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§2.8 Differential Equations with a Parameter

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§2.8(i) Classification of Cases

Many special functions satisfy an equation of the form

2.8.1 d2w/dz2=(u2f(z)+g(z))w,

in which u is a real or complex parameter, and asymptotic solutions are needed for large |u| that are uniform with respect to z in a point set D in or . For example, u can be the order of a Bessel function or degree of an orthogonal polynomial. The form of the asymptotic expansion depends on the nature of the transition points in D, that is, points at which f(z) has a zero or singularity. Zeros of f(z) are also called turning points.

There are three main cases. In Case I there are no transition points in D and g(z) is analytic. In Case II f(z) has a simple zero at z0 and g(z) is analytic at z0. In Case III f(z) has a simple pole at z0 and (z-z0)2g(z) is analytic at z0.

The same approach is used in all three cases. First we apply the Liouville transformation1.13(iv)) to (2.8.1). This introduces new variables W and ξ, related by

2.8.2 W=z˙-1/2w,

dots denoting differentiations with respect to ξ. Then

2.8.3 d2Wdξ2=(u2z˙2f(z)+ψ(ξ))W,

where

2.8.4 ψ(ξ)=z˙2g(z)+z˙1/2d2dξ2(z˙-1/2).

The transformation is now specialized in such a way that: (a) ξ and z are analytic functions of each other at the transition point (if any); (b) the approximating differential equation obtained by neglecting ψ(ξ) (or part of ψ(ξ)) has solutions that are functions of a single variable. The actual choices are as follows:

2.8.5 z˙2f(z) =1,
ξ =f1/2(z)dz,

for Case I,

2.8.6 z˙2f(z) =ξ,
23ξ3/2 =z0zf1/2(t)dt,

for Case II,

2.8.7 z˙2f(z) =1/ξ,
2ξ1/2 =z0zf1/2(t)dt,

for Case III.

The transformed equation has the form

2.8.8 d2W/dξ2=(u2ξm+ψ(ξ))W,

with m=0 (Case I), m=1 (Case II), m=-1 (Case III). In Cases I and II the asymptotic solutions are in terms of the functions that satisfy (2.8.8) with ψ(ξ)=0. These are elementary functions in Case I, and Airy functions (§9.2) in Case II. In Case III the approximating equation is

2.8.9 d2Wdξ2=(u2ξ+ρξ2)W,

where ρ=lim(ξ2ψ(ξ)) as ξ0. Solutions are Bessel functions, or modified Bessel functions, of order ±(1+4ρ)1/2 (§§10.2, 10.25).

For another approach to these problems based on convergent inverse factorial series expansions see Dunster et al. (1993) and Dunster (2001a, 2004).

§2.8(ii) Case I: No Transition Points

The transformed differential equation is

2.8.10 d2W/dξ2=(u2+ψ(ξ))W,

in which ξ ranges over a bounded or unbounded interval or domain Δ, and ψ(ξ) is C or analytic on Δ. The parameter u is assumed to be real and positive. Corresponding to each positive integer n there are solutions Wn,j(u,ξ), j=1,2, that depend on arbitrarily chosen reference points αj, are C or analytic on Δ, and as u

2.8.11 Wn,1(u,ξ) =euξ(s=0n-1As(ξ)us+O(1un)),
ξΔ1(α1),
2.8.12 Wn,2(u,ξ) =e-uξ(s=0n-1(-1)sAs(ξ)us+O(1un)),
ξΔ2(α2),

with A0(ξ)=1 and

2.8.13 As+1(ξ)=-12As(ξ)+12ψ(ξ)As(ξ)dξ,
s=0,1,2,,

(the constants of integration being arbitrary). The expansions (2.8.11) and (2.8.12) are both uniform and differentiable with respect to ξ. The regions of validity Δj(αj) comprise those points ξ that can be joined to αj in Δ by a path 𝒬j along which v is nondecreasing (j=1) or nonincreasing (j=2) as v passes from αj to ξ. In addition, 𝒱𝒬j(A1) and 𝒱𝒬j(An) must be bounded on Δj(αj).

For error bounds, extensions to pure imaginary or complex u, an extension to inhomogeneous differential equations, and examples, see Olver (1997b, Chapter 10). This reference also supplies sufficient conditions to ensure that the solutions Wn,1(u,ξ) and Wn,2(u,ξ) having the properties (2.8.11) and (2.8.12) are independent of n.

§2.8(iii) Case II: Simple Turning Point

The transformed differential equation is

2.8.14 d2W/dξ2=(u2ξ+ψ(ξ))W,

and for simplicity ξ is assumed to range over a finite or infinite interval (α1,α2) with α1<0, α2>0. Again, u>0 and ψ(ξ) is C on (α1,α2). Corresponding to each positive integer n there are solutions Wn,j(u,ξ), j=1,2, that are C on (α1,α2), and as u

2.8.15 Wn,1(u,ξ) =Ai(u2/3ξ)(s=0n-1As(ξ)u2s+O(1u2n-1))+Ai(u2/3ξ)(s=0n-2Bs(ξ)u2s+(4/3)+O(1u2n-1)),
2.8.16 Wn,2(u,ξ) =Bi(u2/3ξ)(s=0n-1As(ξ)u2s+O(1u2n-1))+Bi(u2/3ξ)(s=0n-2Bs(ξ)u2s+(4/3)+O(1u2n-1)).

Here A0(ξ)=1,

2.8.17 Bs(ξ)={12ξ1/20ξ(ψ(v)As(v)-As′′(v))dvv1/2,ξ>0,12(-ξ)1/2ξ0(ψ(v)As(v)-As′′(v))dv(-v)1/2,ξ<0,

and

2.8.18 As+1(ξ)=-12Bs(ξ)+12ψ(ξ)Bs(ξ)dξ,

when s=0,1,2,. For Ai and Bi see §9.2. The expansions (2.8.15) and (2.8.16) are both uniform and differentiable with respect to ξ. These results are valid when 𝒱α1,α2(|ξ|1/2B0) and 𝒱α1,α2(|ξ|1/2Bn-1) are finite.

An alternative way of representing the error terms in (2.8.15) and (2.8.16) is as follows. Let c=-0.36604... be the real root of the equation

2.8.19 Ai(x)=Bi(x)

of smallest absolute value, and define the envelopes of Ai(x) and Bi(x) by

2.8.20 envAi(x)=envBi(x)=(Ai2(x)+Bi2(x))1/2,
-<xc,
2.8.21 envAi(x) =2Ai(x),
envBi(x) =2Bi(x),
cx<.

These envelopes are continuous functions of x, and as u

2.8.22 Wn,1(u,ξ) =Ai(u2/3ξ)s=0n-1As(ξ)u2s+Ai(u2/3ξ)s=0n-2Bs(ξ)u2s+(4/3)+envAi(u2/3ξ)O(1u2n-1),
2.8.23 Wn,2(u,ξ) =Bi(u2/3ξ)s=0n-1As(ξ)u2s+Bi(u2/3ξ)s=0n-2Bs(ξ)u2s+(4/3)+envBi(u2/3ξ)O(1u2n-1),

uniformly with respect to ξ(α1,α2).

For error bounds, more delicate error estimates, extensions to complex ξ and u, zeros, connection formulas, extensions to inhomogeneous equations, and examples, see Olver (1997b, Chapters 11, 13), Olver (1964b), Reid (1974a, b), Boyd (1987), and Baldwin (1991).

For other examples of uniform asymptotic approximations and expansions of special functions in terms of Airy functions see especially §10.20 and §§12.10(vii), 12.10(viii); also §§12.14(ix), 13.20(v), 13.21(iii), 13.21(iv), 15.12(iii), 18.15(iv), 30.9(i), 30.9(ii), 32.11(ii), 32.11(iii), 33.12(i), 33.12(ii), 33.20(iv), 36.12(ii), 36.13.

§2.8(iv) Case III: Simple Pole

The transformed equation (2.8.8) is renormalized as

2.8.24 d2Wdξ2=(u24ξ+ν2-14ξ2+ψ(ξ)ξ)W.

We again assume ξ(α1,α2) with -α1<0, 0<α2. Also, ψ(ξ) is C on (α1,α2), and u>0. The constant ν (=1+4ρ) is real and nonnegative.

There are two cases: ξ(0,α2) and ξ(α1,0). In the former, corresponding to any positive integer n there are solutions Wn,j(u,ξ), j=1,2, that are C on (0,α2), and as u

2.8.25 Wn,1(u,ξ) =ξ1/2Iν(uξ1/2)s=0n-1As(ξ)u2s+ξIν+1(uξ1/2)s=0n-2Bs(ξ)u2s+1+ξ1/2Iν(uξ1/2)O(1u2n-1),
2.8.26 Wn,2(u,ξ) =ξ1/2Kν(uξ1/2)s=0n-1As(ξ)u2s-ξKν+1(uξ1/2)s=0n-2Bs(ξ)u2s+1+ξ1/2Kν(uξ1/2)O(1u2n-1).

Here A0(ξ)=1,

2.8.27 Bs(ξ)=-As(ξ)+1ξ1/20ξ(ψ(v)As(v)-(ν+12)As(v))dvv1/2,
2.8.28 As+1(ξ)=νBs(ξ)-ξBs(ξ)+ψ(ξ)Bs(ξ)dξ,

s=0,1,2,. For Iν and Kν see §10.25(ii). The expansions (2.8.25) and (2.8.26) are both uniform and differentiable with respect to ξ. These results are valid when 𝒱0,α2(ξ1/2B0) and 𝒱0,α2(ξ1/2Bn-1) are finite.

If ξ(α1,0), then there are solutions Wn,j(u,ξ), j=3,4, that are C on (α1,0), and as u

2.8.29 Wn,3(u,ξ)=|ξ|1/2Jν(u|ξ|1/2)(s=0n-1As(ξ)u2s+O(1u2n-1))-|ξ|Jν+1(u|ξ|1/2)(s=0n-2Bs(ξ)u2s+1+O(1u2n-2)),
2.8.30 Wn,4(u,ξ)=|ξ|1/2Yν(u|ξ|1/2)(s=0n-1As(ξ)u2s+O(1u2n-1))-|ξ|Yν+1(u|ξ|1/2)(s=0n-2Bs(ξ)u2s+1+O(1u2n-2)).

Here A0(ξ)=1,

2.8.31 Bs(ξ)=-As(ξ)+1|ξ|1/2ξ0(ψ(v)As(v)-(ν+12)As(v))dv|v|1/2,

s=0,1,2,, and (2.8.28) again applies. For Jν and Yν see §10.2(ii). The expansions (2.8.29) and (2.8.30) are both uniform and differentiable with respect to ξ. These results are valid when 𝒱α1,0(|ξ|1/2B0) and 𝒱α1,0(|ξ|1/2Bn-1) are finite.

Again, an alternative way of representing the error terms in (2.8.29) and (2.8.30) is by means of envelope functions. Let x=Xν be the smallest positive root of the equation

2.8.32 Jν(x)+Yν(x)=0.

Define

2.8.33 envJν(x) =2Jν(x),
envYν(x) =2|Yν(x)|,
0<xXν,
2.8.34 envJν(x)=envYν(x)=(Jν2(x)+Yν2(x))1/2,
Xνx<.

Then as u

2.8.35 Wn,3(u,ξ)=|ξ|1/2Jν(u|ξ|1/2)s=0n-1As(ξ)u2s-|ξ|Jν+1(u|ξ|1/2)s=0n-2Bs(ξ)u2s+1+|ξ|1/2envJν(u|ξ|1/2)O(1u2n-1),
2.8.36 Wn,4(u,ξ)=|ξ|1/2Yν(u|ξ|1/2)s=0n-1As(ξ)u2s-|ξ|Yν+1(u|ξ|1/2)s=0n-2Bs(ξ)u2s+1+|ξ|1/2envYν(u|ξ|1/2)O(1u2n-1),

uniformly with respect to ξ(α1,0).

For error bounds, more delicate error estimates, extensions to complex ξ, ν, and u, zeros, and examples see Olver (1997b, Chapter 12), Boyd (1990a), and Dunster (1990a).

For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv).

§2.8(v) Multiple and Fractional Turning Points

The approach used in preceding subsections for equation (2.8.1) also succeeds when z0 is a multiple or fractional turning point. For the former f(z) has a zero of multiplicity λ=2,3,4, and g(z) is analytic. For the latter (z-z0)-λf(z) and g(z) are both analytic at z0, λ (>-2) being a real constant. In both cases uniform asymptotic approximations are obtained in terms of Bessel functions of order 1/(λ+2). More generally, g(z) can have a simple or double pole at z0. (In the case of the double pole the order of the approximating Bessel functions is fixed but no longer 1/(λ+2).) However, in all cases with λ>-2 and λ0 or ±1, only uniform asymptotic approximations are available, not uniform asymptotic expansions. For results, including error bounds, see Olver (1977c).

For connection formulas for Liouville–Green approximations across these transition points see Olver (1977b, a, 1978).

§2.8(vi) Coalescing Transition Points

Corresponding to the problems for integrals outlined in §§2.3(v), 2.4(v), and 2.4(vi), there are analogous problems for differential equations.

For two coalescing turning points see Olver (1975a, 1976) and Dunster (1996a); in this case the uniform approximants are parabolic cylinder functions. (For envelope functions for parabolic cylinder functions see §14.15(v)).

For a coalescing turning point and double pole see Boyd and Dunster (1986) and Dunster (1990b); in this case the uniform approximants are Bessel functions of variable order.

For a coalescing turning point and simple pole see Nestor (1984) and Dunster (1994b); in this case the uniform approximants are Whittaker functions (§13.14(i)) with a fixed value of the second parameter.

For further examples of uniform asymptotic approximations in terms of parabolic cylinder functions see §§13.20(iii), 13.20(iv), 14.15(v), 15.12(iii), 18.24.

For further examples of uniform asymptotic approximations in terms of Bessel functions or modified Bessel functions of variable order see §§13.21(ii), 14.15(ii), 14.15(iv), 14.20(viii), 30.9(i), 30.9(ii).

For examples of uniform asymptotic approximations in terms of Whittaker functions with fixed second parameter see §18.15(i) and §28.8(iv).

Lastly, for an example of a fourth-order differential equation, see Wong and Zhang (2007).