§28.2 Definitions and Basic Properties

The standard form of Mathieu’s equation with parameters $(a,q)$ is

With $\zeta ={\sin }^{2}z$ we obtain the algebraic form of Mathieu’s equation

This equation has regular singularities at 0 and 1, both with exponents 0 and $\frac{1}{2}$, and an irregular singular point at $\mathrm{\infty }$. With $\zeta =\cos z$ we obtain another algebraic form:

Since (28.2.1) has no finite singularities its solutions are entire functions of $z$. Furthermore, a solution $w$ with given initial constant values of $w$ and $w^{\prime }$ at a point $z_0$ is an entire function of the three variables $z$, $a$, and $q$.

The following three transformations

each leave (28.2.1) unchanged. (28.2.1) possesses a fundamental pair of solutions $w_{\text{I}}(z;a,q),w_{\text{II}}(z;a,q)$ called basic solutions with

$w_{\text{I}}(z;a,q)$ is even and $w_{\text{II}}(z;a,q)$ is odd. Other properties are as follows.

Let $\nu$ be any real or complex constant. Then Mathieu’s equation (28.2.1) has a nontrivial solution $w(z)$ such that

iff ${\mathrm{e}}^{\pi \mathrm{i}\nu }$ is an eigenvalue of the matrix

Equivalently,

This is the characteristic equation of Mathieu’s equation (28.2.1). $\cos \left(\pi \nu \right)$ is an entire function of $a,q^2$. The solutions of (28.2.16) are given by $\nu ={\pi }^{-1}\arccos \left(w_{\text{I}}(\pi ;a,q)\right)$. If the inverse cosine takes its principal value (§4.23(ii)), then $\nu =\widehat{\nu }$, where $0\le \mathrm{\Re }\widehat{\nu }\le 1$. The general solution of (28.2.16) is $\nu =\pm \widehat{\nu }+2n$, where $n\in \mathbb{Z}$. Either $\widehat{\nu }$ or $\nu$ is called a characteristic exponent of (28.2.1). If $\widehat{\nu }=0$ or $1$, or equivalently, $\nu =n$, then $\nu$ 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 $\nu$. (28.2.9), (28.2.16), and (28.2.7) give for each solution $w(z)$ of (28.2.1) the connection formula

Therefore a nontrivial solution $w(z)$ is either a Floquet solution with respect to $\nu$, or $w(z+\pi )-{\mathrm{e}}^{\mathrm{i}\nu \pi }w(z)$ is a Floquet solution with respect to $-\nu$.

If $q\ne 0$, then for a given value of $\nu$ 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 $ℂ$. The coefficients $c_{2n}$ satisfy

Conversely, a nontrivial solution $c_{2n}$ of (28.2.19) that satisfies

leads to a Floquet solution.

For given $\nu$ and $q$, equation (28.2.16) determines an infinite discrete set of values of $a$, the eigenvalues or characteristic values, of Mathieu’s equation. When $\widehat{\nu }=0$ or $1$, the notation for the two sets of eigenvalues corresponding to each $\widehat{\nu }$ is shown in Table 28.2.1, together with the boundary conditions of the associated eigenvalue problem. In Table 28.2.1 $n=0,1,2,\mathrm{\dots }$.

An equivalent formulation is given by

and

where $n=0,1,2,\mathrm{\dots }$. When $q=0$,

Near $q=0$, $a_n\left(q\right)$ and $b_n\left(q\right)$ can be expanded in power series in $q$ (see §28.6(i)); elsewhere they are determined by analytic continuation (see §28.7). For nonnegative real values of $q$, 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 $\pi$ means that the eigenfunction has the property $w(z+\pi )=w(z)$, whereas antiperiod $\pi$ means that $w(z+\pi )=-w(z)$. Even parity means $w(-z)=w(z)$, and odd parity means $w(-z)=-w(z)$.

When $q=0$,

For simple roots $q$ 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 $q=0$ and by continuity for the other values of $q$.

The functions are orthogonal, that is,

For change of sign of $q$ (compare (28.2.4))

For the connection with the basic solutions in §28.2(ii),