§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 $w^{\prime\prime}+(a-2q\cos\left(2z\right))w=0.$ ⓘ Symbols: $\cos\NVar{z}$: 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), 1st item, 2nd item, §28.33(iii), item (a), item (c), item (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), §28.2 and Ch.28

With $\zeta={\sin^{2}}z$ 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), §28.2 and Ch.28

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 $\zeta=\cos z$ 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 and Ch.28

§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), §28.2 and Ch.28

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), item (a), item (c) Permalink: http://dlmf.nist.gov/28.2.E5 Encodings: TeX, pMML, png See also: Annotations for §28.2(ii), §28.2 and Ch.28

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

 28.2.6 $\mathscr{W}\left\{w_{\mbox{\tiny I}},w_{\mbox{\tiny II}}\right\}=1,$ ⓘ Symbols: $\mathscr{W}$: 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 and Ch.28
 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 $\cos\left(\pi\nu\right)=w_{\mbox{\tiny I}}(\pi;a,q)=w_{\mbox{\tiny I}}(\pi;a,-% q).$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: 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), item (c), §28.34(i) Permalink: http://dlmf.nist.gov/28.2.E16 Encodings: TeX, pMML, png See also: Annotations for §28.2(iii), §28.2 and Ch.28

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 $\nu=\pi^{-1}\operatorname{arccos}\left(w_{\mbox{\tiny I}}(\pi;a,q)\right)$. 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 $w(z+\pi)+w(z-\pi)=2\cos\left(\pi\nu\right)w(z).$

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 $w(z)=\sum_{n=-\infty}^{\infty}c_{2n}e^{\mathrm{i}(\nu+2n)z}$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $n$: integer, $z$: complex variable, $\nu$: complex parameter, $w(z)$: Mathieu’s equation solution and $c_{2n}$: coefficients A&S Ref: 20.3.8 Permalink: http://dlmf.nist.gov/28.2.E18 Encodings: TeX, pMML, png See also: Annotations for §28.2(iv), §28.2 and Ch.28

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), §28.2 and Ch.28

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 and Ch.28

leads to a Floquet solution.

§28.2(v) Eigenvalues $a_{n}$, $b_{n}$

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

and

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

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

 28.2.23 $\displaystyle a_{n}\left(0\right)$ $\displaystyle=n^{2},$ $n=0,1,2,\dots$, ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$: 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 and Ch.28 28.2.24 $\displaystyle b_{n}\left(0\right)$ $\displaystyle=n^{2},$ $n=1,2,3,\dots$. ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$: 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), §28.2 and Ch.28

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. Figure 28.2.1: Eigenvalues an⁡(q), bn⁡(q) of Mathieu’s equation as functions of q for 0≤q≤10, n=0,1,2,3,4 (a’s), n=1,2,3,4 (b’s). Magnify

Distribution

 28.2.25 $\begin{array}[]{rl}\mbox{for q>0:}&a_{0} ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation, $b_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation and $q=h^{2}$: parameter Permalink: http://dlmf.nist.gov/28.2.E25 Encodings: TeX, pMML, png See also: Annotations for §28.2(v), §28.2(v), §28.2 and Ch.28

Change of Sign of $q$

 28.2.26 $\displaystyle a_{2n}\left(-q\right)$ $\displaystyle=a_{2n}\left(q\right),$ ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$: 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(v), §28.2 and Ch.28 28.2.27 $\displaystyle a_{2n+1}\left(-q\right)$ $\displaystyle=b_{2n+1}\left(q\right),$ ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation, $b_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation, $q=h^{2}$: parameter and $n$: integer A&S Ref: 20.8.3 Permalink: http://dlmf.nist.gov/28.2.E27 Encodings: TeX, pMML, png See also: Annotations for §28.2(v), §28.2(v), §28.2 and Ch.28 28.2.28 $\displaystyle b_{2n+2}\left(-q\right)$ $\displaystyle=b_{2n+2}\left(q\right).$ ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$: 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(v), §28.2 and Ch.28

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

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

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

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

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

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

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