12.10(ii), §12.10 and Ch.12

where

12.10.15		$$h(\mu )=2^{-\frac{1}{4}{\mu }^2-\frac{1}{4}}{\mathrm{e}}^{-\frac{1}{4}{\mu }^2}{\mu }^{\frac{1}{2}{\mu }^2-\frac{1}{2}},$$
ⓘ Defines: $h(\mu )$: expansion (locally) Symbols: $\mathrm{e}$: base of natural logarithm Referenced by: §12.10(vi) Permalink: http://dlmf.nist.gov/12.10.E15 Encodings: TeX, pMML, png See also: Annotations for §12.10(ii), §12.10 and Ch.12

and the coefficients ${\gamma }_s$ are defined by

12.10.16		$$\mathrm{\Gamma }\left(\frac{1}{2}+z\right)\sim \sqrt{2\pi }{\mathrm{e}}^{-z}{z}^z\sum _{s=0}^{\mathrm{\infty }}\frac{{\gamma }_s}{z^s};$$
ⓘ Defines: ${\gamma }_s$: coefficients (locally) Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $s$: nonnegative integer Permalink: http://dlmf.nist.gov/12.10.E16 Encodings: TeX, pMML, png See also: Annotations for §12.10(ii), §12.10 and Ch.12

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

12.10.17		${\gamma }_0$	$=1,$
		${\gamma }_1$	$=-\frac{1}{24},$
		${\gamma }_2$	$=\frac{1}{1152},$
		${\gamma }_3$	$=\frac{1003}{\mathrm{4\hspace{0.33em}14720}},$
		${\gamma }_4$	$=-\frac{4027}{\mathrm{398\hspace{0.33em}13120}}.$
ⓘ 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 See also: Annotations for §12.10(ii), §12.10 and Ch.12

When $\mu \to \mathrm{\infty }$, asymptotic expansions for the functions $U(-\frac{1}{2}{\mu }^2,-\mu t\sqrt{2})$ and $V(-\frac{1}{2}{\mu }^2,-\mu t\sqrt{2})$ that are uniform for $t\in [1+\delta ,\mathrm{\infty })$ are obtainable by substitution into (12.2.15) and (12.2.16) by means of (12.10.3) and (12.10.5). Similarly for $U^{\prime }(-\frac{1}{2}{\mu }^2,-\mu t\sqrt{2})$ and $V^{\prime }(-\frac{1}{2}{\mu }^2,-\mu t\sqrt{2})$.

As $a\to -\mathrm{\infty }$

12.10.18		$U(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})$	$\sim {\displaystyle \frac{2g(\mu )}{{(1-t^2)}^{\frac{1}{4}}}}\left(\cos \kappa {\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{(-1)}^s{\displaystyle \frac{{\widetilde{𝒜}}_{2s}(t)}{{\mu }^{4s}}}-\sin \kappa {\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{(-1)}^s{\displaystyle \frac{{\widetilde{𝒜}}_{2s+1}(t)}{{\mu }^{4s+2}}}\right),$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $U(a,z)$: parabolic cylinder function, $\sin z$: sine function, $s$: nonnegative integer, $\kappa$, ${\widetilde{𝒜}}_s(t)$: coefficients and $g(\mu )$: expansion Permalink: http://dlmf.nist.gov/12.10.E18 Encodings: TeX, pMML, png See also: Annotations for §12.10(iv), §12.10 and Ch.12
12.10.19		$U^{\prime }(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})$	$\sim \mu \sqrt{2}g(\mu ){(1-t^2)}^{\frac{1}{4}}\left(\sin \kappa {\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{(-1)}^s{\displaystyle \frac{{\widetilde{ℬ}}_{2s}(t)}{{\mu }^{4s}}}+\cos \kappa {\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{(-1)}^s{\displaystyle \frac{{\widetilde{ℬ}}_{2s+1}(t)}{{\mu }^{4s+2}}}\right),$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $U(a,z)$: parabolic cylinder function, $\sin z$: sine function, $s$: nonnegative integer, $\kappa$, ${\widetilde{ℬ}}_s(t)$: coefficients and $g(\mu )$: expansion Permalink: http://dlmf.nist.gov/12.10.E19 Encodings: TeX, pMML, png See also: Annotations for §12.10(iv), §12.10 and Ch.12
12.10.20		$V(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})$	$\sim {\displaystyle \frac{2g(\mu )}{\mathrm{\Gamma }\left(\frac{1}{2}+\frac{1}{2}{\mu }^2\right){(1-t^2)}^{\frac{1}{4}}}}\left(\cos \chi {\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{(-1)}^s{\displaystyle \frac{{\widetilde{𝒜}}_{2s}(t)}{{\mu }^{4s}}}-\sin \chi {\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{(-1)}^s{\displaystyle \frac{{\widetilde{𝒜}}_{2s+1}(t)}{{\mu }^{4s+2}}}\right),$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $V(a,z)$: parabolic cylinder function, $\sin z$: sine function, $s$: nonnegative integer, $\chi$, ${\widetilde{𝒜}}_s(t)$: coefficients and $g(\mu )$: expansion Permalink: http://dlmf.nist.gov/12.10.E20 Encodings: TeX, pMML, png See also: Annotations for §12.10(iv), §12.10 and Ch.12
12.10.21		$V^{\prime }(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})$	$\sim {\displaystyle \frac{\mu \sqrt{2}g(\mu ){(1-t^2)}^{\frac{1}{4}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\frac{1}{2}{\mu }^2\right)}}\left(\sin \chi {\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{(-1)}^s{\displaystyle \frac{{\widetilde{ℬ}}_{2s}(t)}{{\mu }^{4s}}}+\cos \chi {\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{(-1)}^s{\displaystyle \frac{{\widetilde{ℬ}}_{2s+1}(t)}{{\mu }^{4s+2}}}\right),$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $V(a,z)$: parabolic cylinder function, $\sin z$: sine function, $s$: nonnegative integer, $\chi$, ${\widetilde{ℬ}}_s(t)$: coefficients and $g(\mu )$: expansion Permalink: http://dlmf.nist.gov/12.10.E21 Encodings: TeX, pMML, png See also: Annotations for §12.10(iv), §12.10 and Ch.12

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

12.10.22		$\kappa$	$={\mu }^2\eta -\frac{1}{4}\pi ,$
		$\chi$	$={\mu }^2\eta +\frac{1}{4}\pi ,$
ⓘ Defines: $\kappa$ (locally) and $\chi$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter and $\eta$ Permalink: http://dlmf.nist.gov/12.10.E22 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §12.10(iv), §12.10 and Ch.12

where

12.10.23		$$\eta =\frac{1}{2}\arccos t-\frac{1}{2}t\sqrt{1-t^2},$$
ⓘ Defines: $\eta$ (locally) Symbols: $\arccos z$: arccosine function Referenced by: §12.10(vii), §12.14(ix) Permalink: http://dlmf.nist.gov/12.10.E23 Encodings: TeX, pMML, png See also: Annotations for §12.10(iv), §12.10 and Ch.12

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

12.10.24		${\widetilde{𝒜}}_s(t)$	$={\displaystyle \frac{u_s(t)}{{(1-t^2)}^{\frac{3}{2}s}}},$
		${\widetilde{ℬ}}_s(t)$	$={\displaystyle \frac{v_s(t)}{{(1-t^2)}^{\frac{3}{2}s}}};$
ⓘ Defines: ${\widetilde{𝒜}}_s(t)$: coefficients (locally) and ${\widetilde{ℬ}}_s(t)$: coefficients (locally) 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 See also: Annotations for §12.10(iv), §12.10 and Ch.12

compare (12.10.8).

As $a\to \mathrm{\infty }$

12.10.25		$$U(\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim \frac{\overline{g}(\mu ){\mathrm{e}}^{-{\mu }^2\overline{\xi }}}{{(t^2+1)}^{\frac{1}{4}}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\overline{u}}_s(t)}{{(t^2+1)}^{\frac{3}{2}s}}\frac{1}{{\mu }^{2s}},$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $s$: nonnegative integer, $\overline{\xi }$, ${\overline{u}}_s(t)$: solution and $\overline{g}(\mu )$: expansion Permalink: http://dlmf.nist.gov/12.10.E25 Encodings: TeX, pMML, png See also: Annotations for §12.10(v), §12.10 and Ch.12

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

12.10.26		$$\overline{\xi }=\frac{1}{2}t\sqrt{t^2+1}+\frac{1}{2}\ln \left(t+\sqrt{t^2+1}\right),$$
ⓘ Defines: $\overline{\xi }$ (locally) Symbols: $\ln z$: principal branch of logarithm function Permalink: http://dlmf.nist.gov/12.10.E26 Encodings: TeX, pMML, png See also: Annotations for §12.10(v), §12.10 and Ch.12

12.10.27		$${\overline{u}}_s(t)={\mathrm{i}}^s{u}_s(-\mathrm{i}t),$$
ⓘ Defines: ${\overline{u}}_s(t)$: solution (locally) Symbols: $\mathrm{i}$: imaginary unit, $s$: nonnegative integer and $u_s(t)$: solution Permalink: http://dlmf.nist.gov/12.10.E27 Encodings: TeX, pMML, png See also: Annotations for §12.10(v), §12.10 and Ch.12

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

12.10.28		$$\overline{g}(\mu )\sim \frac{1}{\mu \sqrt{2}h(\mu )}\left(1+\frac{1}{2}\sum _{s=1}^{\mathrm{\infty }}{(-1)}^s\frac{{\gamma }_s}{{(\frac{1}{2}{\mu }^2)}^s}\right),$$
ⓘ Defines: $\overline{g}(\mu )$: expansion (locally) Symbols: $\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $h(\mu )$: expansion and ${\gamma }_s$: coefficients Permalink: http://dlmf.nist.gov/12.10.E28 Encodings: TeX, pMML, png See also: Annotations for §12.10(v), §12.10 and Ch.12

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

With the same conditions

12.10.29		$$U^{\prime }(\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim -\frac{\mu }{\sqrt{2}}\overline{g}(\mu ){(t^2+1)}^{\frac{1}{4}}{\mathrm{e}}^{-{\mu }^2\overline{\xi }}\sum _{s=0}^{\mathrm{\infty }}\frac{{\overline{v}}_s(t)}{{(t^2+1)}^{\frac{3}{2}s}}\frac{1}{{\mu }^{2s}},$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $s$: nonnegative integer, $\overline{\xi }$, $\overline{g}(\mu )$: expansion and ${\overline{v}}_s(t)$: solution Permalink: http://dlmf.nist.gov/12.10.E29 Encodings: TeX, pMML, png See also: Annotations for §12.10(v), §12.10 and Ch.12

where

12.10.30		$${\overline{v}}_s(t)={\mathrm{i}}^s{v}_s(-\mathrm{i}t).$$
ⓘ Defines: ${\overline{v}}_s(t)$: solution (locally) Symbols: $\mathrm{i}$: imaginary unit, $s$: nonnegative integer and $v_s(t)$: solution Permalink: http://dlmf.nist.gov/12.10.E30 Encodings: TeX, pMML, png See also: Annotations for §12.10(v), §12.10 and Ch.12

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

12.10.31		$$U(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim \frac{h(\mu ){\mathrm{e}}^{-{\mu }^2\xi }}{{(t^2-1)}^{\frac{1}{4}}}\sum _{s=0}^{\mathrm{\infty }}\frac{{𝖠}_s(\tau )}{{\mu }^{2s}},$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $s$: nonnegative integer, $\xi$, $\tau$, ${𝖠}_s(\tau )$: coefficients and $h(\mu )$: expansion Referenced by: §12.10(vi) Permalink: http://dlmf.nist.gov/12.10.E31 Encodings: TeX, pMML, png See also: Annotations for §12.10(vi), §12.10 and Ch.12

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 See also: Annotations for §12.10(vi), §12.10 and Ch.12

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

12.10.33		$${𝖠}_{s+1}(\tau )=-4{\tau }^2{(\tau +1)}^2\frac{d}{d\tau }{𝖠}_s(\tau )-\frac{1}{4}{\int }_0^{\tau }\left(20u^2+20u+3\right){𝖠}_s(u)du,$$
$s=0,1,2,\mathrm{\dots }$,
ⓘ Defines: ${𝖠}_s(\tau )$: 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 See also: Annotations for §12.10(vi), §12.10 and Ch.12

starting with ${𝖠}_0(\tau )=1$. Explicitly,

12.10.34		${𝖠}_1(\tau )$	$=-\frac{1}{12}\tau (20{\tau }^2+30\tau +9),$
		${𝖠}_2(\tau )$	$=\frac{1}{288}{\tau }^2(6160{\tau }^4+18480{\tau }^3+19404{\tau }^2+8028\tau +945).$
ⓘ Symbols: $\tau$ and ${𝖠}_s(\tau )$: coefficients Permalink: http://dlmf.nist.gov/12.10.E34 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §12.10(vi), §12.10 and Ch.12

The modified expansion (12.10.31) shares the property of (12.10.3) that it applies when $\mu \to \mathrm{\infty }$ uniformly with respect to $t\in [1+\delta ,\mathrm{\infty })$. In addition, it enjoys a double asymptotic property: it holds if either or both $\mu$ and $t$ tend to infinity. Observe that if $t\to \mathrm{\infty }$, then ${𝖠}_s(\tau )=O\left(t^{-2s}\right)$, whereas ${𝒜}_s(t)=O(1)$ or $O\left(t^{-2}\right)$ 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).

For additional information see Temme (2000). See also Olver (1997b, pp. 206–208) and Jones (2006).

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

12.10.35		$U(-{\displaystyle \frac{1}{2}}{\mu }^2,\mu t\sqrt{2})$	$\sim 2{\pi }^{\frac{1}{2}}{\mu }^{\frac{1}{3}}g(\mu )\varphi (\zeta )\left(\mathrm{Ai}\left({\mu }^{\frac{4}{3}}\zeta \right){\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{\displaystyle \frac{A_s(\zeta )}{{\mu }^{4s}}}+{\displaystyle \frac{{\mathrm{Ai}}^{\prime }\left({\mu }^{\frac{4}{3}}\zeta \right)}{{\mu }^{\frac{8}{3}}}}{\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{\displaystyle \frac{B_s(\zeta )}{{\mu }^{4s}}}\right),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $U(a,z)$: parabolic cylinder function, $s$: nonnegative integer, $\zeta$: change of variable, $\varphi (\zeta )$: function, $A_s(\zeta )$: coefficients, $B_s(\zeta )$: coefficients and $g(\mu )$: expansion Referenced by: §12.10(vii), §12.10(viii) Permalink: http://dlmf.nist.gov/12.10.E35 Encodings: TeX, pMML, png See also: Annotations for §12.10(vii), §12.10 and Ch.12
12.10.36		$U^{\prime }(-{\displaystyle \frac{1}{2}}{\mu }^2,\mu t\sqrt{2})$	$\sim {\displaystyle \frac{{(2\pi )}^{\frac{1}{2}}{\mu }^{\frac{2}{3}}g(\mu )}{\varphi (\zeta )}}\left({\displaystyle \frac{\mathrm{Ai}\left({\mu }^{\frac{4}{3}}\zeta \right)}{{\mu }^{\frac{4}{3}}}}{\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{\displaystyle \frac{C_s(\zeta )}{{\mu }^{4s}}}+{\mathrm{Ai}}^{\prime }\left({\mu }^{\frac{4}{3}}\zeta \right){\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{\displaystyle \frac{D_s(\zeta )}{{\mu }^{4s}}}\right),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $U(a,z)$: parabolic cylinder function, $s$: nonnegative integer, $\zeta$: change of variable, $\varphi (\zeta )$: function, $C_s(\zeta )$: coefficients, $D_s(\zeta )$: coefficients and $g(\mu )$: expansion Referenced by: §12.10(vii) Permalink: http://dlmf.nist.gov/12.10.E36 Encodings: TeX, pMML, png See also: Annotations for §12.10(vii), §12.10 and Ch.12
12.10.37		$V(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})$	$\sim {\displaystyle \frac{2{\pi }^{\frac{1}{2}}{\mu }^{\frac{1}{3}}g(\mu )\varphi (\zeta )}{\mathrm{\Gamma }\left(\frac{1}{2}+\frac{1}{2}{\mu }^2\right)}}\left(\mathrm{Bi}\left({\mu }^{\frac{4}{3}}\zeta \right){\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{\displaystyle \frac{A_s(\zeta )}{{\mu }^{4s}}}+{\displaystyle \frac{{\mathrm{Bi}}^{\prime }\left({\mu }^{\frac{4}{3}}\zeta \right)}{{\mu }^{\frac{8}{3}}}}{\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{\displaystyle \frac{B_s(\zeta )}{{\mu }^{4s}}}\right),$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $V(a,z)$: parabolic cylinder function, $s$: nonnegative integer, $\zeta$: change of variable, $\varphi (\zeta )$: function, $A_s(\zeta )$: coefficients, $B_s(\zeta )$: coefficients and $g(\mu )$: expansion Referenced by: §12.10(viii) Permalink: http://dlmf.nist.gov/12.10.E37 Encodings: TeX, pMML, png See also: Annotations for §12.10(vii), §12.10 and Ch.12
12.10.38		$V^{\prime }(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})$	$\sim {\displaystyle \frac{{(2\pi )}^{\frac{1}{2}}{\mu }^{\frac{2}{3}}g(\mu )}{\varphi (\zeta )\mathrm{\Gamma }\left(\frac{1}{2}+\frac{1}{2}{\mu }^2\right)}}\left({\displaystyle \frac{\mathrm{Bi}\left({\mu }^{\frac{4}{3}}\zeta \right)}{{\mu }^{\frac{4}{3}}}}{\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{\displaystyle \frac{C_s(\zeta )}{{\mu }^{4s}}}+{\mathrm{Bi}}^{\prime }\left({\mu }^{\frac{4}{3}}\zeta \right){\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{\displaystyle \frac{D_s(\zeta )}{{\mu }^{4s}}}\right).$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $V(a,z)$: parabolic cylinder function, $s$: nonnegative integer, $\zeta$: change of variable, $\varphi (\zeta )$: function, $C_s(\zeta )$: coefficients, $D_s(\zeta )$: coefficients and $g(\mu )$: expansion Referenced by: §12.10(vii), §12.10(vii) Permalink: http://dlmf.nist.gov/12.10.E38 Encodings: TeX, pMML, png See also: Annotations for §12.10(vii), §12.10 and Ch.12

The variable $\zeta$ is defined by

12.10.39		$\frac{2}{3}{\zeta }^{\frac{3}{2}}$	$=\xi ,1\le t,\text{(}\zeta \ge 0\text{)};$
		$\frac{2}{3}{(-\zeta )}^{\frac{3}{2}}$
	ⓘ 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 See also: Annotations for §12.10(vii), §12.10 and Ch.12

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

12.10.40		$$\varphi (\zeta )={\left(\frac{\zeta }{t^2-1}\right)}^{\frac{1}{4}}.$$
ⓘ Defines: $\varphi (\zeta )$: function (locally) Symbols: $\zeta$: change of variable Referenced by: §12.10(vii) Permalink: http://dlmf.nist.gov/12.10.E40 Encodings: TeX, pMML, png See also: Annotations for §12.10(vii), §12.10 and Ch.12

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		$$t=1+w-\frac{1}{10}{w}^2+\frac{11}{350}{w}^3-\frac{823}{63000}{w}^4+\frac{\mathrm{1\hspace{0.33em}50653}}{\mathrm{242\hspace{0.33em}55000}}{w}^5+\mathrm{\cdots },$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter and $\zeta$: change of variable Referenced by: §12.11(iii), §12.11(iii) Permalink: http://dlmf.nist.gov/12.10.E41 Encodings: TeX, pMML, png See also: Annotations for §12.10(vii), §12.10 and Ch.12

For $g(\mu )$ see (12.10.14). The coefficients $A_s(\zeta )$ and $B_s(\zeta )$ are given by

12.10.42		$A_s(\zeta )$	$={\zeta }^{-3s}{\displaystyle \sum _{m=0}^{2s}}{\beta }_m{(\varphi (\zeta ))}^{6(2s-m)}{u}_{2s-m}(t),$
		${\zeta }^2{B}_s(\zeta )$	$=-{\zeta }^{-3s}{\displaystyle \sum _{m=0}^{2s+1}}{\alpha }_m{(\varphi (\zeta ))}^{6(2s-m+1)}{u}_{2s-m+1}(t),$
ⓘ Symbols: $s$: nonnegative integer, $\zeta$: change of variable, $\varphi (\zeta )$: function, $A_s(\zeta )$: coefficients, $B_s(\zeta )$: coefficients, ${\alpha }_m$: coefficients, ${\beta }_m$: coefficients and $u_s(t)$: solution Referenced by: §10.20(i), §12.10(vii) Permalink: http://dlmf.nist.gov/12.10.E42 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §12.10(vii), §12.10 and Ch.12

where $\varphi (\zeta )$ is as in (12.10.40), $u_k(t)$ is as in §12.10(ii), ${\alpha }_0=1$, and

12.10.43		${\alpha }_m$	$={\displaystyle \frac{(2m+1)(2m+3)\mathrm{\cdots }(6m-1)}{m!{(144)}^m}},$
		${\beta }_m$	$=-{\displaystyle \frac{6m+1}{6m-1}}{\alpha }_m.$
ⓘ 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 See also: Annotations for §12.10(vii), §12.10 and Ch.12

The coefficients $C_s(\zeta )$ and $D_s(\zeta )$ in (12.10.36) and (12.10.38) are given by

12.10.44		$C_s(\zeta )$	$=\chi (\zeta )A_s(\zeta )+A_s^{\prime }(\zeta )+\zeta B_s(\zeta ),$
		$D_s(\zeta )$	$=A_s(\zeta )+\chi (\zeta )B_{s-1}(\zeta )+B_{s-1}^{\prime }(\zeta ),$
ⓘ Symbols: $s$: nonnegative integer, $\zeta$: change of variable, $A_s(\zeta )$: coefficients, $B_s(\zeta )$: coefficients, $C_s(\zeta )$: coefficients, $D_s(\zeta )$: coefficients and $\chi (\zeta )$ Permalink: http://dlmf.nist.gov/12.10.E44 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §12.10(vii), §12.10 and Ch.12

where

12.10.45		$$\chi (\zeta )=\frac{{\varphi }^{\prime }(\zeta )}{\varphi (\zeta )}=\frac{1-2t{(\varphi (\zeta ))}^6}{4\zeta }.$$
ⓘ Defines: $\chi (\zeta )$ (locally) Symbols: $\zeta$: change of variable and $\varphi (\zeta )$: function Permalink: http://dlmf.nist.gov/12.10.E45 Encodings: TeX, pMML, png See also: Annotations for §12.10(vii), §12.10 and Ch.12

Explicitly,

12.10.46		$\zeta C_s(\zeta )$	$=-{\zeta }^{-3s}{\displaystyle \sum _{m=0}^{2s+1}}{\beta }_m{(\varphi (\zeta ))}^{6(2s-m+1)}{v}_{2s-m+1}(t),$
		$D_s(\zeta )$	$={\zeta }^{-3s}{\displaystyle \sum _{m=0}^{2s}}{\alpha }_m{(\varphi (\zeta ))}^{6(2s-m)}{v}_{2s-m}(t),$
ⓘ Symbols: $s$: nonnegative integer, $\zeta$: change of variable, $\varphi (\zeta )$: function, ${\alpha }_m$: coefficients, ${\beta }_m$: coefficients, $C_s(\zeta )$: coefficients, $D_s(\zeta )$: coefficients and $v_s(t)$: solution Referenced by: §10.20(i), §12.10(vii) Permalink: http://dlmf.nist.gov/12.10.E46 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §12.10(vii), §12.10 and Ch.12

where $v_k(t)$ is as in §12.10(ii).