12 Parabolic Cylinder Functions Properties 12.7 Relations to Other Functions 12.9 Asymptotic Expansions for Large Variable

§12.8 Recurrence Relations and Derivatives

ⓘ

Permalink:: http://dlmf.nist.gov/12.8
See also:: Annotations for Ch.12

§12.8(i) Recurrence Relations

ⓘ

Keywords:: parabolic cylinder functions, recurrence relations
Notes:: See Miller (1955, pp. 65).
Permalink:: http://dlmf.nist.gov/12.8.i
See also:: Annotations for §12.8 and Ch.12

12.8.1	$\displaystyle zU\left(a,z\right)-U\left(a-1,z\right)+(a+\tfrac{1}{2})U\left(a+% 1,z\right)$	$\displaystyle=0,$
ⓘ Symbols: $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter A&S Ref: 19.6.4 Referenced by: §12.8(i) Permalink: http://dlmf.nist.gov/12.8.E1 Encodings: TeX, pMML, png See also: Annotations for §12.8(i), §12.8 and Ch.12
12.8.2	$\displaystyle U'\left(a,z\right)+\tfrac{1}{2}zU\left(a,z\right)+(a+\tfrac{1}{2% })U\left(a+1,z\right)$	$\displaystyle=0,$
ⓘ Symbols: $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter A&S Ref: 19.6.1 Permalink: http://dlmf.nist.gov/12.8.E2 Encodings: TeX, pMML, png See also: Annotations for §12.8(i), §12.8 and Ch.12
12.8.3	$\displaystyle U'\left(a,z\right)-\tfrac{1}{2}zU\left(a,z\right)+U\left(a-1,z\right)$	$\displaystyle=0,$
ⓘ Symbols: $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter A&S Ref: 19.6.2 Permalink: http://dlmf.nist.gov/12.8.E3 Encodings: TeX, pMML, png See also: Annotations for §12.8(i), §12.8 and Ch.12
12.8.4	$\displaystyle 2U'\left(a,z\right)+U\left(a-1,z\right)+(a+\tfrac{1}{2})U\left(a% +1,z\right)$	$\displaystyle=0.$
ⓘ Symbols: $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter A&S Ref: 19.6.3 Referenced by: §12.8(i) Permalink: http://dlmf.nist.gov/12.8.E4 Encodings: TeX, pMML, png See also: Annotations for §12.8(i), §12.8 and Ch.12

(12.8.1)–(12.8.4) are also satisfied by $\overline{U}\left(a,z\right)$ .

12.8.5	$\displaystyle zV\left(a,z\right)-V\left(a+1,z\right)+(a-\tfrac{1}{2})V\left(a-% 1,z\right)$	$\displaystyle=0,$
ⓘ Symbols: $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter A&S Ref: 19.6.8 Permalink: http://dlmf.nist.gov/12.8.E5 Encodings: TeX, pMML, png See also: Annotations for §12.8(i), §12.8 and Ch.12
12.8.6	$\displaystyle V'\left(a,z\right)-\tfrac{1}{2}zV\left(a,z\right)-(a-\tfrac{1}{2% })V\left(a-1,z\right)$	$\displaystyle=0,$
ⓘ Symbols: $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter A&S Ref: 19.6.5 Permalink: http://dlmf.nist.gov/12.8.E6 Encodings: TeX, pMML, png See also: Annotations for §12.8(i), §12.8 and Ch.12
12.8.7	$\displaystyle V'\left(a,z\right)+\tfrac{1}{2}zV\left(a,z\right)-V\left(a+1,z\right)$	$\displaystyle=0,$
ⓘ Symbols: $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter A&S Ref: 19.6.6 Permalink: http://dlmf.nist.gov/12.8.E7 Encodings: TeX, pMML, png See also: Annotations for §12.8(i), §12.8 and Ch.12
12.8.8	$\displaystyle 2V'\left(a,z\right)-V\left(a+1,z\right)-(a-\tfrac{1}{2})V\left(a% -1,z\right)$	$\displaystyle=0.$
ⓘ Symbols: $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter A&S Ref: 19.6.7 Permalink: http://dlmf.nist.gov/12.8.E8 Encodings: TeX, pMML, png See also: Annotations for §12.8(i), §12.8 and Ch.12

§12.8(ii) Derivatives

ⓘ

Keywords:: derivatives, parabolic cylinder functions
Notes:: (12.8.9), (12.8.10), (12.8.11), and (12.8.12) can be obtained from (12.5.1), (12.5.6), (12.5.7), and (12.5.9), respectively.
Referenced by:: §12.13(i)
Permalink:: http://dlmf.nist.gov/12.8.ii
See also:: Annotations for §12.8 and Ch.12

For $m=0,1,2,\dots$ ,

12.8.9		$\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^{\frac{1}{4}z^{2}}U\left(a,z% \right)\right)=(-1)^{m}{\left(\tfrac{1}{2}+a\right)_{m}}e^{\frac{1}{4}z^{2}}U% \left(a+m,z\right),$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §12.8(ii) Permalink: http://dlmf.nist.gov/12.8.E9 Encodings: TeX, pMML, png See also: Annotations for §12.8(ii), §12.8 and Ch.12

12.8.10		$\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^{-\frac{1}{4}z^{2}}U\left(a,% z\right)\right)=(-1)^{m}e^{-\frac{1}{4}z^{2}}U\left(a-m,z\right),$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §12.8(ii) Permalink: http://dlmf.nist.gov/12.8.E10 Encodings: TeX, pMML, png See also: Annotations for §12.8(ii), §12.8 and Ch.12

12.8.11		$\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^{\frac{1}{4}z^{2}}V\left(a,z% \right)\right)=e^{\frac{1}{4}z^{2}}V\left(a+m,z\right),$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §12.8(ii) Permalink: http://dlmf.nist.gov/12.8.E11 Encodings: TeX, pMML, png See also: Annotations for §12.8(ii), §12.8 and Ch.12

12.8.12		$\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^{-\frac{1}{4}z^{2}}V\left(a,% z\right)\right)=(-1)^{m}{\left(\tfrac{1}{2}-a\right)_{m}}e^{-\frac{1}{4}z^{2}}% V\left(a-m,z\right).$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §12.8(ii) Permalink: http://dlmf.nist.gov/12.8.E12 Encodings: TeX, pMML, png See also: Annotations for §12.8(ii), §12.8 and Ch.12

12.7 Relations to Other Functions 12.9 Asymptotic Expansions for Large Variable

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.

§12.8 Recurrence Relations and Derivatives

Contents

§12.8(i) Recurrence Relations

§12.8(ii) Derivatives

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.