# §28.12 Definitions and Basic Properties

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

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 $\mathop{\cos\/}\nolimits\!\left(\nu\pi\right)$) 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

 28.12.1 $\mathop{\lambda_{\nu+2n}\/}\nolimits\!\left(0\right)=(\nu+2n)^{2}.$ Symbols: $\mathop{\lambda_{\NVar{\nu+2n}}\/}\nolimits\!\left(\NVar{q}\right)$: 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)

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 $q$,

 28.12.2 $\mathop{\lambda_{\nu}\/}\nolimits\!\left(-q\right)=\mathop{\lambda_{\nu}\/}% \nolimits\!\left(q\right)=\mathop{\lambda_{-\nu}\/}\nolimits\!\left(q\right).$ Symbols: $\mathop{\lambda_{\NVar{\nu+2n}}\/}\nolimits\!\left(\NVar{q}\right)$: 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)

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

 28.12.3 $\mathop{\lambda_{m}\/}\nolimits\!\left(q\right)=\begin{cases}\mathop{a_{m}\/}% \nolimits\!\left(q\right),&m=0,1,\dots,\\ \mathop{b_{-m}\/}\nolimits\!\left(q\right),&m=-1,-2,\dots.\end{cases}$

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 $\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(z,q\right)$

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 $q=0$,

 28.12.4 $\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(z,0\right)=e^{\mathrm{i}\nu z}.$

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

 28.12.5 $\int_{0}^{\pi}\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(x,q\right)\mathop{% \mathrm{me}_{\nu}\/}\nolimits\!\left(-x,q\right)\mathrm{d}x=\pi.$

They have the following pseudoperiodic and orthogonality properties:

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

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

 28.12.8 $\displaystyle\mathop{\mathrm{me}_{-\nu}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(-z,q\right),$ 28.12.9 $\displaystyle\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(z,-q\right)$ $\displaystyle=e^{\mathrm{i}\nu\pi/2}\mathop{\mathrm{me}_{\nu}\/}\nolimits\!% \left(z-\tfrac{1}{2}\pi,q\right),$ 28.12.10 $\displaystyle\overline{\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(z,q\right)}$ $\displaystyle=\mathop{\mathrm{me}_{\bar{\nu}}\/}\nolimits\!\left(-\bar{z},\bar% {q}\right).$ Symbols: $\mathop{\mathrm{me}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $q=h^{2}$: parameter, $z$: complex variable and $\nu$: complex parameter Referenced by: §28.12(ii) Permalink: http://dlmf.nist.gov/28.12.E10 Encodings: TeX, pMML, png See also: Annotations for 28.12(ii)

(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

 28.12.11 $\displaystyle\mathop{\mathrm{me}_{n}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=\sqrt{2}\mathop{\mathrm{ce}_{n}\/}\nolimits\!\left(z,q\right),$ $n=0,1,2,\dots$, $\displaystyle\mathop{\mathrm{me}_{-n}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=-\sqrt{2}\mathrm{i}\mathop{\mathrm{se}_{n}\/}\nolimits\!\left(z,% q\right),$ $n=1,2,\dots$;

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 $\nu\notin\mathbb{Z}$

 28.12.12 $\displaystyle\mathop{\mathrm{ce}_{\nu}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=\tfrac{1}{2}\left(\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(z% ,q\right)+\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(-z,q\right)\right),$ 28.12.13 $\displaystyle\mathop{\mathrm{se}_{\nu}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=-\tfrac{1}{2}\mathrm{i}\left(\mathop{\mathrm{me}_{\nu}\/}% \nolimits\!\left(z,q\right)-\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(-z,q% \right)\right).$

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 $z$,

 28.12.14 $\displaystyle\mathop{\mathrm{ce}_{\nu}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=\mathop{\mathrm{ce}_{\nu}\/}\nolimits\!\left(-z,q\right)=\mathop% {\mathrm{ce}_{-\nu}\/}\nolimits\!\left(z,q\right),$ 28.12.15 $\displaystyle\mathop{\mathrm{se}_{\nu}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=-\mathop{\mathrm{se}_{\nu}\/}\nolimits\!\left(-z,q\right)=-% \mathop{\mathrm{se}_{-\nu}\/}\nolimits\!\left(z,q\right).$

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.