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28 Mathieu Functions and Hill’s EquationHill’s Equation

§28.30 Expansions in Series of Eigenfunctions

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§28.30(i) Real Variable

Let λ^m, m=0,1,2,, be the set of characteristic values (28.29.16) and (28.29.17), arranged in their natural order (see (28.29.18)), and let wm(x), m=0,1,2,, be the eigenfunctions, that is, an orthonormal set of 2π-periodic solutions; thus

28.30.1 wm′′+(λ^m+Q(x))wm =0,
28.30.2 12π02πwm(x)wn(x)x =δm,n.

Then every continuous 2π-periodic function f(x) whose second derivative is square-integrable over the interval [0,2π] can be expanded in a uniformly and absolutely convergent series

28.30.3 f(x)=m=0fmwm(x),

where

28.30.4 fm=12π02πf(x)wm(x)x.

§28.30(ii) Complex Variable

For analogous results to those of §28.19, see Schäfke (1960, 1961b), and Meixner et al. (1980, §1.1.11).