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

Β§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.

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

 12.14.1 $W\left(a,0\right)=2^{-\frac{3}{4}}\left|\frac{\Gamma\left(\tfrac{1}{4}+\tfrac{% 1}{2}ia\right)}{\Gamma\left(\tfrac{3}{4}+\tfrac{1}{2}ia\right)}\right|^{\frac{% 1}{2}},$ β Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathrm{i}$: imaginary unit, $W\left(\NVar{a},\NVar{x}\right)$: 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 Β§12.14(ii), Β§12.14 and Ch.12
 12.14.2 $W'\left(a,0\right)=-2^{-\frac{1}{4}}\left|\frac{\Gamma\left(\tfrac{3}{4}+% \tfrac{1}{2}ia\right)}{\Gamma\left(\tfrac{1}{4}+\tfrac{1}{2}ia\right)}\right|^% {\frac{1}{2}}.$ β Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathrm{i}$: imaginary unit, $W\left(\NVar{a},\NVar{x}\right)$: 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 Β§12.14(ii), Β§12.14 and Ch.12
 12.14.3 $\mathscr{W}\left\{W\left(a,x\right),W\left(a,-x\right)\right\}=1.$ β 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 Β§12.14(ii), Β§12.14 and Ch.12

Β§12.14(iii) Graphs

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

Β§12.14(iv) Connection Formula

 12.14.4 $W\left(a,x\right)=\sqrt{k/2}\,e^{\frac{1}{4}\pi a}\left(e^{i\rho}U\left(ia,xe^% {-\pi i/4}\right)+e^{-i\rho}U\left(-ia,xe^{\pi i/4}\right)\right),$
 12.14.4_5 $W\left(a,-x\right)=\frac{-i}{\sqrt{2k}}\,e^{\frac{1}{4}\pi a}\left(e^{i\rho}U% \left(ia,xe^{-\pi i/4}\right)-e^{-i\rho}U\left(-ia,xe^{\pi i/4}\right)\right),$ β Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $U\left(\NVar{a},\NVar{z}\right)$: parabolic cylinder function, $W\left(\NVar{a},\NVar{x}\right)$: parabolic cylinder function, $x$: real variable, $a$: real or complex parameter, $k$ and $\rho$ Proof sketch: Combine (12.14.4) with (12.2.19). Referenced by: Β§12.14(iv), Erratum (V1.1.3) for Additions Permalink: http://dlmf.nist.gov/12.14.E4_5 Encodings: TeX, pMML, png Addition (effective with 1.1.3): This equation was added. See also: Annotations for Β§12.14(iv), Β§12.14 and Ch.12

where

 12.14.5 $\displaystyle k$ $\displaystyle=\sqrt{1+e^{2\pi a}}-e^{\pi a},$ $\displaystyle 1/k$ $\displaystyle=\sqrt{1+e^{2\pi a}}+e^{\pi a},$ β 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: Β§12.14(x) Permalink: http://dlmf.nist.gov/12.14.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for Β§12.14(iv), Β§12.14 and Ch.12
 12.14.6 $\rho=\tfrac{1}{8}\pi+\tfrac{1}{2}\phi_{2},$ β 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 Β§12.14(iv), Β§12.14 and Ch.12
 12.14.7 $\phi_{2}=\operatorname{ph}\Gamma\left(\tfrac{1}{2}+ia\right),$ β Defines: $\phi_{2}$ (locally) Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathrm{i}$: imaginary unit, $\operatorname{ph}$: phase and $a$: real or complex parameter A&S Ref: 19.17.10 Referenced by: Β§12.14(viii) Permalink: http://dlmf.nist.gov/12.14.E7 Encodings: TeX, pMML, png See also: Annotations for Β§12.14(iv), Β§12.14 and Ch.12

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

Β§12.14(v) Power-Series Expansions

 12.14.8 $W\left(a,x\right)=W\left(a,0\right)w_{1}(a,x)+W'\left(a,0\right)w_{2}(a,x).$ β 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 Β§12.14(v), Β§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 $w_{1}(a,x)=\sum_{n=0}^{\infty}\alpha_{n}(a)\frac{x^{2n}}{(2n)!},$ β 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 Β§12.14(v), Β§12.14 and Ch.12
 12.14.10 $w_{2}(a,x)=\sum_{n=0}^{\infty}\beta_{n}(a)\frac{x^{2n+1}}{(2n+1)!},$ β 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 Β§12.14(v), Β§12.14 and Ch.12

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

 12.14.11 $\displaystyle\alpha_{n+2}$ $\displaystyle=a\alpha_{n+1}-\tfrac{1}{2}(n+1)(2n+1)\alpha_{n},$ $\displaystyle\beta_{n+2}$ $\displaystyle=a\beta_{n+1}-\tfrac{1}{2}(n+1)(2n+3)\beta_{n},$ β 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 Β§12.14(v), Β§12.14 and Ch.12

with

 12.14.12 $\displaystyle\alpha_{0}(a)$ $\displaystyle=1,$ $\displaystyle\alpha_{1}(a)$ $\displaystyle=a,$ $\displaystyle\beta_{0}(a)$ $\displaystyle=1,$ $\displaystyle\beta_{1}(a)$ $\displaystyle=a.$ β 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 Β§12.14(v), Β§12.14 and Ch.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.Β 80), and also (12.14.15) and (12.14.16).

Β§12.14(vi) Integral Representations

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

Β§12.14(vii) Relations to Other Functions

Bessel Functions

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

 12.14.13 $W\left(0,\pm x\right)=2^{-\frac{5}{4}}\sqrt{\pi x}\left(J_{-\frac{1}{4}}\left(% \tfrac{1}{4}x^{2}\right)\mp J_{\frac{1}{4}}\left(\tfrac{1}{4}x^{2}\right)% \right),$ β 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: Β§36.2(ii) Permalink: http://dlmf.nist.gov/12.14.E13 Encodings: TeX, pMML, png See also: Annotations for Β§12.14(vii), Β§12.14(vii), Β§12.14 and Ch.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 Β§13.2(i).

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

 12.14.15 $w_{1}(a,x)=e^{-\frac{1}{4}ix^{2}}M\left(\tfrac{1}{4}-\tfrac{1}{2}ia,\tfrac{1}{% 2},\tfrac{1}{2}ix^{2}\right)=e^{\frac{1}{4}ix^{2}}M\left(\tfrac{1}{4}+\tfrac{1% }{2}ia,\tfrac{1}{2},-\tfrac{1}{2}ix^{2}\right),$
 12.14.16 $w_{2}(a,x)=xe^{-\frac{1}{4}ix^{2}}M\left(\tfrac{3}{4}-\tfrac{1}{2}ia,\tfrac{3}% {2},\tfrac{1}{2}ix^{2}\right)=xe^{\frac{1}{4}ix^{2}}M\left(\tfrac{3}{4}+\tfrac% {1}{2}ia,\tfrac{3}{2},-\tfrac{1}{2}ix^{2}\right).$

Β§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 $\omega=\tfrac{1}{4}x^{2}-a\ln x+\tfrac{1}{4}\pi+\tfrac{1}{2}\phi_{2},$ β 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 Β§12.14(viii), Β§12.14 and Ch.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 $c_{2r}+id_{2r}=\frac{\Gamma\left(2r+\tfrac{1}{2}+ia\right)}{\Gamma\left(\tfrac% {1}{2}+ia\right)}.$ β Defines: $c_{2r}$: coefficients (locally) and $d_{2r}$: coefficients (locally) Symbols: $\Gamma\left(\NVar{z}\right)$: 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 Β§12.14(viii), Β§12.14 and Ch.12

Equivalently,

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

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

The differential equation

 12.14.24 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}t}^{2}}=\mu^{4}(1-t^{2})w$ β 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 Β§12.14(ix), Β§12.14 and Ch.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 Β§12.10(ii), $\sigma$ is defined by

 12.14.27 $\sigma=\mu^{2}\xi+\tfrac{1}{4}\pi,$ β 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 Β§12.14(ix), Β§12.14(ix), Β§12.14 and Ch.12

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

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

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

 12.14.29 $l(\mu)\sim\frac{2^{\frac{1}{4}}}{\mu^{\frac{1}{2}}}\sum_{s=0}^{\infty}\frac{l_% {s}}{\mu^{4s}},$ β Symbols: $\sim$: PoincarΓ© 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 Β§12.14(ix), Β§12.14(ix), Β§12.14 and Ch.12

with

 12.14.30 $\displaystyle l_{0}$ $\displaystyle=1,$ $\displaystyle l_{1}$ $\displaystyle=-\tfrac{1}{1152},$ $\displaystyle l_{2}$ $\displaystyle=-\tfrac{16123}{398\;13120}.$ β 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 Β§12.14(ix), Β§12.14(ix), Β§12.14 and Ch.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 Β§Β§12.10(ii)β12.10(v).

Airy-type Uniform Expansions

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

uniformly for $t\in[-1+\delta,\infty)$, with $\zeta$, $\phi(\zeta)$, $A_{s}(\zeta)$, and $B_{s}(\zeta)$ as in Β§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 ${\overline{\sigma}}=\mu^{2}{\overline{\xi}}+\tfrac{1}{4}\pi,$ β Defines: $\overline{\sigma}$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter and $\overline{\xi}$ Permalink: http://dlmf.nist.gov/12.14.E36 Encodings: TeX, pMML, png See also: Annotations for Β§12.14(ix), Β§12.14(ix), Β§12.14 and Ch.12

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

Β§12.14(x) Modulus and Phase Functions

As noted in Β§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 Β§12.2(vi).

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

Β§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).