The standard form of Mathieu’s equation with parameters is
With we obtain the algebraic form of Mathieu’s equation
This equation has regular singularities at 0 and 1, both with exponents 0 and , and an irregular singular point at . With we obtain another algebraic form:
Since (28.2.1) has no finite singularities its solutions are entire functions of . Furthermore, a solution with given initial constant values of and at a point is an entire function of the three variables , , and .
The following three transformations
each leave (28.2.1) unchanged. (28.2.1) possesses a fundamental pair of solutions called basic solutions with
is even and is odd. Other properties are as follows.
Let be any real or complex constant. Then Mathieu’s equation (28.2.1) has a nontrivial solution such that
iff is an eigenvalue of the matrix
This is the characteristic equation of Mathieu’s equation (28.2.1). is an entire function of . The solutions of (28.2.16) are given by . If the inverse cosine takes its principal value (§4.23(ii)), then , where . The general solution of (28.2.16) is , where . Either or is called a characteristic exponent of (28.2.1). If or , or equivalently, , then is a double root of the characteristic equation, otherwise it is a simple root.
A solution with the pseudoperiodic property (28.2.14) is called a Floquet solution with respect to . (28.2.9), (28.2.16), and (28.2.7) give for each solution of (28.2.1) the connection formula
Therefore a nontrivial solution is either a Floquet solution with respect to , or is a Floquet solution with respect to .
If , then for a given value of the corresponding Floquet solution is unique, except for an arbitrary constant factor (Theorem of Ince; see also 28.5(i)).
The Fourier series of a Floquet solution
converges absolutely and uniformly in compact subsets of . The coefficients satisfy
Conversely, a nontrivial solution of (28.2.19) that satisfies
leads to a Floquet solution.
For given and , equation (28.2.16) determines an infinite discrete set of values of , the eigenvalues or characteristic values, of Mathieu’s equation. When or , the notation for the two sets of eigenvalues corresponding to each is shown in Table 28.2.1, together with the boundary conditions of the associated eigenvalue problem. In Table 28.2.1 .
An equivalent formulation is given by
where . When ,
Near , and can be expanded in power series in (see §28.6(i)); elsewhere they are determined by analytic continuation (see §28.7). For nonnegative real values of , see Figure 28.2.1.
Table 28.2.2 gives the notation for the eigenfunctions corresponding to the eigenvalues in Table 28.2.1. Period means that the eigenfunction has the property , whereas antiperiod means that . Even parity means , and odd parity means .
For simple roots of the corresponding equations (28.2.21) and (28.2.22), the functions are made unique by the normalizations
the ambiguity of sign being resolved by (28.2.29) when and by continuity for the other values of .
The functions are orthogonal, that is,
For change of sign of (compare (28.2.4))
For the connection with the basic solutions in §28.2(ii),