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12 Parabolic Cylinder FunctionsProperties

§12.14 The Function W(a,x)

Contents

§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(a,x) and W(a,-x) form a numerically satisfactory pair of solutions when -<x<.

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

12.14.1 W(a,0)=2-34|Γ(14+12ia)Γ(34+12ia)|12,
12.14.2 W(a,0)=-2-14|Γ(34+12ia)Γ(14+12ia)|12.
12.14.3 𝒲{W(a,x),W(a,-x)}=1.

§12.14(iii) Graphs

For the modulus functions F~(a,x) and G~(a,x) see §12.14(x).

See accompanying text
Figure 12.14.1: k-1/2W(3,x), k1/2W(3,-x), F~(3,x), 0x8. Magnify
See accompanying text
Figure 12.14.2: k-1/2W(3,x), k1/2W(3,-x), G~(3,x), 0x8. Magnify
See accompanying text
Figure 12.14.3: k-1/2W(-3,x), k1/2W(-3,-x), F~(-3,x), 0x8. Magnify
See accompanying text
Figure 12.14.4: k-1/2W(-3,x),k1/2W(-3,-x), G~(-3,x), 0x8. Magnify

§12.14(iv) Connection Formula

12.14.4 W(a,x)=k/2e14πa(eiρU(ia,xe-πi/4)+e-iρU(-ia,xeπi/4)),

where

12.14.5 k =1+e2πa-eπa,
1/k =1+e2πa+eπa,
12.14.6 ρ=18π+12ϕ2,
12.14.7 ϕ2=phΓ(12+ia),

the branch of ph being zero when a=0 and defined by continuity elsewhere.

§12.14(v) Power-Series Expansions

12.14.8 W(a,x)=W(a,0)w1(a,x)+W(a,0)w2(a,x).

Here w1(a,x) and w2(a,x) are the even and odd solutions of (12.2.3):

12.14.9 w1(a,x)=n=0αn(a)x2n(2n)!,
12.14.10 w2(a,x)=n=0βn(a)x2n+1(2n+1)!,

where αn(a) and βn(a) satisfy the recursion relations

12.14.11 αn+2 =aαn+1-12(n+1)(2n+1)αn,
βn+2 =aβn+1-12(n+1)(2n+3)βn,

with

12.14.12 α0(a) =1,
α1(a) =a,
β0(a) =1,
β1(a) =a.

Other expansions, involving cos(14x2) and sin(14x2), can be obtained from (12.4.3) to (12.4.6) by replacing a by -ia and z by xeπi/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(0,±x)=2-54πx(J-14(14x2)J14(14x2)),
12.14.14 ddxW(0,±x)=-2-94xπx(J34(14x2)±J-34(14x2)).

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 w1(a,x)=e-14ix2M(14-12ia,12,12ix2)=e14ix2M(14+12ia,12,-12ix2),
12.14.16 w2(a,x)=xe-14ix2M(34-12ia,32,12ix2)=xe14ix2M(34+12ia,32,-12ix2).

§12.14(viii) Asymptotic Expansions for Large Variable

Write

12.14.17 W(a,x)=2kx(s1(a,x)cosω-s2(a,x)sinω),
12.14.18 W(a,-x)=2kx(s1(a,x)sinω+s2(a,x)cosω),

where

12.14.19 ω=14x2-alnx+14π+12ϕ2,

with ϕ2 given by (12.14.7). Then as x

12.14.20 s1(a,x)1+d21!2x2-c42!22x4-d63!23x6+c84!24x8+,
12.14.21 s2(a,x)-c21!2x2-d42!22x4+c63!23x6+d84!24x8-.

The coefficients c2r and d2r are obtainable by equating real and imaginary parts in

12.14.22 c2r+id2r=Γ(2r+12+ia)Γ(12+ia).

Equivalently,

12.14.23 s1(a,x)+is2(a,x)r=0(-i)r(12+ia)2r2rr!x2r.

§12.14(ix) Uniform Asymptotic Expansions for Large Parameter

The differential equation

12.14.24 d2wdt2=μ4(1-t2)w

follows from (12.2.3), and has solutions W(12μ2,±μt2). For real μ 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 ±1. In the following expansions, obtained from Olver (1959), μ is large and positive, and δ is again an arbitrary small positive constant.

Positive a, 2a<x<

12.14.25 W(12μ2,μt2)2-12e-14πμ2l(μ)(t2-1)14(cosσs=0(-1)s𝒜2s(t)μ4s-sinσs=0(-1)s𝒜2s+1(t)μ4s+2,)
12.14.26 W(12μ2,-μt2)212e14πμ2l(μ)(t2-1)14(sinσs=0(-1)s𝒜2s(t)μ4s+cosσs=0(-1)s𝒜2s+1(t)μ4s+2,)

uniformly for t[1+δ,). Here 𝒜s(t) is as in §12.10(ii), σ is defined by

12.14.27 σ=μ2ξ+14π,

with ξ given by (12.10.7), and

12.14.28 l(μ)=2e18πμ2ei(12ϕ2-18π)g(μe-14πi),

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

12.14.29 l(μ)214μ12s=0lsμ4s,

with

12.14.30 l0 =1,
l1 =-11152,
l2 =-16123398 13120.

Positive a, -2a<x<2a

12.14.31 W(12μ2,μt2)l(μ)eμ2η212e14πμ2(1-t2)14s=0(-1)s𝒜~s(t)μ2s,

uniformly for t[-1+δ,1-δ], with η given by (12.10.23) and 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 W(12μ2,μt2) π12μ13l(μ)212e14πμ2ϕ(ζ)(Bi(-μ43ζ)s=0(-1)sAs(ζ)μ4s+Bi(-μ43ζ)μ83s=0(-1)sBs(ζ)μ4s),
12.14.33 W(12μ2,-μt2) π12μ13l(μ)2-12e-14πμ2ϕ(ζ)(Ai(-μ43ζ)s=0(-1)sAs(ζ)μ4s+Ai(-μ43ζ)μ83s=0(-1)sBs(ζ)μ4s),

uniformly for t[-1+δ,), with ζ, ϕ(ζ), As(ζ), and Bs(ζ) as in §12.10(vii). For the corresponding expansions for the derivatives see Olver (1959).

Negative a, -<x<

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 W(-12μ2,μt2) l(μ)(t2+1)14(cosσ¯s=0(-1)su¯2s(t)(t2+1)3sμ4s-sinσ¯s=0(-1)su¯2s+1(t)(t2+1)3s+32μ4s+2),
12.14.35 W(-12μ2,μt2) -μ2l(μ)(t2+1)14(sinσ¯s=0(-1)sv¯2s(t)(t2+1)3sμ4s+cosσ¯s=0(-1)sv¯2s+1(t)(t2+1)3s+32μ4s+2),

uniformly for t, where

12.14.36 σ¯=μ2ξ¯+14π,

and ξ¯ and the coefficients u¯s(t) and 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 -2a<x<2a in which the solutions are exponential in character. In the oscillatory intervals we write

12.14.37 k-1/2W(a,x)+ik1/2W(a,-x)=F~(a,x)eiθ~(a,x),
12.14.38 k-1/2W(a,x)+ik1/2W(a,-x)=-G~(a,x)eiψ~(a,x),

where k is defined in (12.14.5), and F~(a,x) (>0), θ~(a,x), G~(a,x) (>0), and ψ~(a,x) are real. F~ or G~ is the modulus and θ~ or ψ~ 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(a,x), W(a,x)

For asymptotic expansions of the zeros of W(a,x) and W(a,x), see Olver (1959).