If a nontrivial solution of Mathieu’s equation with has period or , then any linearly independent solution cannot have either period.
Second solutions of (28.2.1) are given by
when , , and by
when , . For , we have
compare §28.2(vi). The functions , are unique.
The factors and in (28.5.1) and (28.5.2) are normalized so that
As with , , , , and . This determines the signs of and . (Other normalizations for and can be found in the literature, but most formulas—including connection formulas—are unaffected since and are invariant.)
See (28.22.12) for and .