The even and odd periodic Mathieu functions - Maple Help

MathieuCE, MathieuSE - The even and odd periodic Mathieu functions

MathieuA, MathieuB - The characteristic value functions

MathieuC, MathieuS - The even and odd general Mathieu functions

MathieuFloquet - Floquet solution of Mathieu's equation

MathieuCPrime, MathieuSPrime, MathieuFloquetPrime, MathieuCEPrime, MathieuSEPrime - The first derivatives of the Mathieu functions

MathieuExponent - The characteristic exponent function

 Calling Sequence MathieuCE(n, q, x) MathieuCEPrime(n, q, x) MathieuSE(n, q, x) MathieuSEPrime(n, q, x) MathieuA(n, q) MathieuB(n, q) MathieuC(a, q, x) MathieuCPrime(a, q, x) MathieuS(a, q, x) MathieuSPrime(a, q, x) MathieuFloquet(a, q, x) MathieuFloquetPrime(a, q, x) MathieuExponent(a, q)

Parameters

 n - algebraic expression (the order or index), understood to be a non-negative integer a, q - algebraic expressions (parameters) x - algebraic expression (argument)

Description

 • The Mathieu functions MathieuC(a, q, x) and MathieuS(a, q, x) are solutions of the Mathieu differential equation:

$y\text{'}\text{'}+\left(a-2q\mathrm{cos}\left(2x\right)\right)y=0$

 MathieuC and MathieuS are even and odd functions of x, respectively.
 • For countably many values of a (as a function of q), MathieuC and MathieuS are 2*Pi-periodic. For $n=0,1,2,...$, MathieuA(n, q) is the nth such characteristic value for MathieuC, and for $n=1,...$, MathieuB(n, q) is the nth characteristic value for MathieuS. The resulting Mathieu functions are:

$\mathrm{MathieuCE}\left(n,q,x\right)=\mathrm{c1}\mathrm{MathieuC}\left(\mathrm{MathieuA}\left(n,q\right),q,x\right)$

$\mathrm{MathieuSE}\left(n,q,x\right)=\mathrm{c2}\mathrm{MathieuS}\left(\mathrm{MathieuB}\left(n,q\right),q,x\right)$

 where $\mathrm{c1}$ and $\mathrm{c2}$ are normalization constants depending on n and q.
 If the index n is even, then both MathieuCE and MathieuSE are Pi-periodic; they are 2*Pi-periodic otherwise. MathieuCE and MathieuSE are even and odd functions of x, respectively.
 • MathieuFloquet(a, q, x) is a Floquet solution of Mathieu's equation. It has the form:

${ⅇ}^{I\mathrm{\nu }x}P\left(x\right)$

 where $\mathrm{\nu }=\mathrm{MathieuExponent}\left(a,q\right)$ is the characteristic exponent and $P\left(x\right)$ is a Pi periodic function.
 • MathieuCPrime, MathieuSPrime, MathieuCEPrime, MathieuSEPrime, and MathieuFloquetPrime are the first derivatives with respect to x of the corresponding Mathieu functions. Note that all higher order derivatives can be written in terms of the 0th and 1st derivatives.
 • The odd and even Mathieu functions are related to the Floquet solution via:

$\mathrm{MathieuC}\left(a,q,x\right)=\frac{1}{2}\frac{\mathrm{MathieuFloquet}\left(a,q,x\right)+\mathrm{MathieuFloquet}\left(a,q,-x\right)}{\mathrm{MathieuFloquet}\left(a,q,0\right)}$

$\mathrm{MathieuS}\left(a,q,x\right)=\frac{1}{2}\frac{\mathrm{MathieuFloquet}\left(a,q,x\right)-\mathrm{MathieuFloquet}\left(a,q,-x\right)}{\mathrm{MathieuFloquetPrime}\left(a,q,0\right)}$

 • The various Mathieu functions are normalized as follows.

$\mathrm{MathieuC}\left(a,q,0\right)=1,\mathrm{MathieuCPrime}\left(a,q,0\right)=0$

$\mathrm{MathieuS}\left(a,q,0\right)=0,\mathrm{MathieuSPrime}\left(a,q,0\right)=1$

${{\int }}_{0}^{2\mathrm{\pi }}{{\mathrm{ce}}_{n}\left(q,x\right)}^{2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x=2\mathrm{\pi }\mathrm{if n = 0}$

${{\int }}_{0}^{2\mathrm{\pi }}{{\mathrm{ce}}_{n}\left(q,x\right)}^{2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x=\mathrm{\pi }\mathrm{if n > 0}$

${{\int }}_{0}^{2\mathrm{\pi }}{{\mathrm{se}}_{n}\left(q,x\right)}^{2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x=\mathrm{\pi }$

${{\int }}_{0}^{2\mathrm{\pi }}{\left|P\left(a,q,x\right)\right|}^{2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x=2\mathrm{\pi }$

 where P is as given in the definition of the Floquet solution.
 The normalizations of MathieuCE, MathieuSE, and their derivatives coincide with the ones in [1] (see references below), except for $n=0$.
 • MathieuExponent is an inverse to both MathieuA and MathieuB in the following sense.

$\mathrm{MathieuExponent}\left(\mathrm{MathieuA}\left(n,q\right),q\right)=n+2k$

$\mathrm{MathieuExponent}\left(\mathrm{MathieuB}\left(n,q\right),q\right)=n+2k$

 where $k$ is an integer.
 • For q = 0, the Mathieu functions assume special values:

$\mathrm{MathieuCE}\left(n,0,x\right)=\mathrm{cos}\left(nx\right)$

$\mathrm{MathieuSE}\left(n,0,x\right)=\mathrm{sin}\left(nx\right)$

$\mathrm{MathieuA}\left(n,0\right)={n}^{2}$

$\mathrm{MathieuB}\left(n,0\right)={n}^{2}$

$\mathrm{MathieuC}\left(a,0,x\right)=\mathrm{cos}\left({a}^{\frac{1}{2}}x\right)$

$\mathrm{MathieuS}\left(a,0,x\right)=\frac{\mathrm{sin}\left({a}^{\frac{1}{2}}x\right)}{{a}^{\frac{1}{2}}}$

$\mathrm{MathieuFloquet}\left(a,0,x\right)=\mathrm{exp}\left({a}^{\frac{1}{2}}Ix\right)$

$\mathrm{MathieuExponent}\left(a,0\right)=\sqrt{a}$

Examples

 > $\mathrm{MathieuC}\left(1.23,4.56,7.89\right)$
 ${147.6289914}$ (1)
 > $\mathrm{MathieuFloquet}\left(0.1,0.2,0.3\right)$
 ${0.8965972662}{+}{0.1161779284}{}{I}$ (2)
 > $\mathrm{MathieuS}\left(a,q,-x\right)$
 ${-}{\mathrm{MathieuS}}{}\left({a}{,}{q}{,}{x}\right)$ (3)
 > $\mathrm{MathieuCE}\left(n,0,x\right)$
 ${\mathrm{cos}}{}\left({n}{}{x}\right)$ (4)
 > $\mathrm{MathieuExponent}\left(a,0\right)$
 $\sqrt{{a}}$ (5)
 > $\mathrm{MathieuSE}\left(n,q,x+\frac{25\mathrm{π}}{11}\right)$
 ${\mathrm{MathieuSE}}{}\left({n}{,}{q}{,}{x}{+}\frac{{3}}{{11}}{}{\mathrm{π}}\right)$ (6)
 > $\frac{\partial }{\partial x}\mathrm{MathieuC}\left(a,q,x\right)$
 ${\mathrm{MathieuCPrime}}{}\left({a}{,}{q}{,}{x}\right)$ (7)
 > $\frac{\partial }{\partial x}$
 $\left({2}{}{q}{}{\mathrm{cos}}{}\left({2}{}{x}\right){-}{a}\right){}{\mathrm{MathieuC}}{}\left({a}{,}{q}{,}{x}\right)$ (8)
 > $\frac{{\partial }^{3}}{\partial {x}^{3}}\mathrm{MathieuCE}\left(n,q,x\right)$
 ${-}{4}{}{q}{}{\mathrm{sin}}{}\left({2}{}{x}\right){}{\mathrm{MathieuCE}}{}\left({n}{,}{q}{,}{x}\right){+}\left({2}{}{q}{}{\mathrm{cos}}{}\left({2}{}{x}\right){-}{\mathrm{MathieuA}}{}\left({n}{,}{q}\right)\right){}{\mathrm{MathieuCEPrime}}{}\left({n}{,}{q}{,}{x}\right)$ (9)
 > $\mathrm{series}\left(\mathrm{MathieuA}\left(1,q\right),q\right)$
 ${1}{+}{q}{-}\frac{{1}}{{8}}{}{{q}}^{{2}}{-}\frac{{1}}{{64}}{}{{q}}^{{3}}{-}\frac{{1}}{{1536}}{}{{q}}^{{4}}{+}\frac{{11}}{{36864}}{}{{q}}^{{5}}{+}{\mathrm{O}}\left({{q}}^{{6}}\right)$ (10)
 > $\mathrm{series}\left(\mathrm{MathieuSE}\left(2,q,x\right),q,3\right)$
 ${\mathrm{sin}}{}\left({2}{}{x}\right){-}\frac{{1}}{{12}}{}{\mathrm{sin}}{}\left({4}{}{x}\right){}{q}{+}\left({-}\frac{{1}}{{288}}{}{\mathrm{sin}}{}\left({2}{}{x}\right){+}\frac{{1}}{{384}}{}{\mathrm{sin}}{}\left({6}{}{x}\right)\right){}{{q}}^{{2}}{+}{\mathrm{O}}\left({{q}}^{{3}}\right)$ (11)
 > $\mathrm{series}\left(\mathrm{MathieuCPrime}\left(2,q,x\right),x,5\right)$
 $\left({2}{}{q}{-}{2}\right){}{x}{+}\left(\frac{{2}}{{3}}{}{{q}}^{{2}}{-}\frac{{8}}{{3}}{}{q}{+}\frac{{2}}{{3}}\right){}{{x}}^{{3}}{+}{\mathrm{O}}\left({{x}}^{{5}}\right)$ (12)