The introduction to the eigenvalues and the functions of general order proceeds as in §§28.2(i), 28.2(ii), and 28.2(iii), except that we now restrict ; equivalently . In consequence, for the Floquet solutions the factor in (28.2.14) is no longer .
For other values of , is determined by analytic continuation. Without loss of generality, from now on we replace by .
For change of signs of and ,
As in §28.7 values of for which (28.2.16) has simple roots are called normal values with respect to . For real values of and all the are real, and is normal. For graphical interpretation see Figure 28.13.1. To complete the definition we require
As a function of with fixed (), is discontinuous at . See Figure 28.13.2.
Two eigenfunctions correspond to each eigenvalue . The Floquet solution with respect to is denoted by . For ,
The other eigenfunction is , a Floquet solution with respect to with . If is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of and by the normalization
They have the following pseudoperiodic and orthogonality properties:
For changes of sign of , , and ,
To complete the definitions of the functions we set
compare (28.12.3). However, these functions are not the limiting values of as .
These functions are real-valued for real , real , and , whereas is complex. When is a rational number, but not an integer, all solutions of Mathieu’s equation are periodic with period .