§28.12 Definitions and Basic Properties

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 $\widehat{\nu }\ne 0,1$; equivalently $\nu \ne n$. In consequence, for the Floquet solutions $w(z)$ the factor ${\mathrm{e}}^{\pi \mathrm{i}\nu }$ 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 ${\lambda }_{\nu +2n}\left(q\right)$, $n=0,\pm 1,\pm 2,\mathrm{\dots }$. When $q=0$ Equation (28.2.16) has simple roots, given by

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 $q$,

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 $q$ ($\ne 0$), ${\lambda }_{\nu }\left(q\right)$ is discontinuous at $\nu =\pm 1,\pm 2,\mathrm{\dots }$. See Figure 28.13.2.

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

The other eigenfunction is ${\mathrm{me}}_{\nu }(-z,q)$, 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 ${\mathrm{me}}_{\nu }$ functions we set

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

These functions are real-valued for real $\nu$, real $q$, and $z=x$, whereas ${\mathrm{me}}_{\nu }(x,q)$ 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 ${\mathrm{ce}}_{\nu }(z,q)$ and ${\mathrm{se}}_{\nu }(z,q)$ as $\nu \to n$ $(\ne 0)$ are not the functions ${\mathrm{ce}}_n(z,q)$ and ${\mathrm{se}}_n(z,q)$ defined in §28.2(vi). Compare e.g. Figure 28.13.3.