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12 Parabolic Cylinder FunctionsProperties

§12.2 Differential Equations

Contents

§12.2(i) Introduction

PCFs are solutions of the differential equation

12.2.1 d2wdz2+(az2+bz+c)w=0,

with three distinct standard forms

12.2.2 d2wdz2-(14z2+a)w=0,
12.2.3 d2wdz2+(14z2-a)w=0,
12.2.4 d2wdz2+(ν+12-14z2)w=0.

Each of these equations is transformable into the others. Standard solutions are U(a,±z), V(a,±z), U¯(a,±x) (not complex conjugate), U(-a,±iz) for (12.2.2); W(a,±x) for (12.2.3); Dν(±z) for (12.2.4), where

12.2.5 Dν(z)=U(-12-ν,z).

All solutions are entire functions of z and entire functions of a or ν.

For real values of z (=x), numerically satisfactory pairs of solutions (§2.7(iv)) of (12.2.2) are U(a,x) and V(a,x) when x is positive, or U(a,-x) and V(a,-x) when x is negative. For (12.2.3) W(a,x) and W(a,-x) comprise a numerically satisfactory pair, for all x. The solutions W(a,±x) are treated in §12.14.

In , for j=0,1,2,3, U((-1)j-1a,(-i)j-1z) and U((-1)ja,(-i)jz) comprise a numerically satisfactory pair of solutions in the half-plane 14(2j-3)πphz14(2j+1)π.

§12.2(ii) Values at z=0

12.2.6 U(a,0) =π212a+14Γ(34+12a),
12.2.7 U(a,0) =-π212a-14Γ(14+12a),
12.2.8 V(a,0) =π212a+14(Γ(34-12a))2Γ(14+12a),
12.2.9 V(a,0) =π212a+34(Γ(14-12a))2Γ(34+12a).

§12.2(iii) Wronskians

12.2.10 𝒲{U(a,z),V(a,z)}=2/π,
12.2.11 𝒲{U(a,z),U(a,-z)}=2πΓ(12+a),
12.2.12 𝒲{U(a,z),U(-a,±iz)}=ie±iπ(12a+14).

§12.2(iv) Reflection Formulas

For n=0,1,,

12.2.13 U(-n-12,-z)=(-1)nU(-n-12,z),
12.2.14 V(n+12,-z)=(-1)nV(n+12,z).

§12.2(v) Connection Formulas

12.2.17 2πU(-a,±iz)=Γ(12+a)(eiπ(12a-14)U(a,z)+e±iπ(12a-14)U(a,-z)).
12.2.18 2πU(a,z)=Γ(12-a)(eiπ(12a+14)U(-a,±iz)+e±iπ(12a+14)U(-a,iz)),
12.2.19 U(a,z)=±ie±iπaU(a,-z)+2πΓ(12+a)e±iπ(12a-14)U(-a,±iz).
12.2.20 V(a,z)=iΓ(12-a)U(a,z)+2πeiπ(12a-14)U(-a,±iz).

§12.2(vi) Solution U¯(a,x); Modulus and Phase Functions

When z (=x) is real the solution U¯(a,x) is defined by

12.2.21 U¯(a,x)=Γ(12-a)V(a,x),

unless a=12,32,, in which case U¯(a,x) is undefined. Its importance is that when a is negative and |a| is large, U(a,x) and U¯(a,x) asymptotically have the same envelope (modulus) and are 12π out of phase in the oscillatory interval -2-a<x<2-a. Properties of U¯(a,x) follow immediately from those of V(a,x) via (12.2.21).

In the oscillatory interval we define

12.2.22 U(a,x)+iU¯(a,x)=F(a,x)eiθ(a,x),
12.2.23 U(a,x)+iU¯(a,x)=-G(a,x)eiψ(a,x),

where 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.

For properties of the modulus and phase functions, including differential equations, see Miller (1955, pp. 72–73). For graphs of the modulus functions see §12.3(i).