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28 Mathieu Functions and Hill’s EquationMathieu Functions of Integer Order

§28.2 Definitions and Basic Properties


§28.2(i) Mathieu’s Equation

The standard form of Mathieu’s equation with parameters (a,q) is

28.2.1 w′′+(a-2qcos(2z))w=0.

With ζ=sin2z we obtain the algebraic form of Mathieu’s equation

28.2.2 ζ(1-ζ)w′′+12(1-2ζ)w+14(a-2q(1-2ζ))w=0.

This equation has regular singularities at 0 and 1, both with exponents 0 and 12, and an irregular singular point at . With ζ=cosz we obtain another algebraic form:

28.2.3 (1-ζ2)w′′-ζw+(a+2q-4qζ2)w=0.

§28.2(ii) Basic Solutions wI, wII

Since (28.2.1) has no finite singularities its solutions are entire functions of z. Furthermore, a solution w with given initial constant values of w and w at a point z0 is an entire function of the three variables z, a, and q.

The following three transformations

28.2.4 z -z;
z z±π;
z z±12π,q

each leave (28.2.1) unchanged. (28.2.1) possesses a fundamental pair of solutions wI(z;a,q),wII(z;a,q) called basic solutions with

28.2.5 [wI(0;a,q)wII(0;a,q)wI(0;a,q)wII(0;a,q)]=[1001].

wI(z;a,q) is even and wII(z;a,q) is odd. Other properties are as follows.

28.2.6 𝒲{wI,wII}=1,
28.2.7 wI(z±π;a,q) =wI(π;a,q)wI(z;a,q)±wI(π;a,q)wII(z;a,q),
28.2.8 wII(z±π;a,q) =±wII(π;a,q)wI(z;a,q)+wII(π;a,q)wII(z;a,q),
28.2.9 wI(π;a,q) =wII(π;a,q),
28.2.10 wI(π;a,q)-1 =2wI(12π;a,q)wII(12π;a,q),
28.2.11 wI(π;a,q)+1 =2wI(12π;a,q)wII(12π;a,q),
28.2.12 wI(π;a,q) =2wI(12π;a,q)wI(12π;a,q),
28.2.13 wII(π;a,q) =2wII(12π;a,q)wII(12π;a,q).

§28.2(iii) Floquet’s Theorem and the Characteristic Exponents

Let ν be any real or complex constant. Then Mathieu’s equation (28.2.1) has a nontrivial solution w(z) such that

28.2.14 w(z+π)=eπiνw(z),

iff eπiν is an eigenvalue of the matrix

28.2.15 [wI(π;a,q)wII(π;a,q)wI(π;a,q)wII(π;a,q)].


28.2.16 cos(πν)=wI(π;a,q)=wI(π;a,-q).

This is the characteristic equation of Mathieu’s equation (28.2.1). cos(πν) is an entire function of a,q2. The solutions of (28.2.16) are given by ν=π-1arccos(wI(π;a,q)). If the inverse cosine takes its principal value (§4.23(ii)), then ν=ν^, where 0ν^1. The general solution of (28.2.16) is ν=±ν^+2n, where n. Either ν^ or ν is called a characteristic exponent of (28.2.1). If ν^=0 or 1, or equivalently, ν=n, then ν is a double root of the characteristic equation, otherwise it is a simple root.

§28.2(iv) Floquet Solutions

A solution with the pseudoperiodic property (28.2.14) is called a Floquet solution with respect to ν. (28.2.9), (28.2.16), and (28.2.7) give for each solution w(z) of (28.2.1) the connection formula

28.2.17 w(z+π)+w(z-π)=2cos(πν)w(z).

Therefore a nontrivial solution w(z) is either a Floquet solution with respect to ν, or w(z+π)-eiνπw(z) is a Floquet solution with respect to -ν.

If q0, then for a given value of ν the corresponding Floquet solution is unique, except for an arbitrary constant factor (Theorem of Ince; see also 28.5(i)).

The Fourier series of a Floquet solution

28.2.18 w(z)=n=-c2nei(ν+2n)z

converges absolutely and uniformly in compact subsets of . The coefficients c2n satisfy

28.2.19 qc2n+2-(a-(ν+2n)2)c2n+qc2n-2=0,

Conversely, a nontrivial solution c2n of (28.2.19) that satisfies

28.2.20 limn±|c2n|1/|n|=0

leads to a Floquet solution.

§28.2(v) Eigenvalues an, bn

For given ν and q, equation (28.2.16) determines an infinite discrete set of values of a, the eigenvalues or characteristic values, of Mathieu’s equation. When ν^=0 or 1, the notation for the two sets of eigenvalues corresponding to each ν^ is shown in Table 28.2.1, together with the boundary conditions of the associated eigenvalue problem. In Table 28.2.1 n=0,1,2,.

Table 28.2.1: Eigenvalues of Mathieu’s equation.
ν^ Boundary Conditions Eigenvalues
0 w(0)=w(12π)=0 a2n(q)
1 w(0)=w(12π)=0 a2n+1(q)
1 w(0)=w(12π)=0 b2n+1(q)
0 w(0)=w(12π)=0 b2n+2(q)

An equivalent formulation is given by

28.2.21 wI(12π;a,q)=0,a=a2n(q),wI(12π;a,q)=0,a=a2n+1(q),


28.2.22 wII(12π;a,q)=0,a=b2n+1(q),wII(12π;a,q)=0,a=b2n+2(q),

where n=0,1,2,. When q=0,

28.2.23 an(0) =n2,
28.2.24 bn(0) =n2,

Near q=0, an(q) and bn(q) can be expanded in power series in q (see §28.6(i)); elsewhere they are determined by analytic continuation (see §28.7). For nonnegative real values of q, see Figure 28.2.1.

See accompanying text
Figure 28.2.1: Eigenvalues an(q), bn(q) of Mathieu’s equation as functions of q for 0q10, n=0,1,2,3,4 (a’s), n=1,2,3,4 (b’s). Magnify


28.2.25 for q>0:a0<b1<a1<b2<a2<b3<,for q<0:a0<a1<b1<b2<a2<a3<.

Change of Sign of q

28.2.26 a2n(-q) =a2n(q),
28.2.27 a2n+1(-q) =b2n+1(q),
28.2.28 b2n+2(-q) =b2n+2(q).

§28.2(vi) Eigenfunctions

Table 28.2.2 gives the notation for the eigenfunctions corresponding to the eigenvalues in Table 28.2.1. Period π means that the eigenfunction has the property w(z+π)=w(z), whereas antiperiod π means that w(z+π)=-w(z). Even parity means w(-z)=w(z), and odd parity means w(-z)=-w(z).

Table 28.2.2: Eigenfunctions of Mathieu’s equation.
Eigenvalues Eigenfunctions Periodicity Parity
a2n(q) ce2n(z,q) Period π Even
a2n+1(q) ce2n+1(z,q) Antiperiod π Even
b2n+1(q) se2n+1(z,q) Antiperiod π Odd
b2n+2(q) se2n+2(z,q) Period π Odd

When q=0,

28.2.29 ce0(z,0) =1/2,
cen(z,0) =cos(nz),
sen(z,0) =sin(nz),

For simple roots q of the corresponding equations (28.2.21) and (28.2.22), the functions are made unique by the normalizations

28.2.30 02π(cen(x,q))2dx =π,
02π(sen(x,q))2dx =π,

the ambiguity of sign being resolved by (28.2.29) when q=0 and by continuity for the other values of q.

The functions are orthogonal, that is,

28.2.31 02πcem(x,q)cen(x,q)dx =0,
28.2.32 02πsem(x,q)sen(x,q)dx =0,
28.2.33 02πcem(x,q)sen(x,q)dx =0.

For change of sign of q (compare (28.2.4))

28.2.34 ce2n(z,-q) =(-1)nce2n(12π-z,q),
28.2.35 ce2n+1(z,-q) =(-1)nse2n+1(12π-z,q),
28.2.36 se2n+1(z,-q) =(-1)nce2n+1(12π-z,q),
28.2.37 se2n+2(z,-q) =(-1)nse2n+2(12π-z,q).

For the connection with the basic solutions in §28.2(ii),

28.2.38 cen(z,q)cen(0,q) =wI(z;an(q),q),
28.2.39 sen(z,q)sen(0,q) =wII(z;bn(q),q),