28 Mathieu Functions and Hill’s Equation Mathieu Functions of Integer Order 28.1 Special Notation 28.3 Graphics

§28.2 Definitions and Basic Properties

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Permalink:: http://dlmf.nist.gov/28.2
See also:: Annotations for Ch.28

§28.2(i) Mathieu’s Equation
§28.2(ii) Basic Solutions $w_{\mbox{\rm\tiny I}}$ , $w_{\mbox{\rm\tiny II}}$
§28.2(iii) Floquet’s Theorem and the Characteristic Exponents
§28.2(iv) Floquet Solutions
§28.2(v) Eigenvalues $a_{n}$ , $b_{n}$
§28.2(vi) Eigenfunctions

§28.2(i) Mathieu’s Equation

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Keywords:: Mathieu functions, Mathieu’s equation, algebraic form, differential equation, singularities, standard form
Notes:: See McLachlan (1947, p. 10).
Referenced by:: §28.12(i)
Permalink:: http://dlmf.nist.gov/28.2.i
See also:: Annotations for §28.2 and Ch.28

The standard form of Mathieu’s equation with parameters is

28.2.1 $w^{\prime\prime}+(a-2q\cos\left(2z\right))w=0.$

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Symbols:: $\cos\NVar{z}$ : cosine function, $q=h^{2}$ : parameter, : complex variable, : parameter and : Mathieu’s equation solution
A&S Ref:: 20.3.1 20.1.1 (in slightly different form)
Referenced by:: §1.18(vii), 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 $\zeta(1-\zeta)w^{\prime\prime}+\tfrac{1}{2}\left(1-2\zeta)w^{\prime}+\tfrac{1}% {4}(a-2q(1-2\zeta)\right)w=0.$

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Symbols:: $q=h^{2}$ : parameter, : parameter, : 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 $(1-\zeta^{2})w^{\prime\prime}-\zeta w^{\prime}+\left(a+2q-4q\zeta^{2}\right)w=0.$

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Symbols:: $q=h^{2}$ : parameter, : parameter, : 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 $w_{\mbox{\rm\tiny I}}$ , $w_{\mbox{\rm\tiny II}}$

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Keywords:: Mathieu functions, Mathieu’s equation, analytic properties, basic solutions, transformations
Notes:: See Arscott (1964b, §2.1.1) and Meixner and Schäfke (1954, §§2.11–2.12).
Referenced by:: §28.12(i), §28.2(vi), §28.29(ii), §28.29(ii), §28.4(vii)
Permalink:: http://dlmf.nist.gov/28.2.ii
See also:: Annotations for §28.2 and Ch.28

Since (28.2.1) has no finite singularities its solutions are entire functions of . Furthermore, a solution with given initial constant values of and $w^{\prime}$ at a point $z_{0}$ is an entire function of the three variables , , and .

The following three transformations

28.2.4

$z\to-z;$

$z\to z\pm\pi;$

$z\to z\pm\tfrac{1}{2}\pi,q\to-q;$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $q=h^{2}$ : parameter and : 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 $\begin{bmatrix}w_{\mbox{\tiny I}}(0;a,q)&w_{\mbox{\tiny II}}(0;a,q)\\ w^{\prime}_{\mbox{\tiny I}}(0;a,q)&w^{\prime}_{\mbox{\tiny II}}(0;a,q)\end{% bmatrix}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}.$

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Symbols:: $q=h^{2}$ : parameter, : parameter, : 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 $\mathscr{W}\left\{w_{\mbox{\tiny I}},w_{\mbox{\tiny II}}\right\}=1,$

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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 $w_{\mbox{\tiny I}}(z\pm\pi;a,q)=w_{\mbox{\tiny I}}(\pi;a,q)w_{\mbox{\tiny I}}(% z;a,q)\pm w^{\prime}_{\mbox{\tiny I}}(\pi;a,q)w_{\mbox{\tiny II}}(z;a,q),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $q=h^{2}$ : parameter, : complex variable, : parameter, : Mathieu’s equation solution, $w_{\mbox{\tiny I}}(z;a,q)$ : fundamental solution and $w_{\mbox{\tiny II}}(z;a,q)$ : fundamental solution
Referenced by:: §28.2(iv)
Permalink:: http://dlmf.nist.gov/28.2.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(ii), §28.2 and Ch.28

28.2.8 $w_{\mbox{\tiny II}}(z\pm\pi;a,q)=\pm w_{\mbox{\tiny II}}(\pi;a,q)w_{\mbox{% \tiny I}}(z;a,q)+w^{\prime}_{\mbox{\tiny II}}(\pi;a,q)w_{\mbox{\tiny II}}(z;a,% q),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $q=h^{2}$ : parameter, : complex variable, : parameter, : Mathieu’s equation solution, $w_{\mbox{\tiny I}}(z;a,q)$ : fundamental solution and $w_{\mbox{\tiny II}}(z;a,q)$ : fundamental solution
Permalink:: http://dlmf.nist.gov/28.2.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(ii), §28.2 and Ch.28

28.2.9 $w_{\mbox{\tiny I}}(\pi;a,q)=w^{\prime}_{\mbox{\tiny II}}(\pi;a,q),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $q=h^{2}$ : parameter, : parameter, : Mathieu’s equation solution and $w_{\mbox{\tiny I}}(z;a,q)$ : fundamental solution
Referenced by:: §28.2(iv)
Permalink:: http://dlmf.nist.gov/28.2.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(ii), §28.2 and Ch.28

28.2.10 $w_{\mbox{\tiny I}}(\pi;a,q)-1=2w^{\prime}_{\mbox{\tiny I}}(\tfrac{1}{2}\pi;a,q% )w_{\mbox{\tiny II}}(\tfrac{1}{2}\pi;a,q),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $q=h^{2}$ : parameter, : parameter, : Mathieu’s equation solution, $w_{\mbox{\tiny I}}(z;a,q)$ : fundamental solution and $w_{\mbox{\tiny II}}(z;a,q)$ : fundamental solution
Permalink:: http://dlmf.nist.gov/28.2.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(ii), §28.2 and Ch.28

28.2.11 $w_{\mbox{\tiny I}}(\pi;a,q)+1=2w_{\mbox{\tiny I}}(\tfrac{1}{2}\pi;a,q)w^{% \prime}_{\mbox{\tiny II}}(\tfrac{1}{2}\pi;a,q),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $q=h^{2}$ : parameter, : parameter, : Mathieu’s equation solution and $w_{\mbox{\tiny I}}(z;a,q)$ : fundamental solution
Permalink:: http://dlmf.nist.gov/28.2.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(ii), §28.2 and Ch.28

28.2.12 $w^{\prime}_{\mbox{\tiny I}}(\pi;a,q)=2w_{\mbox{\tiny I}}(\tfrac{1}{2}\pi;a,q)w% ^{\prime}_{\mbox{\tiny I}}(\tfrac{1}{2}\pi;a,q),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $q=h^{2}$ : parameter, : parameter, : Mathieu’s equation solution and $w_{\mbox{\tiny I}}(z;a,q)$ : fundamental solution
Permalink:: http://dlmf.nist.gov/28.2.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(ii), §28.2 and Ch.28

28.2.13 $w_{\mbox{\tiny II}}(\pi;a,q)=2w_{\mbox{\tiny II}}(\tfrac{1}{2}\pi;a,q)w^{% \prime}_{\mbox{\tiny II}}(\tfrac{1}{2}\pi;a,q).$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $q=h^{2}$ : parameter, : parameter, : Mathieu’s equation solution and $w_{\mbox{\tiny II}}(z;a,q)$ : fundamental solution
Permalink:: http://dlmf.nist.gov/28.2.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(ii), §28.2 and Ch.28

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

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Keywords:: Floquet’s theorem, Mathieu’s equation, characteristic equation, characteristic exponents
Notes:: See Arscott (1964b, §2.2), Erdélyi et al. (1955, pp. 98–100), and Meixner and Schäfke (1954, §2.13).
Referenced by:: §28.12(i)
Permalink:: http://dlmf.nist.gov/28.2.iii
See also:: Annotations for §28.2 and Ch.28

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

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, : complex variable, $\nu$ : complex parameter and : Mathieu’s equation solution
Referenced by:: §28.12(i), §28.2(iv)
Permalink:: http://dlmf.nist.gov/28.2.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(iii), §28.2 and Ch.28

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

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $q=h^{2}$ : parameter, : parameter, : Mathieu’s equation solution, $w_{\mbox{\tiny I}}(z;a,q)$ : fundamental solution and $w_{\mbox{\tiny II}}(z;a,q)$ : fundamental solution
Permalink:: http://dlmf.nist.gov/28.2.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(iii), §28.2 and Ch.28

Equivalently,

28.2.16 $\cos\left(\pi\nu\right)=w_{\mbox{\tiny I}}(\pi;a,q)=w_{\mbox{\tiny I}}(\pi;a,-% q).$

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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, : 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

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Keywords:: Floquet solutions, Fourier series, Fourier-series expansions, Ince’s theorem, Mathieu functions, Mathieu’s equation, Theorem of Ince, of Mathieu’s equation, pseudoperiodic solutions, pseudoperiodicity, uniqueness
Notes:: See Meixner and Schäfke (1954, §2.13 and §2.21). For a proof of the Theorem of Ince see Arscott (1964b, §2.4).
Permalink:: http://dlmf.nist.gov/28.2.iv
See also:: Annotations for §28.2 and Ch.28

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 of (28.2.1) the connection formula

28.2.17 $w(z+\pi)+w(z-\pi)=2\cos\left(\pi\nu\right)w(z).$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, : complex variable, $\nu$ : complex parameter and : Mathieu’s equation solution
Permalink:: http://dlmf.nist.gov/28.2.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(iv), §28.2 and Ch.28

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

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Symbols:: $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, : integer, : complex variable, $\nu$ : complex parameter, : 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 ${qc_{2n+2}-\left(a-(\nu+2n)^{2}\right)c_{2n}+qc_{2n-2}=0,}$ $n\in\mathbb{Z}$ .

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Symbols:: $\in$ : element of, $\mathbb{Z}$ : set of all integers, $q=h^{2}$ : parameter, : integer, $\nu$ : complex parameter, : 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 $\lim_{n\to\pm\infty}|c_{2n}|^{1/|n|}=0$

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Symbols:: : 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 $a_{n}$ , $b_{n}$

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Defines:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation
Keywords:: Mathieu’s equation, eigenvalues (or characteristic values), graphics, notation
Notes:: See Erdélyi et al. (1955, §16.4), Meixner and Schäfke (1954, pp. 108–109), and Meixner et al. (1980, p. 88). Figure 28.2.1 was produced at NIST.
Referenced by:: §28.31(ii), §28.34(ii)
Permalink:: http://dlmf.nist.gov/28.2.v
See also:: Annotations for §28.2 and Ch.28

For given $\nu$ and , equation (28.2.16) determines an infinite discrete set of values of , 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 $n=0,1,2,\dots$ .

Table 28.2.1: Eigenvalues of Mathieu’s equation.

$\widehat{\nu}$	Boundary Conditions	Eigenvalues
0	$w^{\prime}(0)=w^{\prime}(\tfrac{1}{2}\pi)=0$	$a_{2n}\left(q\right)$
1	$w^{\prime}(0)=w(\tfrac{1}{2}\pi)=0$	$a_{2n+1}\left(q\right)$
1	$w(0)=w^{\prime}(\tfrac{1}{2}\pi)=0$	$b_{2n+1}\left(q\right)$
0	$w(0)=w(\tfrac{1}{2}\pi)=0$	$b_{2n+2}\left(q\right)$

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Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\pi$ : the ratio of the circumference of a circle to its diameter, $q=h^{2}$ : parameter, : integer, $\nu$ : complex parameter and : Mathieu’s equation solution
Referenced by:: §28.2(v), §28.2(vi)
Permalink:: http://dlmf.nist.gov/28.2.T1
See also:: Annotations for §28.2(v), §28.2 and Ch.28

An equivalent formulation is given by

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

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Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\pi$ : the ratio of the circumference of a circle to its diameter, $q=h^{2}$ : parameter, : integer, : parameter, : Mathieu’s equation solution and $w_{\mbox{\tiny I}}(z;a,q)$ : fundamental solution
Referenced by:: §28.2(vi), §28.29(ii), §28.7
Permalink:: http://dlmf.nist.gov/28.2.E21
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(v), §28.2 and Ch.28

and

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

ⓘ

Symbols:: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\pi$ : the ratio of the circumference of a circle to its diameter, $q=h^{2}$ : parameter, : integer, : parameter, : Mathieu’s equation solution and $w_{\mbox{\tiny II}}(z;a,q)$ : fundamental solution
Referenced by:: §28.2(vi), §28.29(ii), §28.7
Permalink:: http://dlmf.nist.gov/28.2.E22
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(v), §28.2 and Ch.28

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

28.2.23 $a_{n}\left(0\right)=n^{2},$ $n=0,1,2,\dots$ ,

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Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and : 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 $b_{n}\left(0\right)=n^{2},$ $n=1,2,3,\dots$ .

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Symbols:: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and : 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 , $a_{n}\left(q\right)$ and $b_{n}\left(q\right)$ 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.

Figure 28.2.1: Eigenvalues $a_{n}\left(q\right)$ , $b_{n}\left(q\right)$ of Mathieu’s equation as functions of

for $0\leq q\leq 10$ ,

(

’s),

(

’s). Magnify

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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, : integer and : parameter
A&S Ref:: 20.1 (Figure)
Referenced by:: Figure 28.13.1, Figure 28.13.1, §28.17, §28.2(v), §28.2(v), §28.8(i)
Permalink:: http://dlmf.nist.gov/28.2.F1
Encodings:: pdf, png
See also:: Annotations for §28.2(v), §28.2 and Ch.28

Distribution

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Keywords:: Mathieu’s equation, distribution, eigenvalues (or characteristic values)
See also:: Annotations for §28.2(v), §28.2 and Ch.28

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

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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

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Keywords:: Mathieu’s equation, eigenvalues (or characteristic values), reflection properties in
See also:: Annotations for §28.2(v), §28.2 and Ch.28

28.2.26 $a_{2n}\left(-q\right)=a_{2n}\left(q\right),$

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Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter and : 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 $a_{2n+1}\left(-q\right)=b_{2n+1}\left(q\right),$

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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 : 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 $b_{2n+2}\left(-q\right)=b_{2n+2}\left(q\right).$

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Symbols:: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter and : 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

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Defines:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function and $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function
Keywords:: Mathieu functions, Mathieu’s equation, antiperiodicity, basic solutions, basic solutions of Mathieu’s equation, normalization, of integer order, orthogonality, parity, periodicity, reflection properties in , relation to eigenfunctions, relations to other functions, values at
A&S Ref:: 20.8.4
Notes:: See Meixner and Schäfke (1954, §2.71).
Referenced by:: §28.12(iii), §28.31(ii), §28.5(i), §29.5
Permalink:: http://dlmf.nist.gov/28.2.vi
See also:: Annotations for §28.2 and Ch.28

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 , and odd parity means .

Table 28.2.2: Eigenfunctions of Mathieu’s equation.

Eigenvalues	Eigenfunctions	Periodicity	Parity
$a_{2n}\left(q\right)$	$\operatorname{ce}_{2n}\left(z,q\right)$	Period $\pi$	Even
$a_{2n+1}\left(q\right)$	$\operatorname{ce}_{2n+1}\left(z,q\right)$	Antiperiod $\pi$	Even
$b_{2n+1}\left(q\right)$	$\operatorname{se}_{2n+1}\left(z,q\right)$	Antiperiod $\pi$	Odd
$b_{2n+2}\left(q\right)$	$\operatorname{se}_{2n+2}\left(z,q\right)$	Period $\pi$	Odd

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $q=h^{2}$ : parameter, : integer and : complex variable
Referenced by:: §28.2(vi)
Permalink:: http://dlmf.nist.gov/28.2.T2
See also:: Annotations for §28.2(vi), §28.2 and Ch.28

When ,

28.2.29

$\operatorname{ce}_{0}\left(z,0\right)=1/\sqrt{2},$

$\operatorname{ce}_{n}\left(z,0\right)=\cos\left(nz\right),$

$\operatorname{se}_{n}\left(z,0\right)=\sin\left(nz\right)$ , $n=1,2,3,\dots$ .

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, : integer and : complex variable
Referenced by:: §28.2(vi), §28.5(i)
Permalink:: http://dlmf.nist.gov/28.2.E29
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §28.2(vi), §28.2 and Ch.28

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

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

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

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $q=h^{2}$ : parameter, : integer and : real variable
A&S Ref:: 20.5.3
Permalink:: http://dlmf.nist.gov/28.2.E30
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §28.2(vi), §28.2 and Ch.28

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

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : integer, $q=h^{2}$ : parameter, : integer and : real variable
Permalink:: http://dlmf.nist.gov/28.2.E31
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(vi), §28.2 and Ch.28

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

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Symbols:: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : integer, $q=h^{2}$ : parameter, : integer and : real variable
Permalink:: http://dlmf.nist.gov/28.2.E32
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(vi), §28.2 and Ch.28

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

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : integer, $q=h^{2}$ : parameter, : integer and : real variable
Permalink:: http://dlmf.nist.gov/28.2.E33
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(vi), §28.2 and Ch.28

For change of sign of (compare (28.2.4))

28.2.34 $\operatorname{ce}_{2n}\left(z,-q\right)=(-1)^{n}\operatorname{ce}_{2n}\left(% \tfrac{1}{2}\pi-z,q\right),$

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $q=h^{2}$ : parameter, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/28.2.E34
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(vi), §28.2 and Ch.28

28.2.35 $\operatorname{ce}_{2n+1}\left(z,-q\right)=(-1)^{n}\operatorname{se}_{2n+1}% \left(\tfrac{1}{2}\pi-z,q\right),$

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $q=h^{2}$ : parameter, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/28.2.E35
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(vi), §28.2 and Ch.28

28.2.36 $\operatorname{se}_{2n+1}\left(z,-q\right)=(-1)^{n}\operatorname{ce}_{2n+1}% \left(\tfrac{1}{2}\pi-z,q\right),$

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $q=h^{2}$ : parameter, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/28.2.E36
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(vi), §28.2 and Ch.28

28.2.37 $\operatorname{se}_{2n+2}\left(z,-q\right)=(-1)^{n}\operatorname{se}_{2n+2}% \left(\tfrac{1}{2}\pi-z,q\right).$

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Symbols:: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $q=h^{2}$ : parameter, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/28.2.E37
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(vi), §28.2 and Ch.28

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

28.2.38 $\frac{\operatorname{ce}_{n}\left(z,q\right)}{\operatorname{ce}_{n}\left(0,q% \right)}=w_{\mbox{\tiny I}}(z;a_{n}\left(q\right),q),$ $n=0,1,\dots$ ,

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter, : integer, : complex variable and $w_{\mbox{\tiny I}}(z;a,q)$ : fundamental solution
Permalink:: http://dlmf.nist.gov/28.2.E38
Encodings:: TeX, pMML, png
See also:: Annotations for §28.2(vi), §28.2 and Ch.28

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

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