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12 Parabolic Cylinder FunctionsProperties

§12.14 The Function W⁑(a,x)

  1. Β§12.14(i) Introduction
  2. Β§12.14(ii) Values at z=0 and Wronskian
  3. Β§12.14(iii) Graphs
  4. Β§12.14(iv) Connection Formula
  5. Β§12.14(v) Power-Series Expansions
  6. Β§12.14(vi) Integral Representations
  7. Β§12.14(vii) Relations to Other Functions
  8. Β§12.14(viii) Asymptotic Expansions for Large Variable
  9. Β§12.14(ix) Uniform Asymptotic Expansions for Large Parameter
  10. Β§12.14(x) Modulus and Phase Functions
  11. Β§12.14(xi) Zeros of W⁑(a,x), W′⁑(a,x)

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⁑(a,x) and W⁑(a,βˆ’x) form a numerically satisfactory pair of solutions when βˆ’βˆž<x<∞.

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

12.14.1 W⁑(a,0)=2βˆ’34⁒|Γ⁑(14+12⁒i⁒a)Γ⁑(34+12⁒i⁒a)|12,
12.14.2 W′⁑(a,0)=βˆ’2βˆ’14⁒|Γ⁑(34+12⁒i⁒a)Γ⁑(14+12⁒i⁒a)|12.
12.14.3 𝒲⁑{W⁑(a,x),W⁑(a,βˆ’x)}=1.

Β§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: kβˆ’1/2⁒W⁑(3,x), k1/2⁒W⁑(3,βˆ’x), F~⁑(3,x), 0≀x≀8. Magnify
See accompanying text
Figure 12.14.2: kβˆ’1/2⁒W′⁑(3,x), k1/2⁒W′⁑(3,βˆ’x), G~⁑(3,x), 0≀x≀8. Magnify
See accompanying text
Figure 12.14.3: kβˆ’1/2⁒W⁑(βˆ’3,x), k1/2⁒W⁑(βˆ’3,βˆ’x), F~⁑(βˆ’3,x), 0≀x≀8. Magnify
See accompanying text
Figure 12.14.4: kβˆ’1/2⁒W′⁑(βˆ’3,x),k1/2⁒W′⁑(βˆ’3,βˆ’x), G~⁑(βˆ’3,x), 0≀x≀8. Magnify

Β§12.14(iv) Connection Formula

12.14.4 W⁑(a,x)=k/2⁒e14⁒π⁒a⁒(ei⁒ρ⁒U⁑(i⁒a,x⁒eβˆ’Ο€β’i/4)+eβˆ’i⁒ρ⁒U⁑(βˆ’i⁒a,x⁒eπ⁒i/4)),
12.14.4_5 W⁑(a,βˆ’x)=βˆ’i2⁒k⁒e14⁒π⁒a⁒(ei⁒ρ⁒U⁑(i⁒a,x⁒eβˆ’Ο€β’i/4)βˆ’eβˆ’i⁒ρ⁒U⁑(βˆ’i⁒a,x⁒eπ⁒i/4)),


12.14.5 k =1+e2⁒π⁒aβˆ’eπ⁒a,
1/k =1+e2⁒π⁒a+eπ⁒a,
12.14.6 ρ=18⁒π+12⁒ϕ2,
12.14.7 Ο•2=ph⁑Γ⁑(12+i⁒a),

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

Β§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)=βˆ‘n=0∞αn⁑(a)⁒x2⁒n(2⁒n)!,
12.14.10 w2⁑(a,x)=βˆ‘n=0∞βn⁑(a)⁒x2⁒n+1(2⁒n+1)!,

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

12.14.11 Ξ±n+2 =a⁒αn+1βˆ’12⁒(n+1)⁒(2⁒n+1)⁒αn,
Ξ²n+2 =a⁒βn+1βˆ’12⁒(n+1)⁒(2⁒n+3)⁒βn,


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

Other expansions, involving cos⁑(14⁒x2) and sin⁑(14⁒x2), can be obtained from (12.4.3) to (12.4.6) by replacing a by βˆ’i⁒a and z by x⁒eπ⁒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⁑(0,Β±x)=2βˆ’54⁒π⁒x⁒(Jβˆ’14⁑(14⁒x2)βˆ“J14⁑(14⁒x2)),
12.14.14 ddx⁑W⁑(0,Β±x)=βˆ’2βˆ’94⁒x⁒π⁒x⁒(J34⁑(14⁒x2)Β±Jβˆ’34⁑(14⁒x2)).

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 w1⁑(a,x)=eβˆ’14⁒i⁒x2⁒M⁑(14βˆ’12⁒i⁒a,12,12⁒i⁒x2)=e14⁒i⁒x2⁒M⁑(14+12⁒i⁒a,12,βˆ’12⁒i⁒x2),
12.14.16 w2⁑(a,x)=x⁒eβˆ’14⁒i⁒x2⁒M⁑(34βˆ’12⁒i⁒a,32,12⁒i⁒x2)=x⁒e14⁒i⁒x2⁒M⁑(34+12⁒i⁒a,32,βˆ’12⁒i⁒x2).

Β§12.14(viii) Asymptotic Expansions for Large Variable


12.14.17 W⁑(a,x)=2⁒kx⁒(s1⁑(a,x)⁒cosβ‘Ο‰βˆ’s2⁑(a,x)⁒sin⁑ω),
12.14.18 W⁑(a,βˆ’x)=2k⁒x⁒(s1⁑(a,x)⁒sin⁑ω+s2⁑(a,x)⁒cos⁑ω),


12.14.19 Ο‰=14⁒x2βˆ’a⁒ln⁑x+14⁒π+12⁒ϕ2,

with Ο•2 given by (12.14.7). Then as xβ†’βˆž

12.14.20 s1⁑(a,x)∼1+d21!⁒2⁒x2βˆ’c42!⁒22⁒x4βˆ’d63!⁒23⁒x6+c84!⁒24⁒x8+β‹―,
12.14.21 s2⁑(a,x)βˆΌβˆ’c21!⁒2⁒x2βˆ’d42!⁒22⁒x4+c63!⁒23⁒x6+d84!⁒24⁒x8βˆ’β‹―.

The coefficients c2⁒r and d2⁒r are obtainable by equating real and imaginary parts in

12.14.22 c2⁒r+i⁒d2⁒r=Γ⁑(2⁒r+12+i⁒a)Γ⁑(12+i⁒a).


12.14.23 s1⁑(a,x)+i⁒s2⁑(a,x)βˆΌβˆ‘r=0∞(βˆ’i)r⁒(12+i⁒a)2⁒r2r⁒r!⁒x2⁒r.

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

The differential equation

12.14.24 d2wdt2=ΞΌ4⁒(1βˆ’t2)⁒w

follows from (12.2.3), and has solutions W⁑(12⁒μ2,±μ⁒t⁒2). For real ΞΌ and t oscillations occur outside the t-interval [βˆ’1,1]. Airy-type uniform asymptotic expansions can be used to include either one of the turning points Β±1. In the following expansions, obtained from Olver (1959), ΞΌ is large and positive, and Ξ΄ is again an arbitrary small positive constant.

Positive a, 2⁒a<x<∞

12.14.25 W⁑(12⁒μ2,μ⁒t⁒2)∼2βˆ’12⁒eβˆ’14⁒π⁒μ2⁒l⁑(ΞΌ)(t2βˆ’1)14⁒(cosβ‘Οƒβ’βˆ‘s=0∞(βˆ’1)sβ’π’œ2⁒s⁑(t)ΞΌ4⁒sβˆ’sinβ‘Οƒβ’βˆ‘s=0∞(βˆ’1)sβ’π’œ2⁒s+1⁑(t)ΞΌ4⁒s+2),
12.14.26 W⁑(12⁒μ2,βˆ’ΞΌβ’t⁒2)∼212⁒e14⁒π⁒μ2⁒l⁑(ΞΌ)(t2βˆ’1)14⁒(sinβ‘Οƒβ’βˆ‘s=0∞(βˆ’1)sβ’π’œ2⁒s⁑(t)ΞΌ4⁒s+cosβ‘Οƒβ’βˆ‘s=0∞(βˆ’1)sβ’π’œ2⁒s+1⁑(t)ΞΌ4⁒s+2),

uniformly for t∈[1+Ξ΄,∞). Here π’œs⁑(t) is as in Β§12.10(ii), Οƒ is defined by

12.14.27 Οƒ=ΞΌ2⁒ξ+14⁒π,

with ΞΎ given by (12.10.7), and

12.14.28 l⁑(ΞΌ)=2⁒e18⁒π⁒μ2⁒ei⁒(12⁒ϕ2βˆ’18⁒π)⁒g⁑(μ⁒eβˆ’14⁒π⁒i),

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

12.14.29 l⁑(ΞΌ)∼214ΞΌ12β’βˆ‘s=0∞lsΞΌ4⁒s,


12.14.30 l0 =1,
l1 =βˆ’11152,
l2 =βˆ’16123398 13120.

Positive a, βˆ’2⁒a<x<2⁒a

12.14.31 W⁑(12⁒μ2,μ⁒t⁒2)∼l⁑(ΞΌ)⁒eΞΌ2⁒η212⁒e14⁒π⁒μ2⁒(1βˆ’t2)14β’βˆ‘s=0∞(βˆ’1)sβ’π’œ~s⁑(t)ΞΌ2⁒s,

uniformly for t∈[βˆ’1+Ξ΄,1βˆ’Ξ΄], with Ξ· given by (12.10.23) and π’œ~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⁑(12⁒μ2,μ⁒t⁒2) βˆΌΟ€12⁒μ13⁒l⁑(ΞΌ)212⁒e14⁒π⁒μ2⁒ϕ⁑(ΞΆ)⁒(Bi⁑(βˆ’ΞΌ43⁒΢)β’βˆ‘s=0∞(βˆ’1)s⁒As⁑(ΞΆ)ΞΌ4⁒s+Bi′⁑(βˆ’ΞΌ43⁒΢)ΞΌ83β’βˆ‘s=0∞(βˆ’1)s⁒Bs⁑(ΞΆ)ΞΌ4⁒s),
12.14.33 W⁑(12⁒μ2,βˆ’ΞΌβ’t⁒2) βˆΌΟ€12⁒μ13⁒l⁑(ΞΌ)2βˆ’12⁒eβˆ’14⁒π⁒μ2⁒ϕ⁑(ΞΆ)⁒(Ai⁑(βˆ’ΞΌ43⁒΢)β’βˆ‘s=0∞(βˆ’1)s⁒As⁑(ΞΆ)ΞΌ4⁒s+Ai′⁑(βˆ’ΞΌ43⁒΢)ΞΌ83β’βˆ‘s=0∞(βˆ’1)s⁒Bs⁑(ΞΆ)ΞΌ4⁒s),

uniformly for t∈[βˆ’1+Ξ΄,∞), with ΞΆ, ϕ⁑(ΞΆ), As⁑(ΞΆ), and Bs⁑(ΞΆ) as in Β§12.10(vii). For the corresponding expansions for the derivatives see Olver (1959).

Negative a, βˆ’βˆž<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⁑(βˆ’12⁒μ2,μ⁒t⁒2) ∼l⁑(ΞΌ)(t2+1)14⁒(cosβ‘ΟƒΒ―β’βˆ‘s=0∞(βˆ’1)s⁒uΒ―2⁒s⁑(t)(t2+1)3⁒s⁒μ4⁒sβˆ’sinβ‘ΟƒΒ―β’βˆ‘s=0∞(βˆ’1)s⁒uΒ―2⁒s+1⁑(t)(t2+1)3⁒s+32⁒μ4⁒s+2),
12.14.35 W′⁑(βˆ’12⁒μ2,μ⁒t⁒2) βˆΌβˆ’ΞΌ2⁒l⁑(ΞΌ)⁒(t2+1)14⁒(sinβ‘ΟƒΒ―β’βˆ‘s=0∞(βˆ’1)s⁒vΒ―2⁒s⁑(t)(t2+1)3⁒s⁒μ4⁒s+cosβ‘ΟƒΒ―β’βˆ‘s=0∞(βˆ’1)s⁒vΒ―2⁒s+1⁑(t)(t2+1)3⁒s+32⁒μ4⁒s+2),

uniformly for tβˆˆβ„, where

12.14.36 σ¯=ΞΌ2⁒ξ¯+14⁒π,

and ξ¯ and the coefficients u¯s⁑(t) and v¯s⁑(t) as in §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 βˆ’2⁒a<x<2⁒a in which the solutions are exponential in character. In the oscillatory intervals we write

12.14.37 kβˆ’1/2⁒W⁑(a,x)+i⁒k1/2⁒W⁑(a,βˆ’x)=F~⁑(a,x)⁒ei⁒θ~⁑(a,x),
12.14.38 kβˆ’1/2⁒W′⁑(a,x)+i⁒k1/2⁒W′⁑(a,βˆ’x)=βˆ’G~⁑(a,x)⁒ei⁒ψ~⁑(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.Β 87–88). For graphs of the modulus functions see Β§12.14(iii).

Β§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).