§12.2 Differential Equations

PCFs are solutions of the differential equation

12.2.1		$$\frac{d^{2}w}{{dz}^2}+\left(a{z}^2+bz+c\right)w=0,$$
ⓘ Defines: $a$: parameter (locally), $b$: parameter (locally) and $c$: parameter (locally) Symbols: $\frac{df}{dx}$: 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), §12.2 and Ch.12

with three distinct standard forms

12.2.2		$$\frac{d^{2}w}{{dz}^2}-\left(\frac{1}{4}{z}^2+a\right)w=0,$$
ⓘ 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 Encodings: TeX, pMML, png See also: Annotations for §12.2(i), §12.2 and Ch.12

12.2.3		$$\frac{d^{2}w}{{dz}^2}+\left(\frac{1}{4}{z}^2-a\right)w=0,$$
ⓘ 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(ix), §12.14(v), §12.14(x), §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 and Ch.12

12.2.4		$$\frac{d^{2}w}{{dz}^2}+\left(\nu +\frac{1}{2}-\frac{1}{4}{z}^2\right)w=0.$$
ⓘ 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 Encodings: TeX, pMML, png See also: Annotations for §12.2(i), §12.2 and Ch.12

Each of these equations is transformable into the others. Standard solutions are $U(a,\pm z)$, $V(a,\pm z)$, $\overline{U}(a,\pm x)$ (not complex conjugate), $U(-a,\pm \mathrm{i}z)$ for (12.2.2); $W(a,\pm x)$ for (12.2.3); $D_{\nu }\left(\pm z\right)$ for (12.2.4), where

12.2.5		$$D_{\nu }\left(z\right)=U(-\frac{1}{2}-\nu ,z).$$
ⓘ Symbols: $D_{\nu }\left(z\right)$: parabolic cylinder function, $U(a,z)$: parabolic cylinder function, $z$: complex variable and $\nu$: real or complex parameter Permalink: http://dlmf.nist.gov/12.2.E5 Encodings: TeX, pMML, png See also: Annotations for §12.2(i), §12.2 and Ch.12

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 $U(a,x)$ and $V(a,x)$ when $x$ is positive, or $U(a,-x)$ and $V(a,-x)$ when $x$ is negative. For (12.2.3) $W(a,x)$ and $W(a,-x)$ comprise a numerically satisfactory pair, for all $x\in ℝ$. The solutions $W(a,\pm x)$ are treated in §12.14.

In $ℂ$, for $j=0,1,2,3$, $U({(-1)}^{j-1}a,{(-\mathrm{i})}^{j-1}z)$ and $U({(-1)}^{j}a,{(-\mathrm{i})}^{j}z)$ comprise a numerically satisfactory pair of solutions in the half-plane $\frac{1}{4}(2j-3)\pi \le \mathrm{ph}z\le \frac{1}{4}(2j+1)\pi$.

12.2.6		$U(a,0)$	$={\displaystyle \frac{\sqrt{\pi }}{2^{\frac{1}{2}a+\frac{1}{4}}\mathrm{\Gamma }\left(\frac{3}{4}+\frac{1}{2}a\right)}},$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $U(a,z)$: 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 and Ch.12
12.2.7		$U^{\prime }(a,0)$	$=-{\displaystyle \frac{\sqrt{\pi }}{2^{\frac{1}{2}a-\frac{1}{4}}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}a\right)}},$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $U(a,z)$: 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 See also: Annotations for §12.2(ii), §12.2 and Ch.12
12.2.8		$V(a,0)$	$={\displaystyle \frac{\pi 2^{\frac{1}{2}a+\frac{1}{4}}}{{\left(\mathrm{\Gamma }\left(\frac{3}{4}-\frac{1}{2}a\right)\right)}^2\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}a\right)}},$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $V(a,z)$: 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 See also: Annotations for §12.2(ii), §12.2 and Ch.12
12.2.9		$V^{\prime }(a,0)$	$={\displaystyle \frac{\pi 2^{\frac{1}{2}a+\frac{3}{4}}}{{\left(\mathrm{\Gamma }\left(\frac{1}{4}-\frac{1}{2}a\right)\right)}^2\mathrm{\Gamma }\left(\frac{3}{4}+\frac{1}{2}a\right)}}.$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $V(a,z)$: 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 and Ch.12

12.2.10		$$𝒲\left\{U(a,z),V(a,z)\right\}=\sqrt{2/\pi },$$
ⓘ Symbols: $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $U(a,z)$: parabolic cylinder function, $V(a,z)$: 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 See also: Annotations for §12.2(iii), §12.2 and Ch.12

12.2.11		$$𝒲\left\{U(a,z),U(a,-z)\right\}=\frac{\sqrt{2\pi }}{\mathrm{\Gamma }\left(\frac{1}{2}+a\right)},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter Permalink: http://dlmf.nist.gov/12.2.E11 Encodings: TeX, pMML, png See also: Annotations for §12.2(iii), §12.2 and Ch.12

12.2.12		$$𝒲\left\{U(a,z),U(-a,\pm \mathrm{i}z)\right\}=\mp \mathrm{i}{\mathrm{e}}^{\pm \mathrm{i}\pi (\frac{1}{2}a+\frac{1}{4})}.$$
ⓘ Symbols: $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $U(a,z)$: 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 See also: Annotations for §12.2(iii), §12.2 and Ch.12

For $n=0,1,\mathrm{\dots }$,

12.2.13		$$U(-n-\frac{1}{2},-z)={(-1)}^{n}U(-n-\frac{1}{2},z),$$
ⓘ Symbols: $U(a,z)$: 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 and Ch.12

12.2.14		$$V(n+\frac{1}{2},-z)={(-1)}^{n}V(n+\frac{1}{2},z).$$
ⓘ Symbols: $V(a,z)$: 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 and Ch.12

12.2.15		$$U(a,-z)=-\sin \left(\pi a\right)U(a,z)+\frac{\pi }{\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}V(a,z),$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $U(a,z)$: parabolic cylinder function, $V(a,z)$: parabolic cylinder function, $\sin z$: sine function, $z$: complex variable and $a$: real or complex parameter Referenced by: §12.10(iii), §12.10(viii), §12.9(i) Permalink: http://dlmf.nist.gov/12.2.E15 Encodings: TeX, pMML, png See also: Annotations for §12.2(v), §12.2 and Ch.12

12.2.16		$$V(a,-z)=\frac{\cos \left(\pi a\right)}{\mathrm{\Gamma }\left(\frac{1}{2}-a\right)}U(a,z)+\sin \left(\pi a\right)V(a,z).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $U(a,z)$: parabolic cylinder function, $V(a,z)$: parabolic cylinder function, $\sin z$: sine function, $z$: complex variable and $a$: real or complex parameter Referenced by: §12.10(iii), §12.10(viii), §12.9(i) Permalink: http://dlmf.nist.gov/12.2.E16 Encodings: TeX, pMML, png See also: Annotations for §12.2(v), §12.2 and Ch.12

12.2.17		$$\sqrt{2\pi }U(-a,\pm \mathrm{i}z)=\mathrm{\Gamma }\left(\frac{1}{2}+a\right)\left({\mathrm{e}}^{\mp \mathrm{i}\pi (\frac{1}{2}a-\frac{1}{4})}U(a,z)+{\mathrm{e}}^{\pm \mathrm{i}\pi (\frac{1}{2}a-\frac{1}{4})}U(a,-z)\right).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $U(a,z)$: 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 See also: Annotations for §12.2(v), §12.2 and Ch.12

12.2.18		$$\sqrt{2\pi }U(a,z)=\mathrm{\Gamma }\left(\frac{1}{2}-a\right)\left({\mathrm{e}}^{\mp \mathrm{i}\pi (\frac{1}{2}a+\frac{1}{4})}U(-a,\pm \mathrm{i}z)+{\mathrm{e}}^{\pm \mathrm{i}\pi (\frac{1}{2}a+\frac{1}{4})}U(-a,\mp \mathrm{i}z)\right),$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $U(a,z)$: 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 See also: Annotations for §12.2(v), §12.2 and Ch.12

12.2.19		$$U(a,z)=\pm \mathrm{i}{\mathrm{e}}^{\pm \mathrm{i}\pi a}U(a,-z)+\frac{\sqrt{2\pi }}{\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{\mathrm{e}}^{\pm \mathrm{i}\pi (\frac{1}{2}a-\frac{1}{4})}U(-a,\pm \mathrm{i}z).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter Permalink: http://dlmf.nist.gov/12.2.E19 Encodings: TeX, pMML, png See also: Annotations for §12.2(v), §12.2 and Ch.12

12.2.20		$$V(a,z)=\frac{\mp \mathrm{i}}{\mathrm{\Gamma }\left(\frac{1}{2}-a\right)}U(a,z)+\sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\mp \mathrm{i}\pi (\frac{1}{2}a-\frac{1}{4})}U(-a,\pm \mathrm{i}z).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $U(a,z)$: parabolic cylinder function, $V(a,z)$: 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 See also: Annotations for §12.2(v), §12.2 and Ch.12

When $z$ $(=x)$ is real the solution $\overline{U}(a,x)$ is defined by

12.2.21		$$\overline{U}(a,x)=\mathrm{\Gamma }\left(\frac{1}{2}-a\right)V(a,x),$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\overline{U}(a,x)$: parabolic cylinder function, $V(a,z)$: 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 See also: Annotations for §12.2(vi), §12.2 and Ch.12

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

In the oscillatory interval we define

12.2.22		$$U(a,x)+\mathrm{i}\overline{U}(a,x)=F(a,x){\mathrm{e}}^{\mathrm{i}\theta (a,x)},$$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $U(a,z)$: parabolic cylinder function, $\overline{U}(a,x)$: 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 Encodings: TeX, pMML, png See also: Annotations for §12.2(vi), §12.2 and Ch.12

12.2.23		$$U^{\prime }(a,x)+\mathrm{i}{\overline{U}}^{\prime }(a,x)=-G(a,x){\mathrm{e}}^{\mathrm{i}\psi (a,x)},$$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $U(a,z)$: parabolic cylinder function, $\overline{U}(a,x)$: 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 See also: Annotations for §12.2(vi), §12.2 and Ch.12

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).