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

§28.11 Expansions in Series of Mathieu Functions

Let f(z) be a 2π-periodic function that is analytic in an open doubly-infinite strip S that contains the real axis, and q be a normal value (§28.7). Then

28.11.1 f(z)=α0ce0(z,q)+n=1(αncen(z,q)+βnsen(z,q)),


28.11.2 αn =1π02πf(x)cen(x,q)dx,
βn =1π02πf(x)sen(x,q)dx.

The series (28.11.1) converges absolutely and uniformly on any compact subset of the strip S. See Meixner and Schäfke (1954, §2.28), and for expansions in the case of the exceptional values of q see Meixner et al. (1980, p. 33).


With the notation of §28.4,

28.11.3 1=2n=0A02n(q)ce2n(z,q),
28.11.4 cos2mz=n=0A2m2n(q)ce2n(z,q),
28.11.5 cos(2m+1)z =n=0A2m+12n+1(q)ce2n+1(z,q),
28.11.6 sin(2m+1)z =n=0B2m+12n+1(q)se2n+1(z,q),
28.11.7 sin(2m+2)z =n=0B2m+22n+2(q)se2n+2(z,q).