§28.12 Definitions and Basic Properties

§28.12(i) Eigenvalues

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 given (or ) and , equation (28.2.16) determines an infinite discrete set of values of , denoted by , . When Equation (28.2.16) has simple roots, given by

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.

§28.12(ii) Eigenfunctions

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:

28.12.6
28.12.7.

For changes of sign of , , and ,

(28.12.10) is not valid for cuts on the real axis in the -plane for special complex values of ; but it remains valid for small ; compare §28.7.

To complete the definitions of the functions we set

compare (28.12.3). However, these functions are not the limiting values of as .

§28.12(iii) Functions , , when

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 .

For change of signs of and ,

Again, the limiting values of and as are not the functions and defined in §28.2(vi). Compare e.g. Figure 28.13.3.