Mathieu function: Difference between revisions

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{{Short description|Mathematical function}}

{{Distinguish|Massieu function}}

In [[mathematics]], '''Mathieu functions''', sometimes called '''angular Mathieu functions''', are solutions of Mathieu's [[differential equation]]

:<math>

\frac{d^2y}{dx^2} + (a - 2q\cos(2x))y = 0,

</math>

where {{mvar|a, q}} are [[Real number|real]]-valued parameters. Since we may add {{math|&pi;/2}} to {{mvar|x}} to change the sign of {{mvar|q}}, it is a usual convention to set {{math|''q'' ≥ 0}}.

where <math>a</math> and <math>q</math> are parameters. They were first introduced by [[Émile Léonard Mathieu]], who encountered them while studying vibrating elliptical drumheads.<ref>Mathieu (1868).</ref><ref name="MorseandFeshbach">Morse and Feshbach (1953).</ref><ref name="BrimacombeCorlessZamir">Brimacombe, Corless and Zamir (2021)</ref> They have applications in many fields of the physical sciences, such as [[optics]], [[quantum mechanics]], and [[general relativity]]. They tend to occur in problems involving periodic motion, or in the analysis of [[partial differential equation]] [[boundary value problem]]s possessing [[elliptic]] symmetry.<ref name="Gutiérrez-Vega2015">Gutiérrez-Vega (2015).</ref>

where <math>a</math> and <math>q</math> are parameters. They were first introduced by [[Émile Léonard Mathieu]], who encountered them while studying vibrating elliptical drumheads[[drumhead]]s.<ref>Mathieu (1868).</ref><ref name="MorseandFeshbach">Morse and Feshbach (1953).</ref><ref name="BrimacombeCorlessZamir">Brimacombe, Corless and Zamir (2021)</ref> They have applications in many fields of the physical sciences, such as [[optics]], [[quantum mechanics]], and [[general relativity]]. They tend to occur in problems involving periodic motion, or in the analysis of [[partial differential equation]] (PDE) [[boundary value problem]]s possessing [[elliptic]] symmetry.<ref name="Gutiérrez-Vega2015">Gutiérrez-Vega (2015).</ref>

== Definition ==

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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 ===

The Mathieu equation has two parameters. For almost all choices of these parameters, Floquet theory says that any solution either converges to zero or diverges to infinity.

If the Mathieu equation is parameterized as <math>\ddot x + k(1-m \cos(t))x = 0</math>, where <math>k \in \R, m \geq 0</math>, then the regions of stability and instability are separated by the following curves:{{sfn|Butikov|2018}}

<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]

2 \sqrt{\frac{k(k-1)(k-4)}{3 k-8}}, & k>1 .

\end{cases}</math>

== Other types of Mathieu functions ==

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\end{align}

</math>

In particular, <math>X_{2r}</math> is finite whereas <math>Y_{2r}</math> diverges. Writing <math>A_{2r} = c_1 X_{2r} + c_2 Y_{2r}</math>, we therefore see that in order for the Fourier series representation of <math>\text{ce}_{2n}</math> to converge, <math>a</math> must be chosen such that <math>c_2 = 0.</math>. These choices of <math>a</math> correspond to the characteristic numbers.

In general, however, the solution of a three-term recurrence with variable coefficients

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To overcome these issues, more sophisticated semi-analytical/numerical approaches are required, for instance using a [[continued fraction]] expansion,<ref>McLachlan (1947)</ref><ref name="Arscott_chapIII"/> casting the recurrence as a [[matrix (mathematics)|matrix]] eigenvalue problem,<ref>Chaos-Cador and Ley-Koo (2001)</ref> or implementing a backwards recurrence algorithm.<ref name="Wimp1984"/> The complexity of the three-term recurrence relation is one of the reasons there are few simple formulas and identities involving Mathieu functions.<ref name="Temme2015">Temme (2015), p. 234</ref>

In practice, Mathieu functions and the corresponding characteristic numbers can be calculated using pre-packaged software, such as [[Mathematica]], [[Maple (software)|Maple]], [[MATLAB]], and [[SciPy]]. For small values of <math>q</math> and low order <math>n</math>, they can also be expressed perturbatively as [[power series]] of <math>q</math>, which can be useful in physical applications.<ref>Müller-Kirsten (2012), pp. 420-428</ref>

=== Second kind ===

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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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:<math>

\begin{align}

&\int_{}int_0^{2\pi} \text{ce}_n \text{ce}_m \,dx = \int_0^{2\pi} \text{se}_n \text{se}_m \, dx = \delta_{nm} \pi \\

&\int_0^{2\pi} \text{ce}_n \text{se}_m \,dx = 0

\end{align}

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For the even and odd periodic Mathieu functions <math> ce, se</math> and the associated characteristic numbers <math>a</math> one can also derive asymptotic expansions for large <math>q</math>.<ref>McLachlan (1947), p. 237; Dingle and Müller (1962); Müller (1962); Dingle and Müller(1964)</ref> For the characteristic numbers in particular, one has with <math>N</math> approximately an odd integer, i.e. <math>N\approx N_0= 2n+1, n =1,2,3,...,</math>

:<math>

\begin{align}

a(N) = -2q + 2q^{1/2}N -\frac{1}{2^3}(N^2+1) -\frac{1}{2^7q^{1/2}}N(N^2 +3) -\frac{1}{2^{12}q}(5N^4 + 34N^2 +9)</math>

:<math>\;\;\;\;\;\;\;\;a(N) ={}& -2q + 2q^{1/2}N -\frac{1}{2^3}(N^2+1) -\frac{171}q{2^7q^{31/2}}N(33N^4 +410NN^2 +4053) -\frac{1}{2^{2012}q^2}(63N^6 + 1260N5N^4 + 2943N34N^2 +418079) +\\

& -\frac{1}{2^{17}q^{3/2}}N(33N^4 +410N^2 +405) -\frac{1}{2^{20}q^2}(63N^6 + 1260N^4 + 2943N^2 +41807) +

\mathcal{O}(q^{-5/2})

\end{align}

</math>

Observe the symmetry here in replacing <math> q^{1/2}</math> and <math>N</math> by <math>-q^{1/2}</math> and <math>- N</math>, which is a significant feature of the expansion. Terms of this expansion have been obtained explicitly up to and including the term of order <math>|q|^{-7/2}</math>.<ref name="Dingle and Müller 1962">Dingle and Müller (1962)</ref> Here <math>N</math> is only approximately an odd integer because in the limit of <math>q\rightarrowto \infty</math> all minimum segments of the periodic potential <math> \cos 2x</math> become effectively independent harmonic oscillators (hence <math>N_0</math> an odd integer). By decreasing <math>q</math>, tunneling through the barriers becomes possible (in physical language), leading to a splitting of the characteristic numbers <math> a \rightarrowto a_{\mp}</math> (in quantum mechanics called eigenvalues) corresponding to even and odd periodic Mathieu functions. This splitting is obtained with boundary conditions<ref name="Dingle and Müller 1962"/> (in quantum mechanics this provides the splitting of the eigenvalues into energy bands).<ref name="Müller-Kirsten 2012">Müller-Kirsten (2012)</ref> The boundary conditions are:

:<math> \biggleft(\frac{dce_{N_0-1}}{dx}\biggright)_{\pi/2}=0,\;\; ce_{N_0}(\pi/2)=0, \;\; \biggleft(\frac{dse_{N_0}}{dx}\biggright)_{\pi/2}=0, \;\; se_{N_0+1}(\pi/2)=0.</math>

Imposing these boundary conditions on the asymptotic periodic Mathieu functions associated with the above expansion for <math>a</math> one obtains

:<math> N-N_0 = \mp 2\biggleft(\frac{2}{\pi}\biggright)^{1/2} \frac{(16q^{1/2})^{N_0/2}e^{-4q^{1/2}}}{[\frac{1}{2}(N_0-1)]!}\biggleft[1-\frac{3(N_0^2+1)}{2^6q^{1/2}} + \frac{1}{2^{13}q}(9N_0^4 -40N_0^3 +18N_0^2 -136N_0 + 9) + ...\dots \biggright].</math>

The corresponding characteristic numbers or eigenvalues then follow by expansion, i.e.

:<math> a(N) = a(N_0) + (N-N_0) \biggleft(\frac{\partial a}{\partial N}\biggright)_{N_0} + ...\cdots . </math>

Insertion of the appropriate expressions above yields the result

:<math>\begin{align}

:<math>a(N)\rightarrow a_{\mp}(N_0)= -2q + 2q^{1/2}N_0 -\frac{1}{2^3}(N^2_0+1) - \frac{1}{2^7q^{1/2}}N_0(N_0^2+3) -

a(N)\to a_{\mp}(N_0) = {} & -2q + 2q^{1/2}NN_0 -\frac{1}{2^3}(N^22_0+1) - \frac{1}{2^7q^{1/2}}NN_0(NN_0^2 +3) -\frac{1}{2^{12}q}(5N^4 + 34N^2 +9)</math>

\frac{1}{2^{12}q}(5N_0^4+34N_0^2 +9) - ... </math>

:<math>\;\;\;\;\;\;frac{1}{2^{12}q}(5N_0^4+34N_0^2 +9) - \;\;\;\;\;\;cdots \;\;
& \mp \frac{(16q^{1/2})^{N_0/2+1}e^{-4q^{1/2}}}{(8\pi)^{1/2}[\frac{1}{2}(N_0-1)]!}\bigg[1-\frac{N_0}{2^6q^{1/2}}(3N_0^2+8N_0+3) + ...\cdots\bigg].

\end{align}

</math>

For <math>N_0=1,3,5,...\dots</math> these are the eigenvalues associated with the even Mathieu eigenfunctions <math> ce_{N_0} </math> or <math>ce_{N_0-1}</math> (i.e. with upper, minus sign) and odd Mathieu eigenfunctions <math>se_{N_0+1}</math> or

<math>se_{N_0}</math> (i.e. with lower, plus sign). The explicit and normalised expansions of the eigenfunctions can be found in <ref name="Dingle and Müller 1962"/> or.<ref name="Müller-Kirsten 2012"/>

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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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*Continuity of derivative across the interfocal line: <math>\psi_{\mu}(0, \nu) = -\psi_{\mu}(0, -\nu)</math>

For given <math>k</math>, this restricts the solutions to those of the form <math>\text{Ce}_{n}(\mu, q)\text{ce}_n(\nu, q)</math> and <math>\text{Se}_{n}(\mu, q)\text{se}_n(\nu, q)</math>, where <math>q = c^2 k^2 / 24</math>. This is the same as restricting allowable values of <math>a</math>, for given <math>k</math>. Restrictions on <math>k</math> then arise due to imposition of physical conditions on some bounding surface, such as an elliptic boundary defined by <math>\mu = \mu_0 > 0</math>. For instance, clamping the membrane at <math>\mu = \mu_0</math> imposes <math>\psi(\mu_0, \nu) = 0</math>, which in turn requires

:<math>

\begin{align}

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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 |last2=Bognár |first2=G. |last3=Mátyás |first3=L. |last4=Guedda |first4=M. |last5=Hriczó |first5=K. |date=2020 |publisher=Springer |isbn=9783030563226 |editor-last=Pinelas |editor-first=S. |pages=239–254 |chapter=Analytic Traveling-Wave 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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: <math> y = r^{1/2} \varphi, r=\gamma e^z, \gamma = \frac{ig}{h}, h^2 = ikg, h = e^{I\pi/4}(kg)^{1/2}. </math>

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= Annalen der Physik|volume=86 |issue=10 |pages=319–324 |doi=10.1002/andp.19283911006|bibcode=1928AnP...391..319S |url-access=subscription }}</ref>

==See also==

*[[Almost Mathieu operator]]

*[[List of mathematical functions]]

*[[Bessel function]]

*[[Hill differential equation]]

*[[Inverted pendulum]]

*[[Lamé function]]

*[[List of mathematical functions]]

*[[Monochromatic electromagnetic plane wave]]

*[[Inverted pendulum]]

==Notes==

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==References==

{{Refbegin|30em}}

*{{cite book|last1 = Arscott| first1 = Felix| author-link = Felix Arscott| title = Periodic differential equations: an introduction to Mathieu, Lamé, and allied functions| publisher = Pergamon Press| year = 1964| url = https://books.google.com/books?id=jfjiBQAAQBAJ&pg=PP1| isbn = 9781483164885}}

* {{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 |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 }}

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* {{cite book | author=McLachlan, N. W. | title=Theory and application of Mathieu functions| publisher=Oxford University Press | year=1951}} ''Note: Reprinted lithographically in Great Britain at the University Press, Oxford, 1951 from corrected sheets of the (1947) first edition.''

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* {{cite journal|first1=R.B.|last1=Dingle|first2=H.J.W.|last2=Müller|title=Asymptotic Expansions of Mathieu Functions and their Characteristic Numbers|journal=[[Journal für die reine und angewandte Mathematik]]|volume=1962|year=1962|issue=211|pages=11–32|issn=0075-4102|doi=10.1515/crll.1962.211.11|s2cid=117516747 }}

* {{cite journal|first1=H.J.W.|last1=Müller|title=On Asymptotic Expansions of Mathieu Functions|journal=[[Journal für die reine und angewandte Mathematik]]|volume=1962|year=1962|issue=211|pages=179–190|issn=0075-4102|doi=10.1515/crll.1962.211.179|s2cid=118909645 }}

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* {{cite book |last1 = Wimp|first1 = Jet | year = 1984| title = Computation with Recurrence Relations | publisher = Pitman Publishing| isbn = 0-273-08508-5|pages=83–84}}

*{{dlmf|first=G.|last=Wolf|id=28|title=Mathieu Functions and Hill’s Equation}}

* {{cite journal|lastlast1=Brimacombe|firstfirst1=Chris|last2=Corless|first2=Robert M.|last3=Zamir|first3=Mair|date=2021|title=Computation and Applications of Mathieu Functions: A Historical Perspective|url=https://epubs.siam.org/doi/10.1137/20M135786X|journal=SIAM Review|language=en|volume=63|issue=4|pages=653–720|doi=10.1137/20M135786X|s2cid=220969117 |issn=0036-1445|doi-access=free|arxiv=2008.01812}}

{{Refend}}