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

§28.12 Definitions and Basic Properties

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§28.12(i) Eigenvalues \mathop{\lambda_{{\nu+2n}}\/}\nolimits\!\left(q\right)

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Notes:
See Meixner and Schäfke (1954, §2.22).
Defines:
\mathop{\lambda_{{\nu+2n}}\/}\nolimits\!\left(q\right): eigenvalues of Mathieu equation
Keywords:
Mathieu’s equation
Referenced by:
§28.19, §28.20(iii), §28.34(ii)
Permalink:
http://dlmf.nist.gov/28.12.i

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 \widehat{\nu}\neq 0,1; equivalently \nu\neq n. In consequence, for the Floquet solutions w(z) the factor e^{{\pi i\nu}} in (28.2.14) is no longer \pm 1.

For given \nu (or \mathop{\cos\/}\nolimits\!\left(\nu\pi\right)) and q, equation (28.2.16) determines an infinite discrete set of values of a, denoted by \mathop{\lambda_{{\nu+2n}}\/}\nolimits\!\left(q\right), n=0,\pm 1,\pm 2,\dots. When q=0 Equation (28.2.16) has simple roots, given by

For other values of q, \mathop{\lambda_{{\nu+2n}}\/}\nolimits\!\left(q\right) is determined by analytic continuation. Without loss of generality, from now on we replace \nu+2n by \nu.

For change of signs of \nu and q,

As in §28.7 values of q for which (28.2.16) has simple roots \lambda are called normal values with respect to \nu. For real values of \nu and q all the \mathop{\lambda_{{\nu}}\/}\nolimits\!\left(q\right) are real, and q is normal. For graphical interpretation see Figure 28.13.1. To complete the definition we require

As a function of \nu with fixed q (\neq 0), \mathop{\lambda_{{\nu}}\/}\nolimits\!\left(q\right) is discontinuous at \nu=\pm 1,\pm 2,\dots. See Figure 28.13.2.

§28.12(ii) Eigenfunctions \mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(z,q\right)

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Notes:
See Arscott (1964b, §6.5) and Meixner and Schäfke (1954, pp. 111 and 114).
Defines:
\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right): Mathieu function
Keywords:
Mathieu functions, pseudoperiodic solutions
Referenced by:
‣ §28.8(iv)
Permalink:
http://dlmf.nist.gov/28.12.ii

Two eigenfunctions correspond to each eigenvalue a=\mathop{\lambda_{{\nu}}\/}\nolimits\!\left(q\right). The Floquet solution with respect to \nu is denoted by \mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(z,q\right). For q=0,

The other eigenfunction is \mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(-z,q\right), a Floquet solution with respect to -\nu with a=\mathop{\lambda_{{\nu}}\/}\nolimits\!\left(q\right). 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

They have the following pseudoperiodic and orthogonality properties:

28.12.6\mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(z+\pi,q\right)=e^{{\pi i\nu}}% \mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(z,q\right),
28.12.7{\int_{0}^{\pi}\mathop{\mathrm{me}_{{\nu+2m}}\/}\nolimits\!\left(x,q\right)% \mathop{\mathrm{me}_{{\nu+2n}}\/}\nolimits\!\left(-x,q\right)dx=0,}m\neq n.

For changes of sign of \nu, q, and z,

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

To complete the definitions of the \mathop{\mathrm{me}_{{\nu}}\/}\nolimits functions we set

compare (28.12.3). However, these functions are not the limiting values of \mathop{\mathrm{me}_{{\pm\nu}}\/}\nolimits\!\left(z,q\right) as \nu\to n (\neq 0).

§28.12(iii) Functions \mathop{\mathrm{ce}_{{\nu}}\/}\nolimits\!\left(z,q\right), \mathop{\mathrm{se}_{{\nu}}\/}\nolimits\!\left(z,q\right), when \nu\notin\Integer

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Notes:
See Meixner and Schäfke (1954, p. 115).
Keywords:
Mathieu functions
Permalink:
http://dlmf.nist.gov/28.12.iii

These functions are real-valued for real \nu, real q, and z=x, whereas \mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(x,q\right) is complex. When \nu=s/m is a rational number, but not an integer, all solutions of Mathieu’s equation are periodic with period 2m\pi.

For change of signs of \nu and z,

Again, the limiting values of \mathop{\mathrm{ce}_{{\nu}}\/}\nolimits(z,q) and \mathop{\mathrm{se}_{{\nu}}\/}\nolimits(z,q) as \nu\to n (\neq 0) are not the functions \mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right) and \mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right) defined in §28.2(vi). Compare e.g. Figure 28.13.3.

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