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28 Mathieu Functions and Hill’s EquationHill’s Equation

§28.31 Equations of Whittaker–Hill and Ince

  1. §28.31(i) Whittaker–Hill Equation
  2. §28.31(ii) Equation of Ince; Ince Polynomials
  3. §28.31(iii) Paraboloidal Wave Functions

§28.31(i) Whittaker–Hill Equation

Hill’s equation with three terms

28.31.1 W′′+(A+Bcos(2z)12(kc)2cos(4z))W=0

and constant values of A,B,k, and c, is called the Equation of Whittaker–Hill. It has been discussed in detail by Arscott (1967) for k2<0, and by Urwin and Arscott (1970) for k2>0.

§28.31(ii) Equation of Ince; Ince Polynomials

When k2<0, we substitute

28.31.2 ξ2 =4k2c2,
A =η18ξ2,
B =(p+1)ξ,
W(z) =w(z)exp(14ξcos(2z)),

in (28.31.1). The result is the Equation of Ince:

28.31.3 w′′+ξsin(2z)w+(ηpξcos(2z))w=0.

Formal 2π-periodic solutions can be constructed as Fourier series; compare §28.4:

28.31.4 we,s(z) ==0A2+scos(2+s)z,
28.31.5 wo,s(z) ==0B2+ssin(2+s)z,

where the coefficients satisfy

28.31.6 2ηA0+(2+p)ξA2 =0,
pξA0+(4η)A2+(12p+2)ξA4 =0,
(12p+1)ξA22+(42η)A2+(12p++1)ξA2+2 =0,
28.31.7 (1η+(12p+12)ξ)A1+(12p+32)ξA3 =0,
(12p+12)ξA21+((2+1)2η)A2+1+(12p++32)ξA2+3 =0,
28.31.8 (1η(12p+12)ξ)B1+(12p+32)ξB3 =0,
(12p+12)ξB21+((2+1)2η)B2+1+(12p++32)ξB2+3 =0,
28.31.9 (4η)B2+(12p+2)ξB4 =0,
(12p+1)ξB22+(42η)B2+(12p++1)ξB2+2 =0,

When p is a nonnegative integer, the parameter η can be chosen so that solutions of (28.31.3) are trigonometric polynomials, called Ince polynomials. They are denoted by

28.31.10 C2n2m(z,ξ)with p=2n,C2n+12m+1(z,ξ)with p=2n+1,
28.31.11 S2n+12m+1(z,ξ)with p=2n+1,S2n+22m+2(z,ξ)with p=2n+2,

and m=0,1,,n in all cases.

The values of η corresponding to Cpm(z,ξ), Spm(z,ξ) are denoted by apm(ξ), bpm(ξ), respectively. They are real and distinct, and can be ordered so that Cpm(z,ξ) and Spm(z,ξ) have precisely m zeros, all simple, in 0z<π. The normalization is given by

28.31.12 1π02π(Cpm(x,ξ))2dx=1π02π(Spm(x,ξ))2dx=1,

ambiguities in sign being resolved by requiring Cpm(x,ξ) and Spm(x,ξ) to be continuous functions of x and positive when x=0.

For ξ0, with x fixed,

28.31.13 Cp0(x,ξ) 1/2,
Cpm(x,ξ) cos(mx),
Spm(x,ξ) sin(mx),
apm(ξ),bpm(ξ) m2.

If p and ξ0 in such a way that pξ2q, then in the notation of §§28.2(v) and 28.2(vi)

28.31.14 Cpm(x,ξ) cem(x,q),
Spm(x,ξ) sem(x,q),
28.31.15 apm(ξ) am(q),
bpm(ξ) bm(q).

For proofs and further information, including convergence of the series (28.31.4), (28.31.5), see Arscott (1967).

§28.31(iii) Paraboloidal Wave Functions

With (28.31.10) and (28.31.11),

28.31.16 hcpm(z,ξ)=e14ξcos(2z)Cpm(z,ξ),
28.31.17 hspm(z,ξ)=e14ξcos(2z)Spm(z,ξ),

are called paraboloidal wave functions. They satisfy the differential equation

28.31.18 w′′+(η18ξ2(p+1)ξcos(2z)+18ξ2cos(4z))w=0,

with η=apm(ξ), η=bpm(ξ), respectively.

For change of sign of ξ,

28.31.19 hc2n2m(z,ξ) =(1)mhc2n2m(12πz,ξ),
hc2n+12m+1(z,ξ) =(1)mhs2n+12m+1(12πz,ξ),


28.31.20 hs2n+12m+1(z,ξ) =(1)mhc2n+12m+1(12πz,ξ),
hs2n+22m+2(z,ξ) =(1)mhs2n+22m+2(12πz,ξ).

For m1m2,

28.31.21 02πhcpm1(x,ξ)hcpm2(x,ξ)dx=02πhspm1(x,ξ)hspm2(x,ξ)dx=0.

More important are the double orthogonality relations for p1p2 or m1m2 or both, given by

28.31.22 u0u02πhcp1m1(u,ξ)hcp1m1(v,ξ)hcp2m2(u,ξ)×hcp2m2(v,ξ)(cos(2u)cos(2v))dvdu=0,


28.31.23 u0u02πhsp1m1(u,ξ)hsp1m1(v,ξ)hsp2m2(u,ξ)×hsp2m2(v,ξ)(cos(2u)cos(2v))dvdu=0,

and also for all p1,p2,m1,m2, given by

28.31.24 u0u02πhcp1m1(u,ξ)hcp1m1(v,ξ)hsp2m2(u,ξ)×hsp2m2(v,ξ)(cos(2u)cos(2v))dvdu=0,

where (u0,u)=(0,i) when ξ>0, and (u0,u)=(12π,12π+i) when ξ<0.

For proofs and further integral equations see Urwin (1964, 1965).

Asymptotic Behavior

For ξ>0, the functions hcpm(z,ξ), hspm(z,ξ) behave asymptotically as multiples of exp(14ξcos(2z))(cosz)p as z±i. All other periodic solutions behave as multiples of exp(14ξcos(2z))(cosz)p2.

For ξ>0, the functions hcpm(z,ξ), hspm(z,ξ) behave asymptotically as multiples of exp(14ξcos(2z))(cosz)p2 as z12π±i. All other periodic solutions behave as multiples of exp(14ξcos(2z))(cosz)p.