12 Parabolic Cylinder Functions Properties 12.8 Recurrence Relations and Derivatives 12.10 Uniform Asymptotic Expansions for Large Parameter

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

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : base of exponential function, : : factorial, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, : complex variable, : nonnegative integer, : 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, : base of exponential function, : : factorial, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, : complex variable, : nonnegative integer, : 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 $\mathop{U\/}\nolimits\!\left(a,z\right)\sim e^{{-\frac{1}{4}z^{2}}}z^{{-a-% \frac{1}{2}}}\sum_{{s=0}}^{\infty}(-1)^{s}\frac{\left(\frac{1}{2}+a\right)_{{2% s}}}{s!(2z^{2})^{s}}\pm i\frac{\sqrt{2\pi}}{\mathop{\Gamma\/}\nolimits\!\left(% \tfrac{1}{2}+a\right)}e^{{\mp i\pi a}}e^{{\frac{1}{4}z^{2}}}z^{{a-\frac{1}{2}}% }\sum_{{s=0}}^{\infty}\frac{\left(\tfrac{1}{2}-a\right)_{{2s}}}{s!(2z^{2})^{s}},$ $\tfrac{1}{4}\pi+\delta\leq\pm\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{5}{4}% \pi-\delta$ ,

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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 12.8 Recurrence Relations and Derivatives 12.10 Uniform Asymptotic Expansions for Large Parameter

§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