# §28.2 Definitions and Basic Properties

## §28.2(i) Mathieu’s Equation

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

 28.2.1 $w^{\prime\prime}+(a-2q\mathop{\cos\/}\nolimits\!\left(2z\right))w=0.$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $q=h^{2}$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution A&S Ref: 20.3.1 20.1.1 (in slightly different form) Referenced by: Figure 28.17.1, Figure 28.17.1, §28.17, §28.17, §28.2(ii), §28.2(ii), §28.2(iii), §28.2(iv), §28.20(i), §28.29(i), §28.32(i), §28.32(i), §28.33(i), §28.33(i), §28.33(iii), (a), (c), (c), §28.5(i), §28.8(iv), §28.8(iv), §28.8(iv), §30.2(iii) Permalink: http://dlmf.nist.gov/28.2.E1 Encodings: TeX, pMML, png See also: Annotations for 28.2(i)

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

 28.2.2 $\zeta(1-\zeta)w^{\prime\prime}+\tfrac{1}{2}\left(1-2\zeta)w^{\prime}+\tfrac{1}% {4}(a-2q(1-2\zeta)\right)w=0.$ Symbols: $q=h^{2}$: parameter, $a$: parameter, $w(z)$: Mathieu’s equation solution and $\zeta$: change of variable Permalink: http://dlmf.nist.gov/28.2.E2 Encodings: TeX, pMML, png See also: Annotations for 28.2(i)

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

 28.2.3 $(1-\zeta^{2})w^{\prime\prime}-\zeta w^{\prime}+\left(a+2q-4q\zeta^{2}\right)w=0.$ Symbols: $q=h^{2}$: parameter, $a$: parameter, $w(z)$: Mathieu’s equation solution and $\zeta$: change of variable A&S Ref: 20.1.7 (in slightly different notation) Permalink: http://dlmf.nist.gov/28.2.E3 Encodings: TeX, pMML, png See also: Annotations for 28.2(i)

## §28.2(ii) Basic Solutions $w_{\mbox{\rm\tiny I}}$, $w_{\mbox{\rm\tiny II}}$

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

 28.2.4 $\displaystyle z$ $\displaystyle\to-z;$ $\displaystyle z$ $\displaystyle\to z\pm\pi;$ $\displaystyle z$ $\displaystyle\to z\pm\tfrac{1}{2}\pi,q\to-q;$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $q=h^{2}$: parameter and $z$: complex variable Referenced by: §28.2(vi) Permalink: http://dlmf.nist.gov/28.2.E4 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 28.2(ii)

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

 28.2.5 $\begin{bmatrix}w_{\mbox{\tiny I}}(0;a,q)&w_{\mbox{\tiny II}}(0;a,q)\\ w^{\prime}_{\mbox{\tiny I}}(0;a,q)&w^{\prime}_{\mbox{\tiny II}}(0;a,q)\end{% bmatrix}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}.$ Symbols: $q=h^{2}$: parameter, $a$: parameter, $w(z)$: Mathieu’s equation solution, $w_{\mbox{\tiny I}}(z;a,q)$: fundamental solution and $w_{\mbox{\tiny II}}(z;a,q)$: fundamental solution A&S Ref: 20.3.9 (in different notation) Referenced by: §28.29(ii), (a), (c) Permalink: http://dlmf.nist.gov/28.2.E5 Encodings: TeX, pMML, png See also: Annotations for 28.2(ii)

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

 28.2.6 $\mathop{\mathscr{W}\/}\nolimits\left\{w_{\mbox{\tiny I}},w_{\mbox{\tiny II}}% \right\}=1,$ Symbols: $\mathop{\mathscr{W}\/}\nolimits$: Wronskian, $w_{\mbox{\tiny I}}(z;a,q)$: fundamental solution and $w_{\mbox{\tiny II}}(z;a,q)$: fundamental solution Referenced by: §28.29(ii) Permalink: http://dlmf.nist.gov/28.2.E6 Encodings: TeX, pMML, png See also: Annotations for 28.2(ii)
 28.2.7 $\displaystyle w_{\mbox{\tiny I}}(z\pm\pi;a,q)$ $\displaystyle=w_{\mbox{\tiny I}}(\pi;a,q)w_{\mbox{\tiny I}}(z;a,q)\pm w^{% \prime}_{\mbox{\tiny I}}(\pi;a,q)w_{\mbox{\tiny II}}(z;a,q),$ 28.2.8 $\displaystyle w_{\mbox{\tiny II}}(z\pm\pi;a,q)$ $\displaystyle=\pm w_{\mbox{\tiny II}}(\pi;a,q)w_{\mbox{\tiny I}}(z;a,q)+w^{% \prime}_{\mbox{\tiny II}}(\pi;a,q)w_{\mbox{\tiny II}}(z;a,q),$ 28.2.9 $\displaystyle w_{\mbox{\tiny I}}(\pi;a,q)$ $\displaystyle=w^{\prime}_{\mbox{\tiny II}}(\pi;a,q),$ 28.2.10 $\displaystyle w_{\mbox{\tiny I}}(\pi;a,q)-1$ $\displaystyle=2w^{\prime}_{\mbox{\tiny I}}(\tfrac{1}{2}\pi;a,q)w_{\mbox{\tiny II% }}(\tfrac{1}{2}\pi;a,q),$ 28.2.11 $\displaystyle w_{\mbox{\tiny I}}(\pi;a,q)+1$ $\displaystyle=2w_{\mbox{\tiny I}}(\tfrac{1}{2}\pi;a,q)w^{\prime}_{\mbox{\tiny II% }}(\tfrac{1}{2}\pi;a,q),$ 28.2.12 $\displaystyle w^{\prime}_{\mbox{\tiny I}}(\pi;a,q)$ $\displaystyle=2w_{\mbox{\tiny I}}(\tfrac{1}{2}\pi;a,q)w^{\prime}_{\mbox{\tiny I% }}(\tfrac{1}{2}\pi;a,q),$ 28.2.13 $\displaystyle w_{\mbox{\tiny II}}(\pi;a,q)$ $\displaystyle=2w_{\mbox{\tiny II}}(\tfrac{1}{2}\pi;a,q)w^{\prime}_{\mbox{\tiny II% }}(\tfrac{1}{2}\pi;a,q).$

## §28.2(iii) Floquet’s Theorem and the Characteristic Exponents

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

 28.2.14 $w(z+\pi)=e^{\pi\mathrm{i}\nu}w(z),$

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

 28.2.15 $\begin{bmatrix}w_{\mbox{\tiny I}}(\pi;a,q)&w_{\mbox{\tiny II}}(\pi;a,q)\\ w^{\prime}_{\mbox{\tiny I}}(\pi;a,q)&w^{\prime}_{\mbox{\tiny II}}(\pi;a,q)\end% {bmatrix}.$

Equivalently,

 28.2.16 $\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)=w_{\mbox{\tiny I}}(\pi;a,q)=w_{% \mbox{\tiny I}}(\pi;a,-q).$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $q=h^{2}$: parameter, $\nu$: complex parameter, $a$: parameter and $w_{\mbox{\tiny I}}(z;a,q)$: fundamental solution A&S Ref: 20.3.10 (in slightly different form) Referenced by: §28.12(i), §28.12(i), §28.12(ii), §28.2(iii), §28.2(iv), §28.2(v), (c), §28.34(i) Permalink: http://dlmf.nist.gov/28.2.E16 Encodings: TeX, pMML, png See also: Annotations for 28.2(iii)

This is the characteristic equation of Mathieu’s equation (28.2.1). $\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)$ is an entire function of $a,q^{2}$. The solutions of (28.2.16) are given by $\nu=\pi^{-1}\mathop{\mathrm{arccos}\/}\nolimits\!\left(w_{\mbox{\tiny I}}(\pi;% a,q)\right)$. If the inverse cosine takes its principal value (§4.23(ii)), then $\nu=\widehat{\nu}$, where $0\leq\Re{\widehat{\nu}}\leq 1$. 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.

## §28.2(iv) Floquet Solutions

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

 28.2.17 $w(z+\pi)+w(z-\pi)=2\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)w(z).$

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

If $q\neq 0$, 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

 28.2.18 $w(z)=\sum_{n=-\infty}^{\infty}c_{2n}e^{\mathrm{i}(\nu+2n)z}$

converges absolutely and uniformly in compact subsets of $\mathbb{C}$. The coefficients $c_{2n}$ satisfy

 28.2.19 ${qc_{2n+2}-\left(a-(\nu+2n)^{2}\right)c_{2n}+qc_{2n-2}=0,}$ $n\in\mathbb{Z}$. Symbols: $\in$: element of, $\mathbb{Z}$: set of all integers, $q=h^{2}$: parameter, $n$: integer, $\nu$: complex parameter, $a$: parameter and $c_{2n}$: coefficients A&S Ref: 20.3.12 20.3.13 Referenced by: §28.2(iv) Permalink: http://dlmf.nist.gov/28.2.E19 Encodings: TeX, pMML, png See also: Annotations for 28.2(iv)

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

 28.2.20 $\lim_{n\to\pm\infty}|c_{2n}|^{1/|n|}=0$ Symbols: $n$: integer and $c_{2n}$: coefficients Permalink: http://dlmf.nist.gov/28.2.E20 Encodings: TeX, pMML, png See also: Annotations for 28.2(iv)

## §28.2(v) Eigenvalues $\mathop{a_{n}\/}\nolimits$, $\mathop{b_{n}\/}\nolimits$

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 $n=0,1,2,\dots$.

An equivalent formulation is given by

 28.2.21 $\begin{array}[]{ll}w^{\prime}_{\mbox{\tiny I}}(\tfrac{1}{2}\pi;a,q)=0,&a=% \mathop{a_{2n}\/}\nolimits\!\left(q\right),\\ w_{\mbox{\tiny I}}(\tfrac{1}{2}\pi;a,q)=0,&a=\mathop{a_{2n+1}\/}\nolimits\!% \left(q\right),\end{array}$

and

 28.2.22 $\begin{array}[]{ll}w^{\prime}_{\mbox{\tiny II}}(\tfrac{1}{2}\pi;a,q)=0,&a=% \mathop{b_{2n+1}\/}\nolimits\!\left(q\right),\\ w_{\mbox{\tiny II}}(\tfrac{1}{2}\pi;a,q)=0,&a=\mathop{b_{2n+2}\/}\nolimits\!% \left(q\right),\end{array}$

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

 28.2.23 $\displaystyle\mathop{a_{n}\/}\nolimits\!\left(0\right)$ $\displaystyle=n^{2},$ $n=0,1,2,\dots$, Symbols: $\mathop{a_{\NVar{n}}\/}\nolimits\!\left(\NVar{q}\right)$: eigenvalues of Mathieu equation and $n$: integer Permalink: http://dlmf.nist.gov/28.2.E23 Encodings: TeX, pMML, png See also: Annotations for 28.2(v) 28.2.24 $\displaystyle\mathop{b_{n}\/}\nolimits\!\left(0\right)$ $\displaystyle=n^{2},$ $n=1,2,3,\dots$. Symbols: $\mathop{b_{\NVar{n}}\/}\nolimits\!\left(\NVar{q}\right)$: eigenvalues of Mathieu equation and $n$: integer Permalink: http://dlmf.nist.gov/28.2.E24 Encodings: TeX, pMML, png See also: Annotations for 28.2(v)

Near $q=0$, $\mathop{a_{n}\/}\nolimits\!\left(q\right)$ and $\mathop{b_{n}\/}\nolimits\!\left(q\right)$ 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.

### Distribution

 28.2.25 $\begin{array}[]{rl}\mbox{for q>0:}&\mathop{a_{0}\/}\nolimits<\mathop{b_{1}\/% }\nolimits<\mathop{a_{1}\/}\nolimits<\mathop{b_{2}\/}\nolimits<\mathop{a_{2}\/% }\nolimits<\mathop{b_{3}\/}\nolimits<\cdots,\\ \mbox{for q<0:}&\mathop{a_{0}\/}\nolimits<\mathop{a_{1}\/}\nolimits<\mathop{% b_{1}\/}\nolimits<\mathop{b_{2}\/}\nolimits<\mathop{a_{2}\/}\nolimits<\mathop{% a_{3}\/}\nolimits<\cdots.\end{array}$

### Change of Sign of $q$

 28.2.26 $\displaystyle\mathop{a_{2n}\/}\nolimits\!\left(-q\right)$ $\displaystyle=\mathop{a_{2n}\/}\nolimits\!\left(q\right),$ Symbols: $\mathop{a_{\NVar{n}}\/}\nolimits\!\left(\NVar{q}\right)$: eigenvalues of Mathieu equation, $q=h^{2}$: parameter and $n$: integer A&S Ref: 20.8.3 Permalink: http://dlmf.nist.gov/28.2.E26 Encodings: TeX, pMML, png See also: Annotations for 28.2(v) 28.2.27 $\displaystyle\mathop{a_{2n+1}\/}\nolimits\!\left(-q\right)$ $\displaystyle=\mathop{b_{2n+1}\/}\nolimits\!\left(q\right),$ 28.2.28 $\displaystyle\mathop{b_{2n+2}\/}\nolimits\!\left(-q\right)$ $\displaystyle=\mathop{b_{2n+2}\/}\nolimits\!\left(q\right).$ Symbols: $\mathop{b_{\NVar{n}}\/}\nolimits\!\left(\NVar{q}\right)$: eigenvalues of Mathieu equation, $q=h^{2}$: parameter and $n$: integer A&S Ref: 20.8.3 Permalink: http://dlmf.nist.gov/28.2.E28 Encodings: TeX, pMML, png See also: Annotations for 28.2(v)

## §28.2(vi) Eigenfunctions

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 $q=0$,

 28.2.29 $\displaystyle\mathop{\mathrm{ce}_{0}\/}\nolimits\!\left(z,0\right)$ $\displaystyle=1/\sqrt{2},$ $\displaystyle\mathop{\mathrm{ce}_{n}\/}\nolimits\!\left(z,0\right)$ $\displaystyle=\mathop{\cos\/}\nolimits\!\left(nz\right),$ $\displaystyle\mathop{\mathrm{se}_{n}\/}\nolimits\!\left(z,0\right)$ $\displaystyle=\mathop{\sin\/}\nolimits\!\left(nz\right)$, $n=1,2,3,\dots$.

For simple roots $q$ of the corresponding equations (28.2.21) and (28.2.22), the functions are made unique by the normalizations

 28.2.30 $\displaystyle\int_{0}^{2\pi}\left(\mathop{\mathrm{ce}_{n}\/}\nolimits\!\left(x% ,q\right)\right)^{2}\mathrm{d}x$ $\displaystyle=\pi,$ $\displaystyle\int_{0}^{2\pi}\left(\mathop{\mathrm{se}_{n}\/}\nolimits\!\left(x% ,q\right)\right)^{2}\mathrm{d}x$ $\displaystyle=\pi,$

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,

 28.2.31 $\displaystyle\int_{0}^{2\pi}\mathop{\mathrm{ce}_{m}\/}\nolimits\!\left(x,q% \right)\mathop{\mathrm{ce}_{n}\/}\nolimits\!\left(x,q\right)\mathrm{d}x$ $\displaystyle=0,$ $n\neq m$, 28.2.32 $\displaystyle\int_{0}^{2\pi}\mathop{\mathrm{se}_{m}\/}\nolimits\!\left(x,q% \right)\mathop{\mathrm{se}_{n}\/}\nolimits\!\left(x,q\right)\mathrm{d}x$ $\displaystyle=0,$ $n\neq m$, 28.2.33 $\displaystyle\int_{0}^{2\pi}\mathop{\mathrm{ce}_{m}\/}\nolimits\!\left(x,q% \right)\mathop{\mathrm{se}_{n}\/}\nolimits\!\left(x,q\right)\mathrm{d}x$ $\displaystyle=0.$

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

 28.2.34 $\displaystyle\mathop{\mathrm{ce}_{2n}\/}\nolimits\!\left(z,-q\right)$ $\displaystyle=(-1)^{n}\mathop{\mathrm{ce}_{2n}\/}\nolimits\!\left(\tfrac{1}{2}% \pi-z,q\right),$ 28.2.35 $\displaystyle\mathop{\mathrm{ce}_{2n+1}\/}\nolimits\!\left(z,-q\right)$ $\displaystyle=(-1)^{n}\mathop{\mathrm{se}_{2n+1}\/}\nolimits\!\left(\tfrac{1}{% 2}\pi-z,q\right),$ 28.2.36 $\displaystyle\mathop{\mathrm{se}_{2n+1}\/}\nolimits\!\left(z,-q\right)$ $\displaystyle=(-1)^{n}\mathop{\mathrm{ce}_{2n+1}\/}\nolimits\!\left(\tfrac{1}{% 2}\pi-z,q\right),$ 28.2.37 $\displaystyle\mathop{\mathrm{se}_{2n+2}\/}\nolimits\!\left(z,-q\right)$ $\displaystyle=(-1)^{n}\mathop{\mathrm{se}_{2n+2}\/}\nolimits\!\left(\tfrac{1}{% 2}\pi-z,q\right).$

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

 28.2.38 $\displaystyle\frac{\mathop{\mathrm{ce}_{n}\/}\nolimits\!\left(z,q\right)}{% \mathop{\mathrm{ce}_{n}\/}\nolimits\!\left(0,q\right)}$ $\displaystyle=w_{\mbox{\tiny I}}(z;\mathop{a_{n}\/}\nolimits\!\left(q\right),q),$ $n=0,1,\dots$, 28.2.39 $\displaystyle\frac{\mathop{\mathrm{se}_{n}\/}\nolimits\!\left(z,q\right)}{% \mathop{\mathrm{se}_{n}\/}\nolimits'\!\left(0,q\right)}$ $\displaystyle=w_{\mbox{\tiny II}}(z;\mathop{b_{n}\/}\nolimits\!\left(q\right),% q),$ $n=1,2,\dots$.