When is replaced by , (28.2.1) becomes the modified Mathieu’s equation:
with its algebraic form
Assume first that is real, is positive, and ; see §28.12(i). Write
Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to as in the respective sectors , being an arbitrary small positive constant. It follows that (28.20.1) has independent and unique solutions , such that
as with , and
as with . See §10.2(ii) for the notation. In addition, there are unique solutions , that are real when is real and have the properties
as with .
For other values of , and the functions , are determined by analytic continuation. Furthermore,