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| In [[mathematics]], the '''Mathieu functions''' are certain [[special functions]] useful for treating a variety of problems in applied mathematics, including:
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| *vibrating elliptical drumheads,
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| *[[quadrupole mass analyzer]]s and quadrupole [[ion traps]] for [[mass spectrometry]]
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| *wave motion in periodic media, such as ultracold atoms in an [[optical lattice]]
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| *the phenomenon of [[parametric oscillator#Parametric resonance|parametric resonance]] in forced [[oscillator]]s,
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| *exact plane wave solutions in [[general relativity]],
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| *the [[Stark effect]] for a rotating [[electric dipole]],
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| *in general, the solution of [[differential equations]] that are [[Separation of variables|separable]] in [[elliptic cylindrical coordinates]].
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| They were introduced by {{harvs|txt|authorlink=Émile Léonard Mathieu|first=Émile Léonard |last=Mathieu|year= 1868}} in the context of the first problem.
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| ==Mathieu equation==
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| The canonical form for '''Mathieu's differential equation''' is
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| :<math> \frac{d^2y}{dx^2}+[a-2q\cos (2x) ]y=0. </math>
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| The Mathieu equation is a [[Hill differential equation|Hill equation]] with only 1 harmonic mode.
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| Closely related is '''Mathieu's modified differential equation'''
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| :<math> \frac{d^2y}{du^2}-[a-2q\cosh (2u) ]y=0 </math>
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| which follows on substitution <math>u=ix</math>.
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| 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.
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| The substitution <math>t=\cos(x)</math> transforms Mathieu's equation to the ''algebraic form'' | |
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| :<math> (1-t^2)\frac{d^2y}{dt^2} - t\, \frac{d y}{dt} + (a + 2q (1- 2t^2)) \, y=0.</math>
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| 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.
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| 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]].
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| ==Floquet solution==
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| 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
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| :<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):
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| [[Image:MathieuFloquet.gif|center]] | |
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| ==Mathieu sine and cosine==
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| 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
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| #takes the value <math>C(a,q,0)=1</math>,
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| #is an [[even function]], hence <math>C^\prime(a,q,0)=0</math>.
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| Similarly, the '''Mathieu sine''' <math>S(a,q,x)</math> is the unique solution which
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| #takes the value <math>S^\prime(a,q,0)=1</math>,
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| #is an [[odd function]], hence <math>S(a,q,0)=0</math>.
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| These are ''real-valued'' functions which are closely related to the Floquet solution:
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| :<math> C(a,q,x) = \frac{F(a,q,x) + F(a,q,-x)}{2 F(a,q,0)}</math>
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| :<math> S(a,q,x) = \frac{F(a,q,x) - F(a,q,-x)}{2 F^\prime(a,q,0)}.</math>
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| The general solution to the Mathieu equation (for fixed a,q) is a linear combination of the Mathieu cosine and Mathieu sine functions.
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| A noteworthy special case is
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| :<math>C(a,0,x) = \cos(\sqrt{a} x), \; S(a,0,x) = \frac{\sin(\sqrt{a} x)}{\sqrt{a}},</math>
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| i.e. when the corresponding [[Helmholtz equation]] problem has circular symmetry.
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| In general, the Mathieu sine and cosine are ''aperiodic''. Nonetheless, for small values of q, we have approximately
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| :<math> C(a,q,x) \approx \cos(\sqrt{a} x), \; \; S(a,q,x) \approx \frac{\sin (\sqrt{a} x)}{\sqrt{a}}.</math>
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| For example:
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| [[Image:MathieuC shortwave.gif|left|thumb|300px|Red: C(0.3,0.1,x).]]
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| [[Image:MathieuCPrime shortwave.gif|left|thumb|300px|Red: C'(0.3,0.1,x).]]
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| <br style="clear:both;"> | |
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| ==Periodic solutions==
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| 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
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| :<math>C \left( a_n(q),q,x \right) = \frac{CE(n,q,x)}{CE(n,q,0)}</math>
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| :<math>S \left( b_n(q),q,x \right) = \frac{SE(n,q,x)}{SE^\prime(n,q,0)}.</math>
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| Here are the first few periodic Mathieu cosine functions for ''q'' = 1:
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| [[Image:MathieuCE.gif|center]]
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| Note that, for example, <math>CE(1,1,x)</math> (green) resembles a cosine function, but with flatter hills and shallower valleys.
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| ==Solutions to the modified Mathieu equation==
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| {{Empty section|date=November 2013}}
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| ==See also==
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| *[[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.
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| *[[Inverted pendulum]]
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| *[[Lamé function]]
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| ==References==
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| * {{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}}
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| * 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)
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| * {{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}}
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| *{{dlmf|first=G.|last=Wolf|id=28|title=Mathieu Functions and Hill’s Equation}}
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| ==External links==
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| * {{springer|title=Mathieu functions|id=p/m062760}}
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| * Timothy Jones, ''[http://www.physics.drexel.edu/~tim/open/mat/mat.html Mathieu's Equations and the Ideal rf-Paul Trap]'' (2006)
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| * {{mathworld|urlname=MathieuFunction |title=Mathieu function}}
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| * ''[http://eqworld.ipmnet.ru/en/solutions/ode/ode0234.pdf Mathieu equation]'', [http://eqworld.ipmnet.ru/en/ EqWorld]
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| *[http://functions.wolfram.com/MathieuandSpheroidalFunctions/ List of equations and identities for Mathieu Functions] functions.wolfram.com
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| *[http://dlmf.nist.gov/28 NIST Digital Library of Mathematical Functions: Mathieu Functions and Hill's Equation]
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| {{DEFAULTSORT:Mathieu Function}}
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| [[Category:Ordinary differential equations]]
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| [[Category:Special functions]]
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From the tumble of 2007, Concerts: Bryan And which had an outstanding listing of , which include Urban. “It’s much like you are acquiring a acceptance to look to the next level, says those designers that have been an element of the Concert toursabove in a larger amount of artists.” It covered as among the most successful excursions in its 10-calendar year historical past.
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