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In [[mathematics]], the '''Mathieu functions''' are certain [[special functions]] useful for treating a variety of problems in applied mathematics, including:
*vibrating elliptical drumheads,
*[[quadrupole mass analyzer]]s and quadrupole [[ion traps]] for [[mass spectrometry]]
*wave motion in periodic media, such as ultracold atoms in an [[optical lattice]]
*the phenomenon of [[parametric oscillator#Parametric resonance|parametric resonance]] in forced [[oscillator]]s,
*exact plane wave solutions in [[general relativity]],
*the [[Stark effect]] for a rotating [[electric dipole]],
*in general, the solution of [[differential equations]] that are [[Separation of variables|separable]] in [[elliptic cylindrical coordinates]].

They were introduced by {{harvs|txt|authorlink=Émile Léonard Mathieu|first=Émile Léonard |last=Mathieu|year= 1868}} in the context of the first problem.

==Mathieu equation==
The canonical form for '''Mathieu's differential equation''' is

:<math> \frac{d^2y}{dx^2}+[a-2q\cos (2x) ]y=0. </math>

The Mathieu equation is a [[Hill differential equation|Hill equation]] with only 1 harmonic mode.

Closely related is '''Mathieu's modified differential equation'''

:<math> \frac{d^2y}{du^2}-[a-2q\cosh (2u) ]y=0 </math>

which follows on substitution <math>u=ix</math>.

The two above equations can be obtained from the [[Helmholtz equation]] in two dimensions, by expressing it in [[elliptical coordinates]] and then separating the two variables.[http://optica.mty.itesm.mx/pmog/Papers/P009.pdf] This is why they are also known as '''angular''' and '''radial Mathieu equation''', respectively.

The substitution <math>t=\cos(x)</math> transforms Mathieu's equation to the ''algebraic form''

:<math> (1-t^2)\frac{d^2y}{dt^2} - t\, \frac{d y}{dt} + (a + 2q (1- 2t^2)) \, y=0.</math>

This has two regular singularities at <math>t = -1,1</math> and one irregular singularity at infinity, which implies that in general (unlike many other special functions), the solutions of Mathieu's equation ''cannot'' be expressed in terms of [[hypergeometric function]]s.

Mathieu's differential equations arise as models in many contexts, including the stability of railroad rails as trains drive over them, seasonally forced population dynamics, the four-dimensional [[wave equation]], and the [[Floquet theory]] of the stability of [[limit cycles]].

==Floquet solution==
According to [[Floquet's theorem]] (or [[Bloch wave|Bloch's theorem]]), for fixed values of a,q, Mathieu's equation admits a ''complex valued'' solution of form
:<math>F(a,q,x) = \exp(i \mu \,x) \, P(a,q,x)</math>
where <math>\mu</math> is a complex number, the ''Mathieu exponent'', and P is a complex valued function which is ''periodic'' in <math>x</math> with period <math>\pi</math>.  However, P is in general ''not'' sinusoidal.  In the example plotted below, <math>a=1, \, q=\frac{1}{5}, \, \mu \approx 1 + 0.0995 i</math> (real part, red; imaginary part; green):
[[Image:MathieuFloquet.gif|center]]

==Mathieu sine and cosine==
For fixed a,q, the '''Mathieu cosine''' <math>C(a,q,x)</math> is a function of <math>x</math> defined as the unique solution of the Mathieu equation which
#takes the value <math>C(a,q,0)=1</math>,
#is an [[even function]], hence <math>C^\prime(a,q,0)=0</math>.
Similarly, the '''Mathieu sine''' <math>S(a,q,x)</math> is the unique solution which
#takes the value <math>S^\prime(a,q,0)=1</math>,
#is an [[odd function]], hence <math>S(a,q,0)=0</math>.

These are ''real-valued'' functions which are closely related to the Floquet solution:
:<math> C(a,q,x) = \frac{F(a,q,x) + F(a,q,-x)}{2 F(a,q,0)}</math>
:<math> S(a,q,x) = \frac{F(a,q,x) - F(a,q,-x)}{2 F^\prime(a,q,0)}.</math>
The general solution to the Mathieu equation (for fixed a,q) is a linear combination of the Mathieu cosine and Mathieu sine functions.

A noteworthy special case is
:<math>C(a,0,x) = \cos(\sqrt{a} x), \; S(a,0,x) = \frac{\sin(\sqrt{a} x)}{\sqrt{a}},</math>
i.e. when the corresponding [[Helmholtz equation]] problem has circular symmetry.

In general, the Mathieu sine and cosine are ''aperiodic''.  Nonetheless, for small values of q, we have approximately
:<math> C(a,q,x) \approx \cos(\sqrt{a} x), \; \; S(a,q,x) \approx \frac{\sin (\sqrt{a} x)}{\sqrt{a}}.</math>
For example:
[[Image:MathieuC shortwave.gif|left|thumb|300px|Red: C(0.3,0.1,x).]]
[[Image:MathieuCPrime shortwave.gif|left|thumb|300px|Red: C'(0.3,0.1,x).]] 
<br style="clear:both;">

==Periodic solutions==
Given <math>q</math>, for countably many special values of <math>a</math>, called ''characteristic values'', the Mathieu equation admits solutions which are periodic with period <math>2\pi</math>.  The characteristic values of the Mathieu cosine, sine functions respectively are written <math>a_n(q), \, b_n(q)</math>, where ''n'' is a [[natural number]].  The periodic special cases of the Mathieu cosine and sine functions are often written <math>CE(n,q,x), \, SE(n,q,x)</math> respectively, although they are traditionally given a different normalization (namely, that their L<sup>2</sup> norm equal <math>\pi</math>).  Therefore, for positive ''q'', we have
:<math>C \left( a_n(q),q,x \right) = \frac{CE(n,q,x)}{CE(n,q,0)}</math>
:<math>S \left( b_n(q),q,x \right) = \frac{SE(n,q,x)}{SE^\prime(n,q,0)}.</math>
Here are the first few periodic Mathieu cosine functions for&nbsp;''q''&nbsp;=&nbsp;1:
[[Image:MathieuCE.gif|center]]
Note that, for example, <math>CE(1,1,x)</math> (green) resembles a cosine function, but with flatter hills and shallower valleys.

==Solutions to the modified Mathieu equation==

{{Empty section|date=November 2013}}

==See also==
*[[Monochromatic electromagnetic plane wave]], an example of an important exact plane wave solution to the [[Einstein field equation]] in [[general relativity]] which is expressed using Mathieu cosine functions.
*[[Inverted pendulum]]
*[[Lamé function]]

==References==
* {{cite journal | author=Mathieu, E. |title=Mémoire sur Le Mouvement Vibratoire d’une Membrane de forme Elliptique |url=http://math-docARRAYjf-grenobleARRAYr/JMPA/ |journal=[[Journal de Mathématiques Pures et Appliquées]] | year=1868 | pages=137–203 | url=http://visualiseur.bnf.fr/ConsulterElementNum?O=NUMM-16412&Deb=145&Fin=211&E=PDF}}
* Gertrude Blanch, "[http://www.math.sfu.ca/~cbm/aands/page_721.htm Chapter 20. Mathieu Functions]", in Milton Abramowitz and Irene A. Stegun, eds., ''[[Abramowitz and Stegun|Handbook of Mathematical Functions]] with Formulas, Graphs, and Mathematical Tables'' (Dover: New York, 1972)
* {{cite book | author=McLachlan, N. W. | title=Theory and application of Mathieu functions | location=New York | publisher=Dover | year=1962 (reprint of 1947 ed.) | id=LCCN 64016333}}
*{{dlmf|first=G.|last=Wolf|id=28|title=Mathieu Functions and Hill’s Equation}}

==External links==
* {{springer|title=Mathieu functions|id=p/m062760}}
* Timothy Jones, ''[http://www.physics.drexel.edu/~tim/open/mat/mat.html Mathieu's Equations and the Ideal rf-Paul Trap]'' (2006)
* {{mathworld|urlname=MathieuFunction |title=Mathieu function}}
* ''[http://eqworld.ipmnet.ru/en/solutions/ode/ode0234.pdf Mathieu equation]'', [http://eqworld.ipmnet.ru/en/ EqWorld]
*[http://functions.wolfram.com/MathieuandSpheroidalFunctions/ List of equations and identities for Mathieu Functions] functions.wolfram.com
*[http://dlmf.nist.gov/28 NIST Digital Library of Mathematical Functions: Mathieu Functions and Hill's Equation]

{{DEFAULTSORT:Mathieu Function}}
[[Category:Ordinary differential equations]]
[[Category:Special functions]]