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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:

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

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

 • The various Mathieu functions are normalized as follows.

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

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

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

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

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

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

 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 $n=0$, where the function implemented in Maple is equal to the function in reference [1] multiplied by sqrt(2). In this way, we have, in Maple, for instance, MathieuCE(0, 1, 0) = 1 while in [1] the right-hand side is equal to $\sqrt{2}$. For details on this normalization, see reference [2].
 • 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)
 > $\mathrm{MathieuSE}\left(n,q,x+\frac{25\mathrm{Pi}}{11}\right)$
 ${\mathrm{MathieuSE}}{}\left({n}{,}{q}{,}{x}{+}\frac{{3}{}{\mathrm{\pi }}}{{11}}\right)$ (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)
 > $\mathrm{series}\left(\mathrm{MathieuA}\left(1,q\right),q\right)$
 ${1}{+}{q}{-}\frac{{1}}{{8}}{}{{q}}^{{2}}{-}\frac{{1}}{{64}}{}{{q}}^{{3}}{-}\frac{{1}}{{1536}}{}{{q}}^{{4}}{+}\frac{{11}}{{36864}}{}{{q}}^{{5}}{+}{O}{}\left({{q}}^{{6}}\right)$ (10)
 > $\mathrm{series}\left(\mathrm{MathieuSE}\left(2,q,x\right),q,3\right)$
 ${\mathrm{sin}}{}\left({2}{}{x}\right){-}\frac{{1}}{{12}}{}{\mathrm{sin}}{}\left({4}{}{x}\right){}{q}{+}\left({-}\frac{{\mathrm{sin}}{}\left({2}{}{x}\right)}{{288}}{+}\frac{{\mathrm{sin}}{}\left({6}{}{x}\right)}{{384}}\right){}{{q}}^{{2}}{+}{O}{}\left({{q}}^{{3}}\right)$ (11)
 > $\mathrm{series}\left(\mathrm{MathieuCPrime}\left(2,q,x\right),x,5\right)$
 $\left({2}{}{q}{-}{2}\right){}{x}{+}\left(\frac{{2}}{{3}}{}{{q}}^{{2}}{-}\frac{{8}}{{3}}{}{q}{+}\frac{{2}}{{3}}\right){}{{x}}^{{3}}{+}{O}{}\left({{x}}^{{5}}\right)$ (12)

References

 [1] Abramowitz, M., and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover Publications.
 [2] Frenkel, D., and Portugal, R. "Algebraic methods to compute Mathieu functions." Journal of Physics A: Mathematical and General, Vol. 34. (2001): 3541-3551.
 [3] McLachlan, N. W. Theory and Applications of Mathieu Functions. Oxford University Press.