# §12.14 The Function $W\left(a,x\right)$

## §12.14(i) Introduction

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. $W\left(a,x\right)$ and $W\left(a,-x\right)$ form a numerically satisfactory pair of solutions when $-\infty.

## §12.14(ii) Values at $z=0$ and Wronskian

 12.14.1 $W\left(a,0\right)=2^{-\frac{3}{4}}\left|\frac{\Gamma\left(\tfrac{1}{4}+\tfrac{% 1}{2}ia\right)}{\Gamma\left(\tfrac{3}{4}+\tfrac{1}{2}ia\right)}\right|^{\frac{% 1}{2}},$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $W\left(\NVar{a},\NVar{x}\right)$: 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 12
 12.14.2 $W'\left(a,0\right)=-2^{-\frac{1}{4}}\left|\frac{\Gamma\left(\tfrac{3}{4}+% \tfrac{1}{2}ia\right)}{\Gamma\left(\tfrac{1}{4}+\tfrac{1}{2}ia\right)}\right|^% {\frac{1}{2}}.$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $W\left(\NVar{a},\NVar{x}\right)$: 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 12
 12.14.3 $\mathscr{W}\left\{W\left(a,x\right),W\left(a,-x\right)\right\}=1.$ ⓘ Symbols: $\mathscr{W}$: Wronskian, $W\left(\NVar{a},\NVar{x}\right)$: 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 12

## §12.14(iii) Graphs

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

## §12.14(iv) Connection Formula

 12.14.4 $W\left(a,x\right)=\sqrt{k/2}\,e^{\frac{1}{4}\pi a}\left(e^{i\rho}U\left(ia,xe^% {-\pi i/4}\right)+e^{-i\rho}U\left(-ia,xe^{\pi i/4}\right)\right),$

where

 12.14.5 $\displaystyle k$ $\displaystyle=\sqrt{1+e^{2\pi a}}-e^{\pi a},$ $\displaystyle 1/k$ $\displaystyle=\sqrt{1+e^{2\pi a}}+e^{\pi a},$ ⓘ Defines: $k$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function 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 12
 12.14.6 $\rho=\tfrac{1}{8}\pi+\tfrac{1}{2}\phi_{2},$ ⓘ Defines: $\rho$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter and $\phi_{2}$ Permalink: http://dlmf.nist.gov/12.14.E6 Encodings: TeX, pMML, png See also: Annotations for 12.14(iv), 12.14 and 12
 12.14.7 $\phi_{2}=\operatorname{ph}\Gamma\left(\tfrac{1}{2}+ia\right),$ ⓘ Defines: $\phi_{2}$ (locally) Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\operatorname{ph}$: 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 12

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

## §12.14(v) Power-Series Expansions

 12.14.8 $W\left(a,x\right)=W\left(a,0\right)w_{1}(a,x)+W'\left(a,0\right)w_{2}(a,x).$ ⓘ Symbols: $W\left(\NVar{a},\NVar{x}\right)$: 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 12

Here $w_{1}(a,x)$ and $w_{2}(a,x)$ are the even and odd solutions of (12.2.3):

 12.14.9 $w_{1}(a,x)=\sum_{n=0}^{\infty}\alpha_{n}(a)\frac{x^{2n}}{(2n)!},$ ⓘ 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 12
 12.14.10 $w_{2}(a,x)=\sum_{n=0}^{\infty}\beta_{n}(a)\frac{x^{2n+1}}{(2n+1)!},$ ⓘ 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 12

where $\alpha_{n}(a)$ and $\beta_{n}(a)$ satisfy the recursion relations

 12.14.11 $\displaystyle\alpha_{n+2}$ $\displaystyle=a\alpha_{n+1}-\tfrac{1}{2}(n+1)(2n+1)\alpha_{n},$ $\displaystyle\beta_{n+2}$ $\displaystyle=a\beta_{n+1}-\tfrac{1}{2}(n+1)(2n+3)\beta_{n},$ ⓘ 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 12

with

 12.14.12 $\displaystyle\alpha_{0}(a)$ $\displaystyle=1,$ $\displaystyle\alpha_{1}(a)$ $\displaystyle=a,$ $\displaystyle\beta_{0}(a)$ $\displaystyle=1,$ $\displaystyle\beta_{1}(a)$ $\displaystyle=a.$ ⓘ 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 12

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

## §12.14(vi) Integral Representations

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).

## §12.14(vii) Relations to Other Functions

### Bessel Functions

For the notation see §10.2(ii). When $x>0$

 12.14.13 $W\left(0,\pm x\right)=2^{-\frac{5}{4}}\sqrt{\pi x}\left(J_{-\frac{1}{4}}\left(% \tfrac{1}{4}x^{2}\right)\mp J_{\frac{1}{4}}\left(\tfrac{1}{4}x^{2}\right)% \right),$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $W\left(\NVar{a},\NVar{x}\right)$: parabolic cylinder function and $x$: real variable A&S Ref: 19.25.4 (corrected) Referenced by: §36.2(ii) Permalink: http://dlmf.nist.gov/12.14.E13 Encodings: TeX, pMML, png See also: Annotations for 12.14(vii), 12.14(vii), 12.14 and 12
 12.14.14 $\frac{\mathrm{d}}{\mathrm{d}x}W\left(0,\pm x\right)=-2^{-\frac{9}{4}}x\sqrt{% \pi x}\left(J_{\frac{3}{4}}\left(\tfrac{1}{4}x^{2}\right)\pm J_{-\frac{3}{4}}% \left(\tfrac{1}{4}x^{2}\right)\right).$

### Confluent Hypergeometric Functions

For the notation see §13.2(i).

The even and odd solutions of (12.2.3) (see §12.14(v)) are given by

 12.14.15 $w_{1}(a,x)=e^{-\frac{1}{4}ix^{2}}M\left(\tfrac{1}{4}-\tfrac{1}{2}ia,\tfrac{1}{% 2},\tfrac{1}{2}ix^{2}\right)=e^{\frac{1}{4}ix^{2}}M\left(\tfrac{1}{4}+\tfrac{1% }{2}ia,\tfrac{1}{2},-\tfrac{1}{2}ix^{2}\right),$ ⓘ Symbols: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of exponential function, $x$: real variable, $a$: real or complex parameter and $w_{1}(a,x)$: even solution A&S Ref: 19.25.1 (modification of) Referenced by: §12.14(v) Permalink: http://dlmf.nist.gov/12.14.E15 Encodings: TeX, pMML, png See also: Annotations for 12.14(vii), 12.14(vii), 12.14 and 12
 12.14.16 $w_{2}(a,x)=xe^{-\frac{1}{4}ix^{2}}M\left(\tfrac{3}{4}-\tfrac{1}{2}ia,\tfrac{3}% {2},\tfrac{1}{2}ix^{2}\right)=xe^{\frac{1}{4}ix^{2}}M\left(\tfrac{3}{4}+\tfrac% {1}{2}ia,\tfrac{3}{2},-\tfrac{1}{2}ix^{2}\right).$ ⓘ Symbols: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of exponential function, $x$: real variable, $a$: real or complex parameter and $w_{2}(a,x)$: odd solution A&S Ref: 19.25.1 (modification of) Referenced by: §12.14(v) Permalink: http://dlmf.nist.gov/12.14.E16 Encodings: TeX, pMML, png See also: Annotations for 12.14(vii), 12.14(vii), 12.14 and 12

## §12.14(viii) Asymptotic Expansions for Large Variable

Write

 12.14.17 $W\left(a,x\right)=\sqrt{\frac{2k}{x}}\left(s_{1}(a,x)\cos\omega-s_{2}(a,x)\sin% \omega\right),$
 12.14.18 $W\left(a,-x\right)=\sqrt{\frac{2}{kx}}\left(s_{1}(a,x)\sin\omega+s_{2}(a,x)% \cos\omega\right),$

where

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

with $\phi_{2}$ given by (12.14.7). Then as $x\to\infty$

 12.14.20 $s_{1}(a,x)\sim 1+\frac{d_{2}}{1!2x^{2}}-\frac{c_{4}}{2!2^{2}x^{4}}-\frac{d_{6}% }{3!2^{3}x^{6}}+\frac{c_{8}}{4!2^{4}x^{8}}+\cdots,$
 12.14.21 $s_{2}(a,x)\sim-\frac{c_{2}}{1!2x^{2}}-\frac{d_{4}}{2!2^{2}x^{4}}+\frac{c_{6}}{% 3!2^{3}x^{6}}+\frac{d_{8}}{4!2^{4}x^{8}}-\cdots.$

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

 12.14.22 $c_{2r}+id_{2r}=\frac{\Gamma\left(2r+\tfrac{1}{2}+ia\right)}{\Gamma\left(\tfrac% {1}{2}+ia\right)}.$ ⓘ Defines: $c_{2r}$: coefficients (locally) and $d_{2r}$: coefficients (locally) Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function 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 12

Equivalently,

 12.14.23 $s_{1}(a,x)+is_{2}(a,x)\sim\sum_{r=0}^{\infty}(-i)^{r}\frac{{\left(\tfrac{1}{2}% +ia\right)_{2r}}}{2^{r}r!x^{2r}}.$

## §12.14(ix) Uniform Asymptotic Expansions for Large Parameter

The differential equation

 12.14.24 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}t}^{2}}=\mu^{4}(1-t^{2})w$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: 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 12

follows from (12.2.3), and has solutions $W\left(\tfrac{1}{2}\mu^{2},\pm\mu t\sqrt{2}\right)$. 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.

### Positive $a$, $2\sqrt{a}

 12.14.25 $W\left(\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)\sim\frac{2^{-\frac{1}{2}}e^{-% \frac{1}{4}\pi\mu^{2}}l(\mu)}{(t^{2}-1)^{\frac{1}{4}}}\left(\cos\sigma\sum_{s=% 0}^{\infty}(-1)^{s}\frac{{\cal{A}}_{2s}(t)}{\mu^{4s}}-\sin\sigma\sum_{s=0}^{% \infty}(-1)^{s}\frac{{\cal{A}}_{2s+1}(t)}{\mu^{4s+2}}\right),$
 12.14.26 $W\left(\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right)\sim\frac{2^{\frac{1}{2}}e^{% \frac{1}{4}\pi\mu^{2}}l(\mu)}{(t^{2}-1)^{\frac{1}{4}}}\left(\sin\sigma\sum_{s=% 0}^{\infty}(-1)^{s}\frac{{\cal{A}}_{2s}(t)}{\mu^{4s}}+\cos\sigma\sum_{s=0}^{% \infty}(-1)^{s}\frac{{\cal{A}}_{2s+1}(t)}{\mu^{4s+2}}\right),$

uniformly for $t\in[1+\delta,\infty)$. Here ${\cal{A}}_{s}(t)$ is as in §12.10(ii), $\sigma$ is defined by

 12.14.27 $\sigma=\mu^{2}\xi+\tfrac{1}{4}\pi,$ ⓘ Defines: $\sigma$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter and $\xi$ Permalink: http://dlmf.nist.gov/12.14.E27 Encodings: TeX, pMML, png See also: Annotations for 12.14(ix), 12.14(ix), 12.14 and 12

with $\xi$ given by (12.10.7), and

 12.14.28 $l(\mu)=\sqrt{2}e^{\frac{1}{8}\pi\mu^{2}}e^{i(\frac{1}{2}\phi_{2}-\frac{1}{8}% \pi)}g(\mu e^{-\frac{1}{4}\pi i}),$ ⓘ Defines: $l(\mu)$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function, $\phi(\zeta)$: function and $g(\mu)$: expansion Permalink: http://dlmf.nist.gov/12.14.E28 Encodings: TeX, pMML, png See also: Annotations for 12.14(ix), 12.14(ix), 12.14 and 12

with $g(\mu)$ as in §12.10(ii). The function $l(\mu)$ has the asymptotic expansion

 12.14.29 $l(\mu)\sim\frac{2^{\frac{1}{4}}}{\mu^{\frac{1}{2}}}\sum_{s=0}^{\infty}\frac{l_% {s}}{\mu^{4s}},$ ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $l(\mu)$ and $l_{s}$: coefficient Permalink: http://dlmf.nist.gov/12.14.E29 Encodings: TeX, pMML, png See also: Annotations for 12.14(ix), 12.14(ix), 12.14 and 12

with

 12.14.30 $\displaystyle l_{0}$ $\displaystyle=1,$ $\displaystyle l_{1}$ $\displaystyle=-\tfrac{1}{1152},$ $\displaystyle l_{2}$ $\displaystyle=-\tfrac{16123}{398\;13120}.$ ⓘ Defines: $l_{s}$: coefficient (locally) Symbols: $s$: nonnegative integer Permalink: http://dlmf.nist.gov/12.14.E30 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 12.14(ix), 12.14(ix), 12.14 and 12

### Positive $a$, $-2\sqrt{a}

 12.14.31 $W\left(\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)\sim\frac{l(\mu)e^{\mu^{2}\eta}% }{2^{\frac{1}{2}}e^{\frac{1}{4}\pi\mu^{2}}(1-t^{2})^{\frac{1}{4}}}\*\sum_{s=0}% ^{\infty}(-1)^{s}\frac{\widetilde{\cal{A}}_{s}(t)}{\mu^{2s}},$

uniformly for $t\in[-1+\delta,1-\delta]$, with $\eta$ given by (12.10.23) and ${\widetilde{\cal A}}_{s}(t)$ given by (12.10.24).

The expansions for the derivatives corresponding to (12.14.25), (12.14.26), and (12.14.31) may be obtained by formal term-by-term differentiation with respect to $t$; compare the analogous results in §§12.10(ii)12.10(v).

### Airy-type Uniform Expansions

 12.14.32 $\displaystyle W\left(\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)$ $\displaystyle\sim\frac{\pi^{\frac{1}{2}}\mu^{\frac{1}{3}}l(\mu)}{2^{\frac{1}{2% }}e^{\frac{1}{4}\pi\mu^{2}}}\phi(\zeta)\left(\mathrm{Bi}\left(-\mu^{\frac{4}{3% }}\zeta\right)\sum_{s=0}^{\infty}(-1)^{s}\frac{A_{s}(\zeta)}{\mu^{4s}}+\frac{% \mathrm{Bi}'\left(-\mu^{\frac{4}{3}}\zeta\right)}{\mu^{\frac{8}{3}}}\sum_{s=0}% ^{\infty}(-1)^{s}\frac{B_{s}(\zeta)}{\mu^{4s}}\right),$ 12.14.33 $\displaystyle W\left(\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right)$ $\displaystyle\sim\frac{\pi^{\frac{1}{2}}\mu^{\frac{1}{3}}l(\mu)}{2^{-\frac{1}{% 2}}e^{-\frac{1}{4}\pi\mu^{2}}}\phi(\zeta)\left(\mathrm{Ai}\left(-\mu^{\frac{4}% {3}}\zeta\right)\sum_{s=0}^{\infty}(-1)^{s}\frac{A_{s}(\zeta)}{\mu^{4s}}+\frac% {\mathrm{Ai}'\left(-\mu^{\frac{4}{3}}\zeta\right)}{\mu^{\frac{8}{3}}}\sum_{s=0% }^{\infty}(-1)^{s}\frac{B_{s}(\zeta)}{\mu^{4s}}\right),$

uniformly for $t\in[-1+\delta,\infty)$, with $\zeta$, $\phi(\zeta)$, $A_{s}(\zeta)$, and $B_{s}(\zeta)$ as in §12.10(vii). For the corresponding expansions for the derivatives see Olver (1959).

### Negative $a$, $-\infty

In this case there are no real turning points, and the solutions of (12.2.3), with $z$ replaced by $x$, oscillate on the entire real $x$-axis.

 12.14.34 $\displaystyle W\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)$ $\displaystyle\sim\frac{l(\mu)}{(t^{2}+1)^{\frac{1}{4}}}\left(\cos\overline{% \sigma}\sum_{s=0}^{\infty}\frac{(-1)^{s}\overline{u}_{2s}(t)}{(t^{2}+1)^{3s}% \mu^{4s}}-\sin\overline{\sigma}\sum_{s=0}^{\infty}\frac{(-1)^{s}\overline{u}_{% 2s+1}(t)}{(t^{2}+1)^{3s+\frac{3}{2}}\mu^{4s+2}}\right),$ 12.14.35 $\displaystyle W'\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)$ $\displaystyle\sim-\frac{\mu}{\sqrt{2}}l(\mu)(t^{2}+1)^{\frac{1}{4}}\left(\sin% \overline{\sigma}\sum_{s=0}^{\infty}\frac{(-1)^{s}\overline{v}_{2s}(t)}{(t^{2}% +1)^{3s}\mu^{4s}}+\cos\overline{\sigma}\sum_{s=0}^{\infty}\frac{(-1)^{s}% \overline{v}_{2s+1}(t)}{(t^{2}+1)^{3s+\frac{3}{2}}\mu^{4s+2}}\right),$

uniformly for $t\in\mathbb{R}$, where

 12.14.36 ${\overline{\sigma}}=\mu^{2}{\overline{\xi}}+\tfrac{1}{4}\pi,$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\xi$ and $\sigma$ Permalink: http://dlmf.nist.gov/12.14.E36 Encodings: TeX, pMML, png See also: Annotations for 12.14(ix), 12.14(ix), 12.14 and 12

and ${\overline{\xi}}$ and the coefficients $\overline{u}_{s}(t)$ and $\overline{v}_{s}(t)$ as in §12.10(v).

## §12.14(x) Modulus and Phase Functions

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 $-2\sqrt{a} in which the solutions are exponential in character. In the oscillatory intervals we write

 12.14.37 $k^{-\ifrac{1}{2}}W\left(a,x\right)+ik^{\ifrac{1}{2}}W\left(a,-x\right)=% \widetilde{F}(a,x)e^{i\widetilde{\theta}(a,x)},$
 12.14.38 $k^{-\ifrac{1}{2}}W'\left(a,x\right)+ik^{\ifrac{1}{2}}W'\left(a,-x\right)=-% \widetilde{G}(a,x)e^{i\widetilde{\psi}(a,x)},$

where $k$ is defined in (12.14.5), and $\widetilde{F}(a,x)$ ($>$0), $\widetilde{\theta}(a,x)$, $\widetilde{G}(a,x)$ ($>$0), and $\widetilde{\psi}(a,x)$ are real. $\widetilde{F}$ or $\widetilde{G}$ is the modulus and $\widetilde{\theta}$ or $\widetilde{\psi}$ 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).

## §12.14(xi) Zeros of $W\left(a,x\right)$, $W'\left(a,x\right)$

For asymptotic expansions of the zeros of $W\left(a,x\right)$ and $W'\left(a,x\right)$, see Olver (1959).