The Fourier series of the periodic Mathieu functions converge absolutely and
uniformly on all compact sets in the -plane. For ,
Ambiguities in sign are resolved by (28.4.13)–(28.4.16)
when , and by continuity for the other values of .
For fixed and fixed ,
For further terms and expansions see Meixner and Schäfke (1954, p. 122)
and McLachlan (1947, §3.33).
As , with fixed () and fixed ,
For the basic solutions and see §28.2(ii).