# §12.2 Differential Equations

## §12.2(i) Introduction

PCFs are solutions of the differential equation

 12.2.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(az^{2}+bz+c\right)w=0,$ Defines: $a$: parameter (locally), $b$: parameter (locally) and $c$: parameter (locally) Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$ and $z$: complex variable A&S Ref: 19.1.1 Permalink: http://dlmf.nist.gov/12.2.E1 Encodings: TeX, pMML, png See also: Annotations for 12.2(i)

with three distinct standard forms

 12.2.2 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-\left(\tfrac{1}{4}z^{2}+a\right)w=0,$ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: 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 Encodings: TeX, pMML, png See also: Annotations for 12.2(i)
 12.2.3 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\tfrac{1}{4}z^{2}-a\right)w=0,$ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: 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 Encodings: TeX, pMML, png See also: Annotations for 12.2(i)
 12.2.4 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\nu+\tfrac{1}{2}-\tfrac{1}{4% }z^{2}\right)w=0.$ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: 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 Encodings: TeX, pMML, png See also: Annotations for 12.2(i)

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

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 $x\in\mathbb{R}$. The solutions $\mathop{W\/}\nolimits\!\left(a,\pm x\right)$ are treated in §12.14.

In $\mathbb{C}$, 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 $z=0$

 12.2.6 $\displaystyle\mathop{U\/}\nolimits\!\left(a,0\right)$ $\displaystyle=\frac{\sqrt{\pi}}{2^{\frac{1}{2}a+\frac{1}{4}}\mathop{\Gamma\/}% \nolimits\!\left(\frac{3}{4}+\frac{1}{2}a\right)},$ Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: 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 See also: Annotations for 12.2(ii) 12.2.7 $\displaystyle\mathop{U\/}\nolimits'\!\left(a,0\right)$ $\displaystyle=-\frac{\sqrt{\pi}}{2^{\frac{1}{2}a-\frac{1}{4}}\mathop{\Gamma\/}% \nolimits\!\left(\frac{1}{4}+\frac{1}{2}a\right)},$ 12.2.8 $\displaystyle\mathop{V\/}\nolimits\!\left(a,0\right)$ $\displaystyle=\frac{\pi 2^{\frac{1}{2}a+\frac{1}{4}}}{\left(\mathop{\Gamma\/}% \nolimits\!\left(\frac{3}{4}-\frac{1}{2}a\right)\right)^{2}\mathop{\Gamma\/}% \nolimits\!\left(\frac{1}{4}+\frac{1}{2}a\right)},$ 12.2.9 $\displaystyle\mathop{V\/}\nolimits'\!\left(a,0\right)$ $\displaystyle=\frac{\pi 2^{\frac{1}{2}a+\frac{3}{4}}}{\left(\mathop{\Gamma\/}% \nolimits\!\left(\frac{1}{4}-\frac{1}{2}a\right)\right)^{2}\mathop{\Gamma\/}% \nolimits\!\left(\frac{3}{4}+\frac{1}{2}a\right)}.$ Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{V\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: 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 Encodings: TeX, pMML, png See also: Annotations for 12.2(ii)

## §12.2(iii) Wronskians

 12.2.10 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{U\/}\nolimits\!\left(a,z\right),% \mathop{V\/}\nolimits\!\left(a,z\right)\right\}=\sqrt{2/\pi},$
 12.2.11 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{U\/}\nolimits\!\left(a,z\right),% \mathop{U\/}\nolimits\!\left(a,-z\right)\right\}=\frac{\sqrt{2\pi}}{\mathop{% \Gamma\/}\nolimits\!\left(\frac{1}{2}+a\right)},$
 12.2.12 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{U\/}\nolimits\!\left(a,z\right),% \mathop{U\/}\nolimits\!\left(-a,\pm iz\right)\right\}=\mp ie^{\pm i\pi(\frac{1% }{2}a+\frac{1}{4})}.$

## §12.2(iv) Reflection Formulas

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: $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: parabolic cylinder function, $z$: complex variable and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/12.2.E13 Encodings: TeX, pMML, png See also: Annotations for 12.2(iv)
 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: $\mathop{V\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: parabolic cylinder function, $z$: complex variable and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/12.2.E14 Encodings: TeX, pMML, png See also: Annotations for 12.2(iv)

## §12.2(v) Connection Formulas

 12.2.15 $\mathop{U\/}\nolimits\!\left(a,-z\right)=-\mathop{\sin\/}\nolimits(\pi a)% \mathop{U\/}\nolimits\!\left(a,z\right)+\frac{\pi}{\mathop{\Gamma\/}\nolimits(% \frac{1}{2}+a)}\mathop{V\/}\nolimits\!\left(a,z\right),$
 12.2.16 $\mathop{V\/}\nolimits\!\left(a,-z\right)=\frac{\mathop{\cos\/}\nolimits(\pi a)% }{\mathop{\Gamma\/}\nolimits(\frac{1}{2}-a)}\mathop{U\/}\nolimits\!\left(a,z% \right)+\mathop{\sin\/}\nolimits(\pi a)\mathop{V\/}\nolimits\!\left(a,z\right).$
 12.2.17 $\sqrt{2\pi}\mathop{U\/}\nolimits\!\left(-a,\pm iz\right)=\mathop{\Gamma\/}% \nolimits\!\left(\tfrac{1}{2}+a\right)\left(e^{\mp i\pi(\frac{1}{2}a-\frac{1}{% 4})}\mathop{U\/}\nolimits\!\left(a,z\right)+e^{\pm i\pi(\frac{1}{2}a-\frac{1}{% 4})}\mathop{U\/}\nolimits\!\left(a,-z\right)\right).$
 12.2.18 $\sqrt{2\pi}\mathop{U\/}\nolimits\!\left(a,z\right)=\mathop{\Gamma\/}\nolimits% \!\left(\tfrac{1}{2}-a\right)\left(e^{\mp i\pi(\frac{1}{2}a+\frac{1}{4})}% \mathop{U\/}\nolimits\!\left(-a,\pm iz\right)+e^{\pm i\pi(\frac{1}{2}a+\frac{1% }{4})}\mathop{U\/}\nolimits\!\left(-a,\mp iz\right)\right),$
 12.2.19 $\mathop{U\/}\nolimits\!\left(a,z\right)=\pm ie^{\pm i\pi a}\mathop{U\/}% \nolimits\!\left(a,-z\right)+\frac{\sqrt{2\pi}}{\mathop{\Gamma\/}\nolimits\!% \left(\tfrac{1}{2}+a\right)}e^{\pm i\pi(\frac{1}{2}a-\frac{1}{4})}\mathop{U\/}% \nolimits\!\left(-a,\pm iz\right).$
 12.2.20 $\mathop{V\/}\nolimits\!\left(a,z\right)=\frac{\mp i}{\mathop{\Gamma\/}% \nolimits(\frac{1}{2}-a)}\mathop{U\/}\nolimits\!\left(a,z\right)+\sqrt{\frac{2% }{\pi}}e^{\mp i\pi(\frac{1}{2}a-\frac{1}{4})}\mathop{U\/}\nolimits\!\left(-a,% \pm iz\right).$

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

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

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

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}. 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 $\mathop{U\/}\nolimits\!\left(a,x\right)+i\mathop{\overline{U}\/}\nolimits\!% \left(a,x\right)=F(a,x)e^{i\theta(a,x)},$
 12.2.23 $\mathop{U\/}\nolimits'\!\left(a,x\right)+i\mathop{\overline{U}\/}\nolimits'\!% \left(a,x\right)=-G(a,x)e^{i\psi(a,x)},$

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).