About the Project
28 Mathieu Functions and Hill’s EquationMathieu Functions of Integer Order

§28.2 Definitions and Basic Properties

  1. §28.2(i) Mathieu’s Equation
  2. §28.2(ii) Basic Solutions wI, wII
  3. §28.2(iii) Floquet’s Theorem and the Characteristic Exponents
  4. §28.2(iv) Floquet Solutions
  5. §28.2(v) Eigenvalues an, bn
  6. §28.2(vi) Eigenfunctions

§28.2(i) Mathieu’s Equation

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

28.2.1 w′′+(a2qcos(2z))w=0.

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

28.2.2 ζ(1ζ)w′′+12(12ζ)w+14(a2q(12ζ))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 (1ζ2)w′′ζw+(a+2q4qζ2)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,qq;

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=π1arccos(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(zπ)=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+qc2n2=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(12πz,q),
28.2.35 ce2n+1(z,q) =(1)nse2n+1(12πz,q),
28.2.36 se2n+1(z,q) =(1)nce2n+1(12πz,q),
28.2.37 se2n+2(z,q) =(1)nse2n+2(12πz,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),