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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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Keywords:
Mathieu functions, definitions
Permalink:
http://dlmf.nist.gov/28.12
See also:
Annotations for Ch.28
Contents
  1. §28.12(i) Eigenvalues \lambda_{\nu+2n}\left(q\right)
  2. §28.12(ii) Eigenfunctions \operatorname{me}_{\nu}\left(z,q\right)
  3. §28.12(iii) Functions \operatorname{ce}_{\nu}\left(z,q\right), \operatorname{se}_{\nu}\left(z,q\right), when \nu\notin\mathbb{Z}

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

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Defines:
\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right): eigenvalues of Mathieu equation
Keywords:
Mathieu’s equation, distribution, eigenvalues (or characteristic values), normal values, notation, reflection properties in q, reflection properties in \nu
Notes:
See Meixner and Schäfke (1954, §2.22).
Referenced by:
§28.19, §28.20(iii), §28.34(ii)
Permalink:
http://dlmf.nist.gov/28.12.i
See also:
Annotations for §28.12 and Ch.28

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\mathrm{i}\nu} in (28.2.14) is no longer \pm 1.

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

28.12.1 \lambda_{\nu+2n}\left(0\right)=(\nu+2n)^{2}.
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Symbols:
\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right): eigenvalues of Mathieu equation, n: integer and \nu: complex parameter
Permalink:
http://dlmf.nist.gov/28.12.E1
Encodings:
TeX, pMML, png
See also:
Annotations for §28.12(i), §28.12 and Ch.28

For other values of q, \lambda_{\nu+2n}\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,

28.12.2 \lambda_{\nu}\left(-q\right)=\lambda_{\nu}\left(q\right)=\lambda_{-\nu}\left(q% \right).
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Symbols:
\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right): eigenvalues of Mathieu equation, q=h^{2}: parameter and \nu: complex parameter
Permalink:
http://dlmf.nist.gov/28.12.E2
Encodings:
TeX, pMML, png
See also:
Annotations for §28.12(i), §28.12 and Ch.28

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 \lambda_{\nu}\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), \lambda_{\nu}\left(q\right) is discontinuous at \nu=\pm 1,\pm 2,\dots. See Figure 28.13.2.

§28.12(ii) Eigenfunctions \operatorname{me}_{\nu}\left(z,q\right)

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

The other eigenfunction is \operatorname{me}_{\nu}\left(-z,q\right), a Floquet solution with respect to -\nu with a=\lambda_{\nu}\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:

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 \operatorname{me}_{\nu} functions we set

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

§28.12(iii) Functions \operatorname{ce}_{\nu}\left(z,q\right), \operatorname{se}_{\nu}\left(z,q\right), when \nu\notin\mathbb{Z}

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Keywords:
Mathieu functions, analytic properties, connection formulas, limiting forms as order tends to integers, reflection properties in q
Notes:
See Meixner and Schäfke (1954, p. 115).
Permalink:
http://dlmf.nist.gov/28.12.iii
See also:
Annotations for §28.12 and Ch.28
28.12.12 \operatorname{ce}_{\nu}\left(z,q\right)=\tfrac{1}{2}\left(\operatorname{me}_{% \nu}\left(z,q\right)+\operatorname{me}_{\nu}\left(-z,q\right)\right),
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Symbols:
\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right): Mathieu function, \operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right): Mathieu function, q=h^{2}: parameter, z: complex variable and \nu: complex parameter
A&S Ref:
20.5.2 (in different form)
Permalink:
http://dlmf.nist.gov/28.12.E12
Encodings:
TeX, pMML, png
See also:
Annotations for §28.12(iii), §28.12 and Ch.28

These functions are real-valued for real \nu, real q, and z=x, whereas \operatorname{me}_{\nu}\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,

28.12.14 \operatorname{ce}_{\nu}\left(z,q\right)=\operatorname{ce}_{\nu}\left(-z,q% \right)=\operatorname{ce}_{-\nu}\left(z,q\right),
28.12.15 \operatorname{se}_{\nu}\left(z,q\right)=-\operatorname{se}_{\nu}\left(-z,q% \right)=-\operatorname{se}_{-\nu}\left(z,q\right).

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

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