§12.9 Asymptotic Expansions for Large Variable

Keywords:: parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.9

§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).
Permalink:: http://dlmf.nist.gov/12.9.i

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

As $z\to\infty$

12.9.1 $\mathop{U\/}\nolimits\!\left(a,z\right)\sim e^{{-\frac{1}{4}z^{2}}}z^{{-a-\frac{1}{2}}}\sum _{{s=0}}^{\infty}(-1)^{s}\frac{\left(\frac{1}{2}+a\right)_{{2s}}}{s!(2z^{2})^{s}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{3}{4}\pi-\delta(<\tfrac{3}{4}\pi)$ ,

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $z$ : 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

12.9.2 $\mathop{V\/}\nolimits\!\left(a,z\right)\sim\sqrt{\frac{2}{\pi}}e^{{\frac{1}{4}z^{2}}}z^{{a-\frac{1}{2}}}\sum _{{s=0}}^{\infty}\frac{\left(\frac{1}{2}-a\right)_{{2s}}}{s!(2z^{2})^{s}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{4}\pi-\delta(<\tfrac{1}{4}\pi)$ .

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $z$ : 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

12.9.3 $\mathop{U\/}\nolimits\!\left(a,z\right)\sim e^{{-\frac{1}{4}z^{2}}}z^{{-a-\frac{1}{2}}}\sum _{{s=0}}^{\infty}(-1)^{s}\frac{\left(\frac{1}{2}+a\right)_{{2s}}}{s!(2z^{2})^{s}}\pm i\frac{\sqrt{2\pi}}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}+a\right)}e^{{\mp i\pi a}}e^{{\frac{1}{4}z^{2}}}z^{{a-\frac{1}{2}}}\sum _{{s=0}}^{\infty}\frac{\left(\tfrac{1}{2}-a\right)_{{2s}}}{s!(2z^{2})^{s}},$ $\tfrac{1}{4}\pi+\delta\leq\pm\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{5}{4}\pi-\delta$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $z$ : 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

12.9.4 $\mathop{V\/}\nolimits\!\left(a,z\right)\sim\sqrt{\frac{2}{\pi}}e^{{\frac{1}{4}z^{2}}}z^{{a-\frac{1}{2}}}\sum _{{s=0}}^{\infty}\frac{\left(\tfrac{1}{2}-a\right)_{{2s}}}{s!(2z^{2})^{s}}\pm\frac{i}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}-a\right)}e^{{-\frac{1}{4}z^{2}}}z^{{-a-\frac{1}{2}}}\sum _{{s=0}}^{\infty}(-1)^{s}\frac{\left(\tfrac{1}{2}+a\right)_{{2s}}}{s!(2z^{2})^{s}},$ $-\tfrac{1}{4}\pi+\delta\leq\pm\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{3}{4}\pi-\delta$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $z$ : 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

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

Keywords:: parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.9.ii

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