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§2.7 Differential Equations


§2.7(i) Regular Singularities: Fuchs–Frobenius Theory

An ordinary point of the differential equation

2.7.1 2wz2+f(z)wz+g(z)w=0

is one at which the coefficients f(z) and g(z) are analytic. All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients.

Other points z0 are singularities of the differential equation. If both (z-z0)f(z) and (z-z0)2g(z) are analytic at z0, then z0 is a regular singularity (or singularity of the first kind). All other singularities are classified as irregular.

In a punctured neighborhood N of a regular singularity z0

2.7.2 f(z) =s=0fs(z-z0)s-1,
g(z) =s=0gs(z-z0)s-2,

with at least one of the coefficients f0, g0, g1 nonzero. Let α1, α2 denote the indices or exponents, that is, the roots of the indicial equation

2.7.3 Q(α)α(α-1)+f0α+g0=0.

Provided that α1-α2 is not zero or an integer, equation (2.7.1) has independent solutions wj(z), j=1,2, such that

2.7.4 wj(z)=(z-z0)αjs=0as,j(z-z0)s,

with a0,j=1, and

2.7.5 Q(αj+s)as,j=-r=0s-1((αj+r)fs-r+gs-r)ar,j,

when s=1,2,3,.

If α1-α2=0,1,2,, then (2.7.4) applies only in the case j=1. But there is an independent solution

2.7.6 w2(z)=(z-z0)α2s=0sα1-α2bs(z-z0)s+cw1(z)ln(z-z0),

The coefficients bs and constant c are again determined by equating coefficients in the differential equation, beginning with c=1 when α1-α2=0, or with b0=1 when α1-α2=1,2,3,.

The radii of convergence of the series (2.7.4), (2.7.6) are not less than the distance of the next nearest singularity of the differential equation from z0.

To include the point at infinity in the foregoing classification scheme, we transform it into the origin by replacing z in (2.7.1) with 1/z; see Olver (1997b, pp. 153–154). For corresponding definitions, together with examples, for linear differential equations of arbitrary order see §§16.8(i)16.8(ii).

§2.7(ii) Irregular Singularities of Rank 1

If the singularities of f(z) and g(z) at z0 are no worse than poles, then z0 has rank -1, where is the least integer such that (z-z0)f(z) and (z-z0)2g(z) are analytic at z0. Thus a regular singularity has rank 0. The most common type of irregular singularity for special functions has rank 1 and is located at infinity. Then

2.7.7 f(z) =s=0fszs,
g(z) =s=0gszs,

these series converging in an annulus |z|>a, with at least one of f0, g0, g1 nonzero.

Formal solutions are

2.7.8 λjzzμjs=0as,jzs,

where λ1, λ2 are the roots of the characteristic equation

2.7.9 λ2+f0λ+g0=0,
2.7.10 μj=-(f1λj+g1)/(f0+2λj),

a0,j=1, and

2.7.11 (f0+2λj)sas,j=(s-μj)(s-1-μj)as-1,j+r=1s(λjfr+1+gr+1-(s-r-μj)fr)as-r,j,

when s=1,2,. The construction fails iff λ1=λ2, that is, when f02=4g0: this case is treated below.

For large s,

2.7.12 as,1 Λ1(λ1-λ2)sj=0aj,2(λ1-λ2)jΓ(s+μ2-μ1-j),
2.7.13 as,2 Λ2(λ2-λ1)sj=0aj,1(λ2-λ1)jΓ(s+μ1-μ2-j),

where Λ1 and Λ2 are constants, and the Jth remainder terms in the sums are O(Γ(s+μ2-μ1-J)) and O(Γ(s+μ1-μ2-J)), respectively (Olver (1994a)). Hence unless the series (2.7.8) terminate (in which case the corresponding Λj is zero) they diverge. However, there are unique and linearly independent solutions wj(z), j=1,2, such that

2.7.14 wj(z)λjz((λ2-λ1)z)μjs=0as,jzs

as z in the sectors

2.7.15 -32π+δph((λ2-λ1)z)32π-δ,
2.7.16 -12π+δph((λ2-λ1)z)52π-δ,

δ being an arbitrary small positive constant.

Although the expansions (2.7.14) apply only in the sectors (2.7.15) and (2.7.16), each solution wj(z) can be continued analytically into any other sector. Typical connection formulas are

2.7.17 w1(z) =2πμ1w1(z-2π)+C1w2(z),
w2(z) =-2πμ2w2(z2π)+C2w1(z),

in which C1, C2 are constants, the so-called Stokes multipliers. In combination with (2.7.14) these formulas yield asymptotic expansions for w1(z) in 12π+δph((λ2-λ1)z)52π-δ, and w2(z) in -32π+δph((λ2-λ1)z)12π-δ. Furthermore,

2.7.18 Λ1 =-(μ2-μ1)πC1/(2π),
Λ2 =C2/(2π).

Note that the coefficients in the expansions (2.7.12), (2.7.13) for the “late” coefficients, that is, as,1, as,2 with s large, are the “early” coefficients aj,2, aj,1 with j small. This phenomenon is an example of resurgence, a classification due to Écalle (1981a, b). See §2.11(v) for other examples.

The exceptional case f02=4g0 is handled by Fabry’s transformation:

2.7.19 w =-f0z/2W,
t =z1/2.

The transformed differential equation either has a regular singularity at t=, or its characteristic equation has unequal roots.

For error bounds for (2.7.14) see Olver (1997b, Chapter 7). For the calculation of Stokes multipliers see Olde Daalhuis and Olver (1995b). For extensions to singularities of higher rank see Olver and Stenger (1965). For extensions to higher-order differential equations see Stenger (1966a, b), Olver (1997a, 1999), and Olde Daalhuis and Olver (1998).

§2.7(iii) Liouville–Green (WKBJ) Approximation

For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows:

Liouville–Green Approximation Theorem

In a finite or infinite interval (a1,a2) let f(x) be real, positive, and twice-continuously differentiable, and g(x) be continuous. Then in (a1,a2) the differential equation

2.7.20 2wx2=(f(x)+g(x))w

has twice-continuously differentiable solutions

2.7.21 w1(x) =f-1/4(x)exp(f1/2(x)x)(1+ϵ1(x)),
2.7.22 w2(x) =f-1/4(x)exp(-f1/2(x)x)(1+ϵ2(x)),

such that

2.7.23 |ϵj(x)|, 12f-1/2(x)|ϵj(x)|exp(12𝒱aj,x(F))-1,

provided that 𝒱aj,x(F)<. Here F(x) is the error-control function

2.7.24 F(x)=(1f1/42x2(1f1/4)-gf1/2)x,

and 𝒱 denotes the variational operator (§2.3(i)). Thus

2.7.25 𝒱aj,x(F)=ajx|(1f1/4(t)2t2(1f1/4(t))-g(t)f1/2(t))t|.

Assuming also 𝒱a1,a2(F)<, we have

2.7.26 w1(x)f-1/4(x)exp(f1/2(x)x),
2.7.27 w2(x)f-1/4(x)exp(-f1/2(x)x),

Suppose in addition |f1/2(x)x| is unbounded as xa1+ and xa2-. Then there are solutions w3(x), w4(x), such that

2.7.28 w3(x)f-1/4(x)exp(f1/2(x)x),
2.7.29 w4(x)f-1/4(x)exp(-f1/2(x)x),

The solutions with the properties (2.7.26), (2.7.27) are unique, but not those with the properties (2.7.28), (2.7.29). In fact, since

2.7.30 w1(x)/w4(x)0,

w1(x) is a recessive (or subdominant) solution as xa1+, and w4(x) is a dominant solution as xa1+. Similarly for w2(x) and w3(x) as xa2-.


2.7.31 2wx2=(x+lnx)w,

We cannot take f=x and g=lnx because gf-1/2x would diverge as x+. Instead set f=x+lnx, g=0. By approximating

2.7.32 f1/2=x1/2+12x-1/2lnx+O(x-3/2(lnx)2),

we arrive at

2.7.33 w2(x) x-(1/4)-xexp(2x1/2-23x3/2),
2.7.34 w3(x) x-(1/4)+xexp(23x3/2-2x1/2),

as x+, w2(x) being recessive and w3(x) dominant.

For other examples, and also the corresponding results when f(x) is negative, see Olver (1997b, Chapter 6), Olver (1980a), Taylor (1978, 1982), and Smith (1986). The first of these references includes extensions to complex variables and reversions for zeros.

§2.7(iv) Numerically Satisfactory Solutions

One pair of independent solutions of the equation

2.7.35 2w/z2=w

is w1(z)=z, w2(z)=-z. Another is w3(z)=coshz, w4(z)=sinhz. In theory either pair may be used to construct any other solution

2.7.36 w(z)=Aw1(z)+Bw2(z),


2.7.37 w(z)=Cw3(z)+Dw4(z),

where A,B,C,D are constants. From the numerical standpoint, however, the pair w3(z) and w4(z) has the drawback that severe numerical cancellation can occur with certain combinations of C and D, for example if C and D are equal, or nearly equal, and z, or z, is large and negative. This kind of cancellation cannot take place with w1(z) and w2(z), and for this reason, and following Miller (1950), we call w1(z) and w2(z) a numerically satisfactory pair of solutions.

The solutions w1(z) and w2(z) are respectively recessive and dominant as z-, and vice versa as z+. This is characteristic of numerically satisfactory pairs. In a neighborhood, or sectorial neighborhood of a singularity, one member has to be recessive. In consequence, if a differential equation has more than one singularity in the extended plane, then usually more than two standard solutions need to be chosen in order to have numerically satisfactory representations everywhere.

In oscillatory intervals, and again following Miller (1950), we call a pair of solutions numerically satisfactory if asymptotically they have the same amplitude and are 12π out of phase.