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
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.
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 ,
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 .
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),