Mathieu function: Difference between revisions

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{{Short description|Mathematical function}}

{{short description|Solutions to a differential equation studied by Émile L. Mathieu}}

{{Distinguish|Massieu function}}

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An equivalent statement of Floquet's theorem is that 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 ''Floquet exponent'' (or sometimes ''Mathieu exponent''), and <math>P</math> is a complex valued function periodic in <math>x</math> with period <math>\pi</math>. An example <math>P(a, q, x)</math> is plotted to the right.

=== Stability in parameter space ===

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<math display=block>m(k) = \begin{cases}

2 \sqrt{\frac{k(k-1)(k-4)}{3 k-8}}, & k<0 ; \\[4pt]

\frac{1}{4} \left[\sqrt{(9-4 k)(13-20 k)}-(9-4 k) \right], & 0<k<\frac{1}{4} ; \\[10pt]

\frac{1}{4} \left[9-4 k \mp \sqrt{(9-4 k)(13-20 k)} \right], & \frac{1}{4}<k<\frac{13}{20} ; \\[6pt]

\sqrt{\frac{2(k-1)(k-4)(k-9)}{k-5}}, & \frac{13}{20}<k<1 ; \\[2pt]

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== Properties ==

There are relatively few analytic expressions and identities involving Mathieu functions. Moreover, unlike many other [[special functions]], the solutions of Mathieu's equation cannot in general be expressed in terms of [[hypergeometric function]]s. This can be seen by transformation of Mathieu's equation to algebraic form, using the change of variable <math>t=\cos(x)</math>:

:<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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== Applications ==

Mathieu's differential equations appear in a wide range of contexts in engineering, physics, and [[applied mathematics]]. Many of these applications fall into one of two general categories: 1) the analysis of partial differential equations in elliptic geometries, and 2) dynamical problems which involve forces that are periodic in either space or time. Examples within both categories are discussed below.

=== Partial differential equations ===

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*the [[Stark effect]] for a rotating [[electric dipole]]

* the [[Floquet theory]] of the stability of [[limit cycles]]

* analytic [[Traveling wave|traveling-wave]] solutions of the [[Kardar–Parisi–Zhang equation|Kardar-Parisi-Zhang]] interface growing equation with periodic noise term<ref>{{Cite book |last1=Barna |first1=Imre Ferenc |title=Differential and Difference Equations with Applications, ICDDEA 2019, Lisbon, Portugal, July 1–5 Conference proceedings |datelast2=2020Bognár |publisherfirst2=SpringerG. |yearlast3=Mátyás |first3=L. |last4=Guedda |first4=M. |last5=Hriczó |first5=K. |date=2020 |publisher=Springer |isbn=9783030563226 |editor-last=Pinelas |editor-first=S. |pages=239239–254 |chapter=Analytic Traveling-253Wave Solutions of the Kardar-Parisi-Zhang Interface Growing Equation with Different Kind of Noise Terms |arxiv=1908.09615 |editor-last2=Graef |editor-first2=John R. |editor-last3=Hilger |editor-first3=S. |editor-last4=Kloeden |editor-first4=P |editor-last5=Shinas |editor-first5=C. }}</ref>

=== Quantum mechanics ===

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By solving the Schrödinger equation (for this particular potential) in terms of solutions of the modified Mathieu equation, scattering properties such as the [[S-matrix]] and the [[Absorptance|absorptivity]] can be obtained.<ref>Müller-Kirsten (2006)</ref>

Originally the Schrödinger equation with cosine function was solved in 1928 by Strutt.<ref>{{Cite journal |last=Strutt |first=M.J.O. |date=1928 |title=Zur Wellenmechanik des Atomgitters |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.19283911006 |journal=Ann. d.Annalen Phys.der Physik|volume=86 |issue=10 |pages=319319–324 |doi=10.1002/andp.19283911006|bibcode=1928AnP...391..319S |url-access=subscription }}</ref>.

==See also==

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* {{Citation|last= Barakat|first=R.| year =1963| title=Diffraction of Plane Waves by an Elliptic Cylinder | journal =The Journal of the Acoustical Society of America |volume = 35 |issue=12| pages=1990–1996| doi =10.1121/1.1918878 |bibcode=1963ASAJ...35.1990B}}

* {{cite book |first1 = Malcolm M.|last1 = Bibby|first2=Andrew F.|last2=Peterson|year = 2014| title = Accurate Computation of Mathieu Functions| publisher = Morgan & Claypool| isbn = 9781627050852| doi = 10.2200/S00526ED1V01Y201307CEM032| s2cid=28354918 }}

* {{Cite journal |last=Butikov |first=Eugene I. |date=April 2018 |title=Analytical expressions for stability regions in the Ince–Strutt diagram of Mathieu equation |url=https://pubs.aip.org/aapt/ajp/article/86/4/257-267/1057663 |journal=American Journal of Physics |language=en |volume=86 |issue=4 |pages=257–267 |doi=10.1119/1.5021895 |bibcode=2018AmJPh..86..257B |issn=0002-9505|doi-access=free }}

* {{Citation|last1= Chaos-Cador|first1= L.|last2= Ley-Koo|first2= E.| year =2002| title= Mathieu functions revisited: matrix evaluation and generating functions| url = https://rmf.smf.mx/ojs/rmf/article/view/3035| journal = Revista mexicana de física|volume =48| issue =1 | pages=67–75}}

* {{cite journal|first1=Robert B.|last1=Dingle|first2=Harald J.W.|last2=Müller|title=The Form of the Coefficients of the Late Terms in Asymptotic Expansions of the Characteristic Numbers of Mathieu and Spheroidal-Wave Functions|journal=[[Journal für die reine und angewandte Mathematik]] |volume=216|year=1964|pages=123–133|issn=0075-4102}}