28 Mathieu Functions and Hill’s EquationMathieu Functions of Integer Order28.8 Asymptotic Expansions for Large $q$28.10 Integral Equations

For real $q$ each of the functions ${\mathrm{ce}}_{2n}\left(z,q\right)$, ${\mathrm{se}}_{2n+1}\left(z,q\right)$, ${\mathrm{ce}}_{2n+1}\left(z,q\right)$, and ${\mathrm{se}}_{2n+2}\left(z,q\right)$ has exactly $n$ zeros in $$. They are continuous in $q$. For $q\to \mathrm{\infty}$ the zeros of ${\mathrm{ce}}_{2n}\left(z,q\right)$ and ${\mathrm{se}}_{2n+1}\left(z,q\right)$ approach asymptotically the zeros of ${\mathit{He}}_{2n}\left({q}^{1/4}\left(\pi -2z\right)\right)$, and the zeros of ${\mathrm{ce}}_{2n+1}\left(z,q\right)$ and ${\mathrm{se}}_{2n+2}\left(z,q\right)$ approach asymptotically the zeros of ${\mathit{He}}_{2n+1}\left({q}^{1/4}\left(\pi -2z\right)\right)$. Here ${\mathit{He}}_{n}\left(z\right)$ denotes the Hermite polynomial of degree $n$ (§18.3). Furthermore, for $q>0$ ${\mathrm{ce}}_{m}\left(z,q\right)$ and ${\mathrm{se}}_{m}\left(z,q\right)$ also have purely imaginary zeros that correspond uniquely to the purely imaginary $z$-zeros of ${J}_{m}\left(2\sqrt{q}\mathrm{cos}z\right)$ (§10.21(i)), and they are asymptotically equal as $q\to 0$ and $\left|\mathrm{\Im}z\right|\to \mathrm{\infty}$. There are no zeros within the strip $$ other than those on the real and imaginary axes.