# §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 $e^{\pi i\nu}$ 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 $\mathop{\lambda_{\nu+2n}\/}\nolimits\!\left(0\right)=(\nu+2n)^{2}.$

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 $\mathop{\lambda_{\nu}\/}\nolimits\!\left(-q\right)=\mathop{\lambda_{\nu}\/}% \nolimits\!\left(q\right)=\mathop{\lambda_{-\nu}\/}\nolimits\!\left(q\right).$

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

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

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 $z$,

 28.12.8 $\displaystyle\mathop{\mathrm{me}_{-\nu}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(-z,q\right),$ 28.12.9 $\displaystyle\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(z,-q\right)$ $\displaystyle=e^{i\nu\pi/2}\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(z-% \tfrac{1}{2}\pi,q\right),$ 28.12.10 $\displaystyle\overline{\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(z,q\right)}$ $\displaystyle=\mathop{\mathrm{me}_{\bar{\nu}}\/}\nolimits\!\left(-\bar{z},\bar% {q}\right).$

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

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 $\nu\notin\Integer$

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

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

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.