§28.9 Zeros

Keywords:: Mathieu functions
Permalink:: http://dlmf.nist.gov/28.9

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

For further details see McLachlan (1947, pp. 234–239) and Meixner and Schäfke (1954, §§2.331, 2.8, 2.81, and 2.85).