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

§28.19 Expansions in Series of meν+2n Functions

Let q be a normal value (§28.12(i)) with respect to ν, and f(z) be a function that is analytic on a doubly-infinite open strip S that contains the real axis. Assume also

28.19.1 f(z+π)=eiνπf(z).


28.19.2 f(z)=n=fnmeν+2n(z,q),


28.19.3 fn=1π0πf(z)meν+2n(z,q)dz.

The series (28.19.2) converges absolutely and uniformly on compact subsets within S.


28.19.4 eiνz=n=c2nν+2n(q)meν+2n(z,q),

where the coefficients are as in §28.14.