The standard form of Mathieu’s equation with parameters is
28.2.1 | |||
With we obtain the algebraic form of Mathieu’s equation
28.2.2 | |||
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:
28.2.3 | |||
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
28.2.4 | ||||
each leave (28.2.1) unchanged. (28.2.1) possesses a fundamental pair of solutions called basic solutions with
28.2.5 | |||
is even and is odd. Other properties are as follows.
28.2.6 | |||
28.2.7 | ||||
28.2.8 | ||||
28.2.9 | ||||
28.2.10 | ||||
28.2.11 | ||||
28.2.12 | ||||
28.2.13 | ||||
Let be any real or complex constant. Then Mathieu’s equation (28.2.1) has a nontrivial solution such that
28.2.14 | |||
iff is an eigenvalue of the matrix
28.2.15 | |||
Equivalently,
28.2.16 | |||
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
28.2.17 | |||
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
28.2.18 | |||
converges absolutely and uniformly in compact subsets of . The coefficients satisfy
28.2.19 | |||
. | |||
Conversely, a nontrivial solution of (28.2.19) that satisfies
28.2.20 | |||
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 .
Boundary Conditions | Eigenvalues | |
---|---|---|
0 | ||
1 | ||
1 | ||
0 |
An equivalent formulation is given by
28.2.21 | |||
and
28.2.22 | |||
where . When ,
28.2.23 | ||||
, | ||||
28.2.24 | ||||
. | ||||
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.
28.2.25 | |||
28.2.26 | ||||
28.2.27 | ||||
28.2.28 | ||||
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 .
Eigenvalues | Eigenfunctions | Periodicity | Parity |
---|---|---|---|
Period | Even | ||
Antiperiod | Even | ||
Antiperiod | Odd | ||
Period | Odd |
When ,
28.2.29 | ||||
, | ||||
. | ||||
For simple roots of the corresponding equations (28.2.21) and (28.2.22), the functions are made unique by the normalizations
28.2.30 | ||||
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,
28.2.31 | ||||
, | ||||
28.2.32 | ||||
, | ||||
28.2.33 | ||||
For change of sign of (compare (28.2.4))
28.2.34 | ||||
28.2.35 | ||||
28.2.36 | ||||
28.2.37 | ||||
For the connection with the basic solutions in §28.2(ii),
28.2.38 | ||||
, | ||||
28.2.39 | ||||
. | ||||