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28 Mathieu Functions and Hill’s EquationMathieu Functions of Integer Order

§28.9 Zeros

For real q each of the functions ce2n(z,q), se2n+1(z,q), ce2n+1(z,q), and se2n+2(z,q) has exactly n zeros in 0<z<12π. They are continuous in q. For q the zeros of ce2n(z,q) and se2n+1(z,q) approach asymptotically the zeros of He2n(q1/4(π-2z)), and the zeros of ce2n+1(z,q) and se2n+2(z,q) approach asymptotically the zeros of He2n+1(q1/4(π-2z)). Here Hen(z) denotes the Hermite polynomial of degree n18.3). Furthermore, for q>0 cem(z,q) and sem(z,q) also have purely imaginary zeros that correspond uniquely to the purely imaginary z-zeros of Jm(2qcosz)10.21(i)), and they are asymptotically equal as q0 and |z|. There are no zeros within the strip |z|<12π 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).