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