§12.9 Asymptotic Expansions for Large Variable

ⓘ

Keywords:: asymptotic expansions for large variable, parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.9
See also:: Annotations for Ch.12

§12.9(i) Poincaré-Type Expansions
§12.9(ii) Bounds and Re-Expansions for the Remainder Terms

§12.9(i) Poincaré-Type Expansions

ⓘ

Notes:: (12.9.1) is obtained from (12.7.14) and (13.7.3). (12.9.2)–(12.9.4) follow from (12.2.18) and (12.2.20). See also Whittaker (1902) and Whittaker and Watson (1927, pp. 348–349).
Referenced by:: Erratum (V1.0.11) for References
Permalink:: http://dlmf.nist.gov/12.9.i
Addition (effective with 1.0.11):: A sentence and reference to Temme (2015, Chapter 11) was added at the end of this subsection.
See also:: Annotations for §12.9 and Ch.12

Throughout this subsection $\delta$ is an arbitrary small positive constant.

As $z\to\infty$

12.9.1		$U\left(a,z\right)\sim e^{-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}\sum_{s=0}^{% \infty}(-1)^{s}\frac{{\left(\frac{1}{2}+a\right)_{2s}}}{s!(2z^{2})^{s}},$
$\|\operatorname{ph}z\|\leq\tfrac{3}{4}\pi-\delta(<\tfrac{3}{4}\pi)$ ,
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\operatorname{ph}$ : phase, $z$ : complex variable, $s$ : nonnegative integer, $a$ : real or complex parameter and $\delta$ : arbitrary small positive constant A&S Ref: 19.8.1 (modification of) Referenced by: §12.9(i), §12.9(ii) Permalink: http://dlmf.nist.gov/12.9.E1 Encodings: TeX, pMML, png See also: Annotations for §12.9(i), §12.9 and Ch.12

12.9.2		$V\left(a,z\right)\sim\sqrt{\frac{2}{\pi}}e^{\frac{1}{4}z^{2}}z^{a-\frac{1}{2}}% \sum_{s=0}^{\infty}\frac{{\left(\frac{1}{2}-a\right)_{2s}}}{s!(2z^{2})^{s}},$
$\|\operatorname{ph}z\|\leq\tfrac{1}{4}\pi-\delta(<\tfrac{1}{4}\pi)$ .
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\operatorname{ph}$ : phase, $z$ : complex variable, $s$ : nonnegative integer, $a$ : real or complex parameter and $\delta$ : arbitrary small positive constant A&S Ref: 19.8.2 (modification of) Referenced by: §12.9(i), §12.9(ii) Permalink: http://dlmf.nist.gov/12.9.E2 Encodings: TeX, pMML, png See also: Annotations for §12.9(i), §12.9 and Ch.12

12.9.3		$U\left(a,z\right)\sim e^{-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}\sum_{s=0}^{% \infty}(-1)^{s}\frac{{\left(\frac{1}{2}+a\right)_{2s}}}{s!(2z^{2})^{s}}\pm i% \frac{\sqrt{2\pi}}{\Gamma\left(\tfrac{1}{2}+a\right)}e^{\mp i\pi a}e^{\frac{1}% {4}z^{2}}z^{a-\frac{1}{2}}\sum_{s=0}^{\infty}\frac{{\left(\tfrac{1}{2}-a\right% )_{2s}}}{s!(2z^{2})^{s}},$
$\tfrac{1}{4}\pi+\delta\leq\pm\operatorname{ph}z\leq\tfrac{5}{4}\pi-\delta$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\operatorname{ph}$ : phase, $z$ : complex variable, $s$ : nonnegative integer, $a$ : real or complex parameter and $\delta$ : arbitrary small positive constant Permalink: http://dlmf.nist.gov/12.9.E3 Encodings: TeX, pMML, png See also: Annotations for §12.9(i), §12.9 and Ch.12

12.9.4		$V\left(a,z\right)\sim\sqrt{\frac{2}{\pi}}e^{\frac{1}{4}z^{2}}z^{a-\frac{1}{2}}% \sum_{s=0}^{\infty}\frac{{\left(\tfrac{1}{2}-a\right)_{2s}}}{s!(2z^{2})^{s}}% \pm\frac{i}{\Gamma\left(\tfrac{1}{2}-a\right)}e^{-\frac{1}{4}z^{2}}z^{-a-\frac% {1}{2}}\sum_{s=0}^{\infty}(-1)^{s}\frac{{\left(\tfrac{1}{2}+a\right)_{2s}}}{s!% (2z^{2})^{s}},$
$-\tfrac{1}{4}\pi+\delta\leq\pm\operatorname{ph}z\leq\tfrac{3}{4}\pi-\delta$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\operatorname{ph}$ : phase, $z$ : complex variable, $s$ : nonnegative integer, $a$ : real or complex parameter and $\delta$ : arbitrary small positive constant Referenced by: §12.9(i) Permalink: http://dlmf.nist.gov/12.9.E4 Encodings: TeX, pMML, png See also: Annotations for §12.9(i), §12.9 and Ch.12

To obtain approximations for $U\left(a,-z\right)$ and $V\left(a,-z\right)$ as $z\to\infty$ combine the results above with (12.2.15) and (12.2.16). See also Temme (2015, Chapter 11).

§12.9(ii) Bounds and Re-Expansions for the Remainder Terms

ⓘ

Keywords:: asymptotic expansions for large variable, exponentially-improved, parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.9.ii
See also:: Annotations for §12.9 and Ch.12

Bounds and re-expansions for the error term in (12.9.1) can be obtained by use of (12.7.14) and §§13.7(ii), 13.7(iii). Corresponding results for (12.9.2) can be obtained via (12.2.20).

§12.9 Asymptotic Expansions for Large Variable

Contents

§12.9(i) Poincaré-Type Expansions

§12.9(ii) Bounds and Re-Expansions for the Remainder Terms