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

§28.12 Definitions and Basic Properties

  1. §28.12(i) Eigenvalues λν+2n(q)
  2. §28.12(ii) Eigenfunctions meν(z,q)
  3. §28.12(iii) Functions ceν(z,q), seν(z,q), when ν

§28.12(i) Eigenvalues λν+2n(q)

The introduction to the eigenvalues and the functions of general order proceeds as in §§28.2(i), 28.2(ii), and 28.2(iii), except that we now restrict ν^0,1; equivalently νn. In consequence, for the Floquet solutions w(z) the factor eπiν in (28.2.14) is no longer ±1.

For given ν (or cos(νπ)) and q, equation (28.2.16) determines an infinite discrete set of values of a, denoted by λν+2n(q), n=0,±1,±2,. When q=0 Equation (28.2.16) has simple roots, given by

28.12.1 λν+2n(0)=(ν+2n)2.

For other values of q, λν+2n(q) is determined by analytic continuation. Without loss of generality, from now on we replace ν+2n by ν.

For change of signs of ν and q,

28.12.2 λν(q)=λν(q)=λν(q).

As in §28.7 values of q for which (28.2.16) has simple roots λ are called normal values with respect to ν. For real values of ν and q all the λν(q) are real, and q is normal. For graphical interpretation see Figure 28.13.1. To complete the definition we require

28.12.3 λm(q)={am(q),m=0,1,,bm(q),m=1,2,.

As a function of ν with fixed q (0), λν(q) is discontinuous at ν=±1,±2,. See Figure 28.13.2.

§28.12(ii) Eigenfunctions meν(z,q)

Two eigenfunctions correspond to each eigenvalue a=λν(q). The Floquet solution with respect to ν is denoted by meν(z,q). For q=0,

28.12.4 meν(z,0)=eiνz.

The other eigenfunction is meν(z,q), a Floquet solution with respect to ν with a=λν(q). If q is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of z and q by the normalization

28.12.5 0πmeν(x,q)meν(x,q)dx=π.

They have the following pseudoperiodic and orthogonality properties:

28.12.6 meν(z+π,q)=eπiνmeν(z,q),
28.12.7 0πmeν+2m(x,q)meν+2n(x,q)dx=0,

For changes of sign of ν, q, and z,

28.12.8 meν(z,q) =meν(z,q),
28.12.9 meν(z,q) =eiνπ/2meν(z12π,q),
28.12.10 meν(z,q)¯ =meν¯(z¯,q¯).

(28.12.10) is not valid for cuts on the real axis in the q-plane for special complex values of ν; but it remains valid for small q; compare §28.7.

To complete the definitions of the meν functions we set

28.12.11 men(z,q) =2cen(z,q),
men(z,q) =2isen(z,q),

compare (28.12.3). However, these functions are not the limiting values of me±ν(z,q) as νn (0).

§28.12(iii) Functions ceν(z,q), seν(z,q), when ν

28.12.12 ceν(z,q) =12(meν(z,q)+meν(z,q)),
28.12.13 seν(z,q) =12i(meν(z,q)meν(z,q)).

These functions are real-valued for real ν, real q, and z=x, whereas meν(x,q) is complex. When ν=s/m is a rational number, but not an integer, all solutions of Mathieu’s equation are periodic with period 2mπ.

For change of signs of ν and z,

28.12.14 ceν(z,q) =ceν(z,q)=ceν(z,q),
28.12.15 seν(z,q) =seν(z,q)=seν(z,q).

Again, the limiting values of ceν(z,q) and seν(z,q) as νn (0) are not the functions cen(z,q) and sen(z,q) defined in §28.2(vi). Compare e.g. Figure 28.13.3.