§12.14 The Function $W(a,x)$

In this section solutions of equation (12.2.3) are considered. This equation is important when $a$ and $z$ $(=x)$ are real, and we shall assume this to be the case. In other cases the general theory of (12.2.2) is available. $W(a,x)$ and $W(a,-x)$ form a numerically satisfactory pair of solutions when .

12.14.1		$$W(a,0)=2^{-\frac{3}{4}}{\left|\frac{\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}a\right)}{\mathrm{\Gamma }\left(\frac{3}{4}+\frac{1}{2}\mathrm{i}a\right)}\right|}^{\frac{1}{2}},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{i}$: imaginary unit, $W(a,x)$: parabolic cylinder function and $a$: real or complex parameter A&S Ref: 19.17.4 (modification of) Permalink: http://dlmf.nist.gov/12.14.E1 Encodings: TeX, pMML, png See also: Annotations for §12.14(ii), §12.14 and Ch.12

12.14.2		$$W^{\prime }(a,0)=-2^{-\frac{1}{4}}{\left|\frac{\mathrm{\Gamma }\left(\frac{3}{4}+\frac{1}{2}\mathrm{i}a\right)}{\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}a\right)}\right|}^{\frac{1}{2}}.$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{i}$: imaginary unit, $W(a,x)$: parabolic cylinder function and $a$: real or complex parameter A&S Ref: 19.17.5 (modification of) Permalink: http://dlmf.nist.gov/12.14.E2 Encodings: TeX, pMML, png See also: Annotations for §12.14(ii), §12.14 and Ch.12

12.14.3		$$𝒲\left\{W(a,x),W(a,-x)\right\}=1.$$
ⓘ Symbols: $𝒲$: Wronskian, $W(a,x)$: parabolic cylinder function, $x$: real variable and $a$: real or complex parameter A&S Ref: 19.18.1 Permalink: http://dlmf.nist.gov/12.14.E3 Encodings: TeX, pMML, png See also: Annotations for §12.14(ii), §12.14 and Ch.12

For the modulus functions $\tilde{F}(a,x)$ and $\tilde{G}(a,x)$ see §12.14(x).

12.14.4		$$W(a,x)=\sqrt{k/2}{\mathrm{e}}^{\frac{1}{4}\pi a}\left({\mathrm{e}}^{\mathrm{i}\rho }U(\mathrm{i}a,x{\mathrm{e}}^{-\pi \mathrm{i}/4})+{\mathrm{e}}^{-\mathrm{i}\rho }U(-\mathrm{i}a,x{\mathrm{e}}^{\pi \mathrm{i}/4})\right),$$
ⓘ Symbols: $\pi$: 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, $W(a,x)$: parabolic cylinder function, $x$: real variable, $a$: real or complex parameter, $k$ and $\rho$ Permalink: http://dlmf.nist.gov/12.14.E4 Encodings: TeX, pMML, png See also: Annotations for §12.14(iv), §12.14 and Ch.12

where

12.14.5		$k$	$=\sqrt{1+{\mathrm{e}}^{2\pi a}}-{\mathrm{e}}^{\pi a},$
		$1/k$	$=\sqrt{1+{\mathrm{e}}^{2\pi a}}+{\mathrm{e}}^{\pi a},$
ⓘ Defines: $k$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $a$: real or complex parameter A&S Ref: 19.17.8 Referenced by: §12.14(x) Permalink: http://dlmf.nist.gov/12.14.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §12.14(iv), §12.14 and Ch.12

12.14.6		$$\rho =\frac{1}{8}\pi +\frac{1}{2}{\varphi }_2,$$
ⓘ Defines: $\rho$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter and ${\varphi }_2$ Permalink: http://dlmf.nist.gov/12.14.E6 Encodings: TeX, pMML, png See also: Annotations for §12.14(iv), §12.14 and Ch.12

12.14.7		$${\varphi }_2=\mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{2}+\mathrm{i}a\right),$$
ⓘ Defines: ${\varphi }_2$ (locally) Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase and $a$: real or complex parameter A&S Ref: 19.17.10 Referenced by: §12.14(viii) Permalink: http://dlmf.nist.gov/12.14.E7 Encodings: TeX, pMML, png See also: Annotations for §12.14(iv), §12.14 and Ch.12

the branch of $\mathrm{ph}$ being zero when $a=0$ and defined by continuity elsewhere.

12.14.8		$$W(a,x)=W(a,0)w_1(a,x)+W^{\prime }(a,0)w_2(a,x).$$
ⓘ Symbols: $W(a,x)$: parabolic cylinder function, $x$: real variable, $a$: real or complex parameter, $w_1(a,x)$: even solution and $w_2(a,x)$: odd solution A&S Ref: 19.17.1 (modification of) Permalink: http://dlmf.nist.gov/12.14.E8 Encodings: TeX, pMML, png See also: Annotations for §12.14(v), §12.14 and Ch.12

Here $w_1(a,x)$ and $w_2(a,x)$ are the even and odd solutions of (12.2.3):

12.14.9		$$w_1(a,x)=\sum _{n=0}^{\mathrm{\infty }}{\alpha }_n(a)\frac{x^{2n}}{(2n)!},$$
ⓘ Defines: $w_1(a,x)$: even solution (locally) Symbols: $!$: factorial (as in $n!$), $x$: real variable, $n$: nonnegative integer, $a$: real or complex parameter and ${\alpha }_n(a)$: recursion A&S Ref: 19.16.1 (modification of) Permalink: http://dlmf.nist.gov/12.14.E9 Encodings: TeX, pMML, png See also: Annotations for §12.14(v), §12.14 and Ch.12

12.14.10		$$w_2(a,x)=\sum _{n=0}^{\mathrm{\infty }}{\beta }_n(a)\frac{x^{2n+1}}{(2n+1)!},$$
ⓘ Defines: $w_2(a,x)$: odd solution (locally) Symbols: $!$: factorial (as in $n!$), $x$: real variable, $n$: nonnegative integer, $a$: real or complex parameter and ${\beta }_n(a)$: recursion A&S Ref: 19.16.2 (modification of) Permalink: http://dlmf.nist.gov/12.14.E10 Encodings: TeX, pMML, png See also: Annotations for §12.14(v), §12.14 and Ch.12

where ${\alpha }_n(a)$ and ${\beta }_n(a)$ satisfy the recursion relations

12.14.11		${\alpha }_{n+2}$	$=a{\alpha }_{n+1}-\frac{1}{2}(n+1)(2n+1){\alpha }_n,$
		${\beta }_{n+2}$	$=a{\beta }_{n+1}-\frac{1}{2}(n+1)(2n+3){\beta }_n,$
ⓘ Defines: ${\alpha }_n(a)$: recursion (locally) and ${\beta }_n(a)$: recursion (locally) Symbols: $n$: nonnegative integer and $a$: real or complex parameter Permalink: http://dlmf.nist.gov/12.14.E11 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §12.14(v), §12.14 and Ch.12

with

12.14.12		${\alpha }_0(a)$	$=1,$
		${\alpha }_1(a)$	$=a,$
		${\beta }_0(a)$	$=1,$
		${\beta }_1(a)$	$=a.$
ⓘ Symbols: $a$: real or complex parameter, ${\alpha }_n(a)$: recursion and ${\beta }_n(a)$: recursion A&S Ref: 19.16.3 (modification of) Permalink: http://dlmf.nist.gov/12.14.E12 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §12.14(v), §12.14 and Ch.12

Other expansions, involving $\cos \left(\frac{1}{4}{x}^2\right)$ and $\sin \left(\frac{1}{4}{x}^2\right)$, can be obtained from (12.4.3) to (12.4.6) by replacing $a$ by $-\mathrm{i}a$ and $z$ by $x{\mathrm{e}}^{\pi \mathrm{i}/4}$; see Miller (1955, p. 80), and also (12.14.15) and (12.14.16).

These follow from the contour integrals of §12.5(ii), which are valid for general complex values of the argument $z$ and parameter $a$. See Miller (1955, p. 26).

Write

12.14.17		$$W(a,x)=\sqrt{\frac{2k}{x}}\left(s_1(a,x)\cos \omega -s_2(a,x)\sin \omega \right),$$
ⓘ Symbols: $\cos z$: cosine function, $W(a,x)$: parabolic cylinder function, $\sin z$: sine function, $x$: real variable, $a$: real or complex parameter, $k$, $\omega$ and $s_n(a,x)$: coefficients A&S Ref: 19.21.2 (modification of) Permalink: http://dlmf.nist.gov/12.14.E17 Encodings: TeX, pMML, png See also: Annotations for §12.14(viii), §12.14 and Ch.12

12.14.18		$$W(a,-x)=\sqrt{\frac{2}{kx}}\left(s_1(a,x)\sin \omega +s_2(a,x)\cos \omega \right),$$
ⓘ Symbols: $\cos z$: cosine function, $W(a,x)$: parabolic cylinder function, $\sin z$: sine function, $x$: real variable, $a$: real or complex parameter, $k$, $\omega$ and $s_n(a,x)$: coefficients A&S Ref: 19.21.3 (modification of) Permalink: http://dlmf.nist.gov/12.14.E18 Encodings: TeX, pMML, png See also: Annotations for §12.14(viii), §12.14 and Ch.12

where

12.14.19		$$\omega =\frac{1}{4}{x}^2-a\ln x+\frac{1}{4}\pi +\frac{1}{2}{\varphi }_2,$$
ⓘ Defines: $\omega$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $x$: real variable, $a$: real or complex parameter and ${\varphi }_2$ Permalink: http://dlmf.nist.gov/12.14.E19 Encodings: TeX, pMML, png See also: Annotations for §12.14(viii), §12.14 and Ch.12

with ${\varphi }_2$ given by (12.14.7). Then as $x\to \mathrm{\infty }$

12.14.20		$$s_1(a,x)\sim 1+\frac{d_2}{1!2x^2}-\frac{c_4}{2!2^2{x}^4}-\frac{d_6}{3!2^3{x}^6}+\frac{c_8}{4!2^4{x}^8}+\mathrm{\cdots },$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $x$: real variable, $a$: real or complex parameter, $c_{2r}$: coefficients, $d_{2r}$: coefficients and $s_n(a,x)$: coefficients A&S Ref: 19.21.5 Permalink: http://dlmf.nist.gov/12.14.E20 Encodings: TeX, pMML, png See also: Annotations for §12.14(viii), §12.14 and Ch.12

12.14.21		$$s_2(a,x)\sim -\frac{c_2}{1!2x^2}-\frac{d_4}{2!2^2{x}^4}+\frac{c_6}{3!2^3{x}^6}+\frac{d_8}{4!2^4{x}^8}-\mathrm{\cdots }.$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $x$: real variable, $a$: real or complex parameter, $c_{2r}$: coefficients, $d_{2r}$: coefficients and $s_n(a,x)$: coefficients A&S Ref: 19.21.6 Permalink: http://dlmf.nist.gov/12.14.E21 Encodings: TeX, pMML, png See also: Annotations for §12.14(viii), §12.14 and Ch.12

The coefficients $c_{2r}$ and $d_{2r}$ are obtainable by equating real and imaginary parts in

12.14.22		$$c_{2r}+\mathrm{i}{d}_{2r}=\frac{\mathrm{\Gamma }\left(2r+\frac{1}{2}+\mathrm{i}a\right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mathrm{i}a\right)}.$$
ⓘ Defines: $c_{2r}$: coefficients (locally) and $d_{2r}$: coefficients (locally) Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{i}$: imaginary unit and $a$: real or complex parameter A&S Ref: 19.21.7 (modification of) Permalink: http://dlmf.nist.gov/12.14.E22 Encodings: TeX, pMML, png See also: Annotations for §12.14(viii), §12.14 and Ch.12

Equivalently,

12.14.23		$$s_1(a,x)+\mathrm{i}{s}_2(a,x)\sim \sum _{r=0}^{\mathrm{\infty }}{(-\mathrm{i})}^r\frac{{\left(\frac{1}{2}+\mathrm{i}a\right)}_{2r}}{2^{r}r!x^{2r}}.$$
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $x$: real variable, $a$: real or complex parameter and $s_n(a,x)$: coefficients A&S Ref: 19.21.8 (modification of) Permalink: http://dlmf.nist.gov/12.14.E23 Encodings: TeX, pMML, png See also: Annotations for §12.14(viii), §12.14 and Ch.12

The differential equation

12.14.24		$$\frac{d^{2}w}{{dt}^2}={\mu }^4(1-t^2)w$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$ Permalink: http://dlmf.nist.gov/12.14.E24 Encodings: TeX, pMML, png See also: Annotations for §12.14(ix), §12.14 and Ch.12

follows from (12.2.3), and has solutions $W(\frac{1}{2}{\mu }^2,\pm \mu t\sqrt{2})$. For real $\mu$ and $t$ oscillations occur outside the $t$-interval $[-1,1]$. Airy-type uniform asymptotic expansions can be used to include either one of the turning points $\pm 1$. In the following expansions, obtained from Olver (1959), $\mu$ is large and positive, and $\delta$ is again an arbitrary small positive constant.

As noted in §12.14(ix), when $a$ is negative the solutions of (12.2.3), with $z$ replaced by $x$, are oscillatory on the whole real line; also, when $a$ is positive there is a central interval in which the solutions are exponential in character. In the oscillatory intervals we write

12.14.37		$$k^{-1/2}W(a,x)+\mathrm{i}{k}^{1/2}W(a,-x)=\tilde{F}(a,x){\mathrm{e}}^{\mathrm{i}\tilde{\theta }(a,x)},$$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $W(a,x)$: parabolic cylinder function, $x$: real variable, $a$: real or complex parameter, $k$, $\tilde{F}(a,x)$: modulus and $\tilde{\theta }(a,x)$: phase Permalink: http://dlmf.nist.gov/12.14.E37 Encodings: TeX, pMML, png See also: Annotations for §12.14(x), §12.14 and Ch.12

12.14.38		$$k^{-1/2}{W}^{\prime }(a,x)+\mathrm{i}{k}^{1/2}{W}^{\prime }(a,-x)=-\tilde{G}(a,x){\mathrm{e}}^{\mathrm{i}\tilde{\psi }(a,x)},$$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $W(a,x)$: parabolic cylinder function, $x$: real variable, $a$: real or complex parameter, $k$, $\tilde{G}(a,x)$: modulus and $\tilde{\psi }(a,x)$: phase Permalink: http://dlmf.nist.gov/12.14.E38 Encodings: TeX, pMML, png See also: Annotations for §12.14(x), §12.14 and Ch.12

where $k$ is defined in (12.14.5), and $\tilde{F}(a,x)$ ($>$0), $\tilde{\theta }(a,x)$, $\tilde{G}(a,x)$ ($>$0), and $\tilde{\psi }(a,x)$ are real. $\tilde{F}$ or $\tilde{G}$ is the modulus and $\tilde{\theta }$ or $\tilde{\psi }$ is the corresponding phase. Compare §12.2(vi).

For properties of the modulus and phase functions, including differential equations and asymptotic expansions for large $x$, see Miller (1955, pp. 87–88). For graphs of the modulus functions see §12.14(iii).

For asymptotic expansions of the zeros of $W(a,x)$ and $W^{\prime }(a,x)$, see Olver (1959).