§12.2 Differential Equations

Keywords:: differential equations, parabolic cylinder functions
Referenced by:: §13.18(iv), §13.20(iii), §13.20(iv), §13.6(iv), §13.8(ii), §14.15(v), §15.12(iii), ¶ ‣ §18.11(i), ¶ ‣ §18.17(iii), §32.10(iv), §32.11(v), §7.18(iv)
Permalink:: http://dlmf.nist.gov/12.2

§12.2(i) Introduction
§12.2(ii) Values at $z=0$
§12.2(iii) Wronskians
§12.2(iv) Reflection Formulas
§12.2(v) Connection Formulas
§12.2(vi) Solution $\mathop{\overline{U}\/}\nolimits\!\left(a,x\right)$ ; Modulus and Phase Functions

§12.2(i) Introduction

Notes:: See Miller (1955, pp. 9–10), Miller (1952), and Olver (1997b, Chapter 5, §3.3).
Defines:: $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function and $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function
Keywords:: differential equations, parabolic cylinder functions
Referenced by:: ¶ ‣ §10.39
Permalink:: http://dlmf.nist.gov/12.2.i

PCFs are solutions of the differential equation

12.2.1 $\frac{{d}^{2}w}{{dz}^{2}}+\left(az^{2}+bz+c\right)w=0,$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $a$ : parameter, $b$ : parameter and $c$ : parameter
A&S Ref:: 19.1.1
Permalink:: http://dlmf.nist.gov/12.2.E1
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with three distinct standard forms

12.2.2 $\frac{{d}^{2}w}{{dz}^{2}}-\left(\tfrac{1}{4}z^{2}+a\right)w=0,$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable and $a$ : parameter
A&S Ref:: 19.1.2
Referenced by:: §12.10(i), §12.14(i), §12.17, §12.2(i), §12.2(i), §12.4, §12.7(iv)
Permalink:: http://dlmf.nist.gov/12.2.E2
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12.2.3 $\frac{{d}^{2}w}{{dz}^{2}}+\left(\tfrac{1}{4}z^{2}-a\right)w=0,$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable and $a$ : parameter
A&S Ref:: 19.1.3
Referenced by:: ¶ ‣ §12.14(vii), ¶ ‣ §12.14(ix), §12.14(i), §12.14(x), §12.14(v), §12.14(ix), §12.17, §12.2(i), §12.2(i)
Permalink:: http://dlmf.nist.gov/12.2.E3
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12.2.4 $\frac{{d}^{2}w}{{dz}^{2}}+\left(\nu+\tfrac{1}{2}-\tfrac{1}{4}z^{2}\right)w=0.$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable and $\nu$ : real or complex parameter
Referenced by:: §12.15, §12.2(i)
Permalink:: http://dlmf.nist.gov/12.2.E4
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Each of these equations is transformable into the others. Standard solutions are $\mathop{U\/}\nolimits\!\left(a,\pm z\right)$ , $\mathop{V\/}\nolimits\!\left(a,\pm z\right)$ , $\mathop{\overline{U}\/}\nolimits\!\left(a,\pm x\right)$ (not complex conjugate), $\mathop{U\/}\nolimits\!\left(-a,\pm iz\right)$ for (12.2.2); $\mathop{W\/}\nolimits\!\left(a,\pm x\right)$ for (12.2.3); $\mathop{D_{{\nu}}\/}\nolimits\!\left(\pm z\right)$ for (12.2.4), where

12.2.5 $\mathop{D_{{\nu}}\/}\nolimits\!\left(z\right)=\mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}-\nu,z\right).$

Symbols:: $\mathop{D_{{\nu}}\/}\nolimits\!\left(z\right)$ : parabolic cylinder function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $z$ : complex variable and $\nu$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/12.2.E5
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All solutions are entire functions of $z$ and entire functions of $a$ or $\nu$ .

For real values of $z$ $(=x)$ , numerically satisfactory pairs of solutions (§2.7(iv)) of (12.2.2) are $\mathop{U\/}\nolimits\!\left(a,x\right)$ and $\mathop{V\/}\nolimits\!\left(a,x\right)$ when $x$ is positive, or $\mathop{U\/}\nolimits\!\left(a,-x\right)$ and $\mathop{V\/}\nolimits\!\left(a,-x\right)$ when $x$ is negative. For (12.2.3) $\mathop{W\/}\nolimits\!\left(a,x\right)$ and $\mathop{W\/}\nolimits\!\left(a,-x\right)$ comprise a numerically satisfactory pair, for all $x\in\Real$ . The solutions $\mathop{W\/}\nolimits\!\left(a,\pm x\right)$ are treated in §12.14.

In $\Complex$ , for $j=0,1,2,3,$ $\mathop{U\/}\nolimits\!\left((-1)^{{j-1}}a,(-i)^{{j-1}}z\right)$ and $\mathop{U\/}\nolimits\!\left((-1)^{j}a,(-i)^{j}z\right)$ comprise a numerically satisfactory pair of solutions in the half-plane $\tfrac{1}{4}(2j-3)\pi\leq\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{1}{4}(2j+1)\pi$ .

§12.2(ii) Values at $z=0$

Notes:: See Miller (1955, p. 17).
Keywords:: parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.2.ii

12.2.6 $\mathop{U\/}\nolimits\!\left(a,0\right)=\frac{\sqrt{\pi}}{2^{{\frac{1}{2}a+\frac{1}{4}}}\mathop{\Gamma\/}\nolimits\!\left(\frac{3}{4}+\frac{1}{2}a\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function and $a$ : real or complex parameter
A&S Ref:: 19.3.5
Referenced by:: §12.4
Permalink:: http://dlmf.nist.gov/12.2.E6
Encodings:: TeX, pMML, png

12.2.7 ${\mathop{U\/}\nolimits^{{\prime}}}\!\left(a,0\right)=-\frac{\sqrt{\pi}}{2^{{\frac{1}{2}a-\frac{1}{4}}}\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{4}+\frac{1}{2}a\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function and $a$ : real or complex parameter
A&S Ref:: 19.3.5
Permalink:: http://dlmf.nist.gov/12.2.E7
Encodings:: TeX, pMML, png

12.2.8 $\mathop{V\/}\nolimits\!\left(a,0\right)=\frac{\pi 2^{{\frac{1}{2}a+\frac{1}{4}}}}{\left(\mathop{\Gamma\/}\nolimits\!\left(\frac{3}{4}-\frac{1}{2}a\right)\right)^{2}\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{4}+\frac{1}{2}a\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function and $a$ : real or complex parameter
A&S Ref:: 19.3.6
Permalink:: http://dlmf.nist.gov/12.2.E8
Encodings:: TeX, pMML, png

12.2.9 ${\mathop{V\/}\nolimits^{{\prime}}}\!\left(a,0\right)=\frac{\pi 2^{{\frac{1}{2}a+\frac{3}{4}}}}{\left(\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{4}-\frac{1}{2}a\right)\right)^{2}\mathop{\Gamma\/}\nolimits\!\left(\frac{3}{4}+\frac{1}{2}a\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function and $a$ : real or complex parameter
A&S Ref:: 19.3.6
Referenced by:: §12.4
Permalink:: http://dlmf.nist.gov/12.2.E9
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§12.2(iii) Wronskians

Notes:: See Miller (1955, p. 64).
Keywords:: parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.2.iii

12.2.10 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{U\/}\nolimits\!\left(a,z\right),\mathop{V\/}\nolimits\!\left(a,z\right)\right\}=\sqrt{2/\pi},$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.4.1
Permalink:: http://dlmf.nist.gov/12.2.E10
Encodings:: TeX, pMML, png

12.2.11 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{U\/}\nolimits\!\left(a,z\right),\mathop{U\/}\nolimits\!\left(a,-z\right)\right\}=\frac{\sqrt{2\pi}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+a\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/12.2.E11
Encodings:: TeX, pMML, png

12.2.12 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{U\/}\nolimits\!\left(a,z\right),\mathop{U\/}\nolimits\!\left(-a,\pm iz\right)\right\}=\mp ie^{{\pm i\pi(\frac{1}{2}a+\frac{1}{4})}}.$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.4.1
Permalink:: http://dlmf.nist.gov/12.2.E12
Encodings:: TeX, pMML, png

§12.2(iv) Reflection Formulas

Notes:: See Miller (1955, pp. 63–64).
Keywords:: parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.2.iv

For $n=0,1,\dots$ ,

12.2.13 $\mathop{U\/}\nolimits\!\left(-n-\tfrac{1}{2},-z\right)=(-1)^{n}\mathop{U\/}\nolimits\!\left(-n-\tfrac{1}{2},z\right),$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $z$ : complex variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/12.2.E13
Encodings:: TeX, pMML, png

12.2.14 $\mathop{V\/}\nolimits\!\left(n+\tfrac{1}{2},-z\right)=(-1)^{n}\mathop{V\/}\nolimits\!\left(n+\tfrac{1}{2},z\right).$

Symbols:: $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $z$ : complex variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/12.2.E14
Encodings:: TeX, pMML, png

§12.2(v) Connection Formulas

Notes:: See Miller (1955, p. 64).
Keywords:: parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.2.v

12.2.15 $\mathop{U\/}\nolimits\!\left(a,-z\right)=-\mathop{\sin\/}\nolimits(\pi a)\mathop{U\/}\nolimits\!\left(a,z\right)+\frac{\pi}{\mathop{\Gamma\/}\nolimits(\frac{1}{2}+a)}\mathop{V\/}\nolimits\!\left(a,z\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable and $a$ : real or complex parameter
Referenced by:: §12.10(iii), §12.10(viii)
Permalink:: http://dlmf.nist.gov/12.2.E15
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12.2.16 $\mathop{V\/}\nolimits\!\left(a,-z\right)=\frac{\mathop{\cos\/}\nolimits(\pi a)}{\mathop{\Gamma\/}\nolimits(\frac{1}{2}-a)}\mathop{U\/}\nolimits\!\left(a,z\right)+\mathop{\sin\/}\nolimits(\pi a)\mathop{V\/}\nolimits\!\left(a,z\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable and $a$ : real or complex parameter
Referenced by:: §12.10(iii), §12.10(viii)
Permalink:: http://dlmf.nist.gov/12.2.E16
Encodings:: TeX, pMML, png

12.2.17 $\sqrt{2\pi}\mathop{U\/}\nolimits\!\left(-a,\pm iz\right)=\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}+a\right)\left(e^{{\mp i\pi(\frac{1}{2}a-\frac{1}{4})}}\mathop{U\/}\nolimits\!\left(a,z\right)+e^{{\pm i\pi(\frac{1}{2}a-\frac{1}{4})}}\mathop{U\/}\nolimits\!\left(a,-z\right)\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.4.6 (modification of)
Permalink:: http://dlmf.nist.gov/12.2.E17
Encodings:: TeX, pMML, png

12.2.18 $\sqrt{2\pi}\mathop{U\/}\nolimits\!\left(a,z\right)=\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}-a\right)\left(e^{{\mp i\pi(\frac{1}{2}a+\frac{1}{4})}}\mathop{U\/}\nolimits\!\left(-a,\pm iz\right)+e^{{\pm i\pi(\frac{1}{2}a+\frac{1}{4})}}\mathop{U\/}\nolimits\!\left(-a,\mp iz\right)\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.4.7 (modification of)
Referenced by:: §12.5(i), §12.9(i)
Permalink:: http://dlmf.nist.gov/12.2.E18
Encodings:: TeX, pMML, png

12.2.19 $\mathop{U\/}\nolimits\!\left(a,z\right)=\pm ie^{{\pm i\pi a}}\mathop{U\/}\nolimits\!\left(a,-z\right)+\frac{\sqrt{2\pi}}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}+a\right)}e^{{\pm i\pi(\frac{1}{2}a-\frac{1}{4})}}\mathop{U\/}\nolimits\!\left(-a,\pm iz\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/12.2.E19
Encodings:: TeX, pMML, png

12.2.20 $\mathop{V\/}\nolimits\!\left(a,z\right)=\frac{\mp i}{\mathop{\Gamma\/}\nolimits(\frac{1}{2}-a)}\mathop{U\/}\nolimits\!\left(a,z\right)+\sqrt{\frac{2}{\pi}}e^{{\mp i\pi(\frac{1}{2}a-\frac{1}{4})}}\mathop{U\/}\nolimits\!\left(-a,\pm iz\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
Referenced by:: §12.9(i), §12.9(ii)
Permalink:: http://dlmf.nist.gov/12.2.E20
Encodings:: TeX, pMML, png

§12.2(vi) Solution $\mathop{\overline{U}\/}\nolimits\!\left(a,x\right)$ ; Modulus and Phase Functions

Notes:: See Miller (1955, p. 72) and Miller (1950).
Defines:: $\mathop{\overline{U}\/}\nolimits\!\left(a,x\right)$ : parabolic cylinder function
Keywords:: parabolic cylinder functions
Referenced by:: §12.14(x)
Permalink:: http://dlmf.nist.gov/12.2.vi

When $z$ $(=x)$ is real the solution $\mathop{\overline{U}\/}\nolimits\!\left(a,x\right)$ is defined by

12.2.21 $\mathop{\overline{U}\/}\nolimits\!\left(a,x\right)=\mathop{\Gamma\/}\nolimits(\tfrac{1}{2}-a)\mathop{V\/}\nolimits\!\left(a,x\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\overline{U}\/}\nolimits\!\left(a,x\right)$ : parabolic cylinder function, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $x$ : real variable and $a$ : real or complex parameter
Referenced by:: §12.2(vi)
Permalink:: http://dlmf.nist.gov/12.2.E21
Encodings:: TeX, pMML, png

unless $a=\tfrac{1}{2},\tfrac{3}{2},\dots$ , in which case $\mathop{\overline{U}\/}\nolimits\!\left(a,x\right)$ is undefined. Its importance is that when $a$ is negative and $|a|$ is large, $\mathop{U\/}\nolimits\!\left(a,x\right)$ and $\mathop{\overline{U}\/}\nolimits\!\left(a,x\right)$ asymptotically have the same envelope (modulus) and are $\tfrac{1}{2}\pi$ out of phase in the oscillatory interval $-2\sqrt{-a}<x<2\sqrt{-a}$ . Properties of $\mathop{\overline{U}\/}\nolimits\!\left(a,x\right)$ follow immediately from those of $\mathop{V\/}\nolimits\!\left(a,x\right)$ via (12.2.21).

In the oscillatory interval we define

12.2.22 $\mathop{U\/}\nolimits\!\left(a,x\right)+i\mathop{\overline{U}\/}\nolimits\!\left(a,x\right)=F(a,x)e^{{i\theta(a,x)}},$

Symbols:: $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\overline{U}\/}\nolimits\!\left(a,x\right)$ : parabolic cylinder function, $x$ : real variable, $a$ : real or complex parameter, $F(a,x)$ : modulus and $\theta(a,x)$ : phase
Referenced by:: ¶ ‣ §12.12
Permalink:: http://dlmf.nist.gov/12.2.E22
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12.2.23 ${\mathop{U\/}\nolimits^{{\prime}}}\!\left(a,x\right)+i{\mathop{\overline{U}\/}\nolimits^{{\prime}}}\!\left(a,x\right)=-G(a,x)e^{{i\psi(a,x)}},$

Symbols:: $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\overline{U}\/}\nolimits\!\left(a,x\right)$ : parabolic cylinder function, $x$ : real variable, $a$ : real or complex parameter, $G(a,x)$ : modulus and $\psi(a,x)$ : phase
Permalink:: http://dlmf.nist.gov/12.2.E23
Encodings:: TeX, pMML, png

where $F(a,x)$ ( $>$ 0), $\theta(a,x)$ , $G(a,x)$ ( $>$ 0), and $\psi(a,x)$ are real. $F$ or $G$ is the modulus and $\theta$ or $\psi$ is the corresponding phase.

For properties of the modulus and phase functions, including differential equations, see Miller (1955, pp. 72–73). For graphs of the modulus functions see §12.3(i).