Digital Library of Mathematical Functions
About the Project
NIST
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)=νπf(z).

Then

28.19.2 f(z)=n=-fnmeν+2n(z,q),

where

28.19.3 fn=1π0πf(z)meν+2n(-z,q)z.

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

Example

28.19.4 νz=n=-c-2nν+2n(q)meν+2n(z,q),

where the coefficients are as in §28.14.