28 Mathieu Functions and Hill’s Equation Mathieu Functions of Integer Order 28.8 Asymptotic Expansions for Large

§28.9 Zeros

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Keywords:: Mathieu functions, zeros
Permalink:: http://dlmf.nist.gov/28.9
See also:: Annotations for Ch.28

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