Mathieu function explained

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

(d ^{2y})/(dx ^{2}) + (a - 2 q\cos( 2 x))y = 0 , where are real-valued parameters. Since we may add to to change the sign of, it is a usual convention to set .

They were first introduced by Émile Léonard Mathieu, who encountered them while studying vibrating elliptical drumheads.^{[1] [2] [3]} 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 problems possessing elliptic symmetry.^[4]

Definition

Mathieu functions

In some usages, Mathieu function refers to solutions of the Mathieu differential equation for arbitrary values of

a and q . When no confusion can arise, other authors use the term to refer specifically to \pi - or 2 \pi -periodic solutions, which exist only for special values of a and q .^[5] More precisely, for given (real) q such periodic solutions exist for an infinite number of values of a , called characteristic numbers, conventionally indexed as two separate sequences a _{n(q)} and b _{n(q)} , for n = 1 , 2 , 3 , \ldots . The corresponding functions are denoted ce _{n(x,} q) and se _{n(x,} q) , respectively. They are sometimes also referred to as cosine-elliptic and sine-elliptic, or Mathieu functions of the first kind.

As a result of assuming that

q is real, both the characteristic numbers and associated functions are real-valued.^[6] ce _{n(x,} q) and se _{n(x,} q) can be further classified by parity and periodicity (both with respect to x ), as follows:^[5]

Function	Parity	Period
ce _{n,} n even	even	\pi
ce _{n,} n odd	even	2 \pi
se _{n,} n even	odd	\pi
se _{n,} n odd	odd	2 \pi

The indexing with the integer

n , besides serving to arrange the characteristic numbers in ascending order, is convenient in that ce _{n(x,} q) and se _{n(x,} q) become proportional to \cos nx and \sin nx as q → 0 . With n being an integer, this gives rise to the classification of ce _{n} and se _{n} as Mathieu functions (of the first kind) of integral order. For general a and q , solutions besides these can be defined, including Mathieu functions of fractional order as well as non-periodic solutions.

Modified Mathieu functions

Closely related are the modified Mathieu functions, also known as radial Mathieu functions, which are solutions of Mathieu's modified differential equation

(d ^{2y})/(dx ^{2}) - (a - 2 q\cosh 2 x)y = 0 , which can be related to the original Mathieu equation by taking x \to \pm {\rm i} x . Accordingly, the modified Mathieu functions of the first kind of integral order, denoted by Ce _{n(x,} q) and Se _{n(x,} q) , are defined from^[7] \begin{align} Ce _{n(x,} q) & = ce _{n({\rm} i} x, q). \\ Se _{n(x,} q) & = - {\rm i} se _{n({\rm} i} x, q). \end{align} These functions are real-valued when x is real.

Normalization

A common normalization,^[8] which will be adopted throughout this article, is to demand

\int ^{2 \pi} _{0} ce _{n(x,} q) ^{2} dx = \int ^{2 \pi} _{0} se _{n(x,} q) ^{2} dx = \pi

as well as require

ce _{n(x,} q) → + \cos nx and se _{n(x,} q) → + \sin nx as q → 0 .

Stability

The Mathieu equation has two parameters. For almost all choices of parameter, by Floquet theory (see next section), any solution either converges to zero or diverges to infinity.

Parametrize Mathieu equation as

\ddot x + k( 1- m \cos(t))x = 0 , where k \in \R, m \geq 0 . The regions of stability and instability are separated by curves ^[9]

Floquet theory

See main article: Floquet theory. Many properties of the Mathieu differential equation can be deduced from the general theory of ordinary differential equations with periodic coefficients, called Floquet theory. The central result is Floquet's theorem:

It is natural to associate the characteristic numbers

a(q) with those values of a which result in \sigma = \pm 1 .^[10] Note, however, that the theorem only guarantees the existence of at least one solution satisfying y(x + \pi) = \sigma y(x) , when Mathieu's equation in fact has two independent solutions for any given a , q . Indeed, it turns out that with a equal to one of the characteristic numbers, Mathieu's equation has only one periodic solution (that is, with period \pi or 2 \pi ), and this solution is one of the ce _{n(x,} q) , se _{n(x,} q) . The other solution is nonperiodic, denoted fe _{n(x,} q) and ge _{n(x,} q) , respectively, and referred to as a Mathieu function of the second kind.^[11] This result can be formally stated as Ince's theorem:

An equivalent statement of Floquet's theorem is that Mathieu's equation admits a complex-valued solution of form

F(a, q, x) = \exp(i \mu x) P(a, q, x), where \mu is a complex number, the Floquet exponent (or sometimes Mathieu exponent), and P is a complex valued function periodic in x with period \pi . An example P(a, q, x) is plotted to the right.

Other types of Mathieu functions

Second kind

Since Mathieu's equation is a second order differential equation, one can construct two linearly independent solutions. Floquet's theory says that if

a is equal to a characteristic number, one of these solutions can be taken to be periodic, and the other nonperiodic. The periodic solution is one of the ce _{n(x,} q) and se _{n(x,} q) , called a Mathieu function of the first kind of integral order. The nonperiodic one is denoted either fe _{n(x,} q) and ge _{n(x,} q) , respectively, and is called a Mathieu function of the second kind (of integral order). The nonperiodic solutions are unstable, that is, they diverge as z → \pm inf ty .^[12]

The second solutions corresponding to the modified Mathieu functions

Ce _{n(x,} q) and Se _{n(x,} q) are naturally defined as Fe _{n(x,} q) = - i fe _{n(xi,} q) and Ge _{n(x,} q) = ge _{n(xi,} q) .

Fractional order

Mathieu functions of fractional order can be defined as those solutions

ce _{p} (x, q) and se _{p} (x, q) , p a non-integer, which turn into \cos p x and \sin p x as q → 0 .^[7] If p is irrational, they are non-periodic; however, they remain bounded as x → inf ty .

An important property of the solutions

ce _{p} (x, q) and se _{p} (x, q) , for p non-integer, is that they exist for the same value of a . In contrast, when p is an integer, ce _{p} (x, q) and se _{p} (x, q) never occur for the same value of a . (See Ince's Theorem above.)

These classifications are summarized in the table below. The modified Mathieu function counterparts are defined similarly.

Classification of Mathieu functions^[13]

Order	First kind	Second kind
Integral	ce _{n(x,} q)	fe _{n(x,} q)
Integral	se _{n(x,} q)	ge _{n(x,} q)
Fractional(p non-integral)	ce _{p} (x, q)	se _{p} (x, q)

Explicit representation and computation

First kind

Mathieu functions of the first kind can be represented as Fourier series:^[5]

\begin{align} ce _{2n} (x, q) & = \sum ^{inf ty} _{r =0} A ^{( 2 n)} _{2 r} (q) \cos ( 2 r x) \\ ce _{2n+1} (x, q) & = \sum ^{inf ty} _{r =0} A ^{( 2 n +1 )} _{2 r +1} (q) \cos \left[ ( 2 r +1 ) x \right] \\ se _{2n+1} (x, q) & = \sum ^{inf ty} _{r =0} B ^{( 2 n +1 )} _{2 r +1} (q) \sin \left[( 2 r +1 ) x\right] \\ se _{2n+2} (x, q) & = \sum ^{inf ty} _{r =0} B ^{( 2 n +2 )} _{2 r +2} (q) \sin \left[( 2 r +2 ) x\right] \\ \end{align} The expansion coefficients A ^{(i)} _{j} (q) and B ^{(i)} _{j} (q) are functions of q but independent of x . By substitution into the Mathieu equation, they can be shown to obey three-term recurrence relations in the lower index. For instance, for each ce _{2n} one finds^[14] \begin{align} a A _{0} - q A _{2} & = 0 \\ (a - 4 ) A _{2} - q (A _{4} + 2 A _{0)} & = 0 \\ (a - 4 r ^{2)} A _{2r} - q (A _{2r+2} + A _{2r-2} ) & = 0 , r \geq 2 \end{align} Being a second-order recurrence in the index 2 r , one can always find two independent solutions X _{2r} and Y _{2r} such that the general solution can be expressed as a linear combination of the two: A _{2r} = c _{1} X _{2r} + c _{2} Y _{2r} . Moreover, in this particular case, an asymptotic analysis^[15] shows that one possible choice of fundamental solutions has the property\begin{align} X _{2r} & = r ^{-2r-1} \left( - (e ^{2} q)/(4) \right) ^{r} \left[ 1 + l{O}(r ^{-1} ) \right] \\ Y _{2r} & = r ^{2r-1} \left( - (4)/(e ^{2} q) \right) ^{r} \left[ 1 + l{O}(r ^{-1} ) \right] \end{align} In particular, X _{2r} is finite whereas Y _{2r} diverges. Writing A _{2r} = c _{1} X _{2r} + c _{2} Y _{2r} , we therefore see that in order for the Fourier series representation of ce _{2n} to converge, a must be chosen such that c _{2} = 0 . These choices of a correspond to the characteristic numbers.

In general, however, the solution of a three-term recurrence with variable coefficientscannot be represented in a simple manner, and hence there is no simple way to determine

a from the condition c _{2} = 0 . Moreover, even if the approximate value of a characteristic number is known, it cannot be used to obtain the coefficients A _{2r} by numerically iterating the recurrence towards increasing r . The reason is that as long as a only approximates a characteristic number, c _{2} is not identically 0 and the divergent solution Y _{2r} eventually dominates for large enough r .

To overcome these issues, more sophisticated semi-analytical/numerical approaches are required, for instance using a continued fraction expansion,^{[16] [5]} casting the recurrence as a matrix eigenvalue problem,^[17] or implementing a backwards recurrence algorithm.^[15] The complexity of the three-term recurrence relation is one of the reasons there are few simple formulas and identities involving Mathieu functions.^[18]

In practice, Mathieu functions and the corresponding characteristic numbers can be calculated using pre-packaged software, such as Mathematica, Maple, MATLAB, and SciPy. For small values of

q and low order n , they can also be expressed perturbatively as power series of q , which can be useful in physical applications.^[19]

Second kind

There are several ways to represent Mathieu functions of the second kind.^[20] One representation is in terms of Bessel functions:^[21]

\begin{align} fe _{2n} (x, q) & =- (\pi \gamma _{n})/(2) \sum ^{inf ty} _{r =0} ( -1 ) ^{r+n} A ^{( 2 n)} _{2 r} ( - q) Im [J ^{ix} _{r(\sqrt{q}e} ) Y ^{- ix} _{r(\sqrt{q}e} )], where \gamma _{n} = \left\{ \begin{array}{cc} \sqrt{ 2 }, & if n = 0 \\ 2 n, & if n \geq 1 \end{array} \right. \\ fe _{2n+1} (x, q) & = (\pi \sqrt{q)/() } \sum_^ (-1)^ A^_(-q) \ \text[J_r(\sqrt{q}e^{ix}) Y_{r+1}(\sqrt{q}e^{-ix}) + J_{r+1}(\sqrt{q}e^{ix}) Y_r(\sqrt{q} e^{-ix})] \\\text_(x, q) &= -\frac \sum_^ (-1)^ B^_(-q) \ \text[J_r(\sqrt{q}e^{ix}) Y_{r+1}(\sqrt{q}e^{-ix}) - J_{r+1}(\sqrt{q}e^{ix}) Y_r(\sqrt{q} e^{-ix})] \\\text_(x, q) &= -\frac \sum_^ (-1)^ B^_(-q) \ \text[J_r(\sqrt{q}e^{ix}) Y_{r+2}(\sqrt{q}e^{-ix}) - J_{r+2}(\sqrt{q}e^{ix}) Y_r(\sqrt{q} e^{-ix})] \endwhere n, q > 0 , and J _{r(x)} and Y _{r(x)} are Bessel functions of the first and second kind.

Modified functions

A traditional approach for numerical evaluation of the modified Mathieu functions is through Bessel function product series.^[22] For large

n and q , the form of the series must be chosen carefully to avoid subtraction errors.^{[23] [24]}

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 functions. This can be seen by transformation of Mathieu's equation to algebraic form, using the change of variable

t = \cos(x) :( 1- t ^{2 ) (d ^{2y})/(dt ^{2})} - t (d y)/(dt) + (a + 2 q ( 1- 2 t ^{2))} y =0 .

Since this equation has an irregular singular point at infinity, it cannot be transformed into an equation of the hypergeometric type.^[18]

Qualitative behavior

For small

q , ce _{n} and se _{n} behave similarly to \cos nx and \sin nx . For arbitrary q , they may deviate significantly from their trigonometric counterparts; however, they remain periodic in general. Moreover, for any real q , ce _{m} (x, q) and se _{m+1} (x, q) have exactly m simple zeros in 0 < x < \pi , and as q → inf ty the zeros cluster about x = \pi / 2 .^{[25] [26]}

For

q > 0 and as x → inf ty the modified Mathieu functions tend to behave as damped periodic functions.

In the following, the

A and B factors from the Fourier expansions for ce _{n} and se _{n} may be referenced (see Explicit representation and computation). They depend on q and n but are independent of x .

Reflections and translations

Due to their parity and periodicity,

ce _{n} and se _{n} have simple properties under reflections and translations by multiples of \pi :^[7] \begin{align} & ce _{n(x} + \pi) = ( -1 ) ^{n} ce _{n(x)} \\ & se _{n(x} + \pi) = ( -1 ) ^{n} se _{n(x)} \\ & ce _{n(x} + \pi / 2 ) = ( -1 ) ^{n} ce _{n(-x} + \pi / 2 ) \\ & se _{n+1} (x + \pi / 2 ) = ( -1 ) ^{n} se _{n+1} ( - x + \pi / 2 ) \end{align}

One can also write functions with negative

q in terms of those with positive q :^{[5] [27]} \begin{align} & ce _{2n+1} (x, - q) = ( -1 ) ^{n} se _{2n+1} ( - x + \pi/ 2 , q) \\ & ce _{2n+2} (x, - q) = ( -1 ) ^{n} ce _{2n+2} ( - x + \pi/ 2 , q) \\ & se _{2n+1} (x, - q) = ( -1 ) ^{n} ce _{2n+1} ( - x + \pi/ 2 , q) \\ & se _{2n+2} (x, - q) = ( -1 ) ^{n} se _{2n+2} ( - x + \pi/ 2 , q) \end{align}

Moreover,

\begin{align} &a _{2n+1} (q) = b _{2n+1} ( - q)\\ &b _{2n+2} (q) = b _{2n+2} ( - q) \end{align}

Orthogonality and completeness

Like their trigonometric counterparts

\cos nx and \sin nx , the periodic Mathieu functions ce _{n(x,} q) and se _{n(x,} q) satisfy orthogonality relations\begin{align} &\int ^{2 \pi} _{0} ce _{n} ce _{m} dx = \int ^{2 \pi} _{0} se _{n} se _{m} dx = \delta _{nm} \pi \\ &\int ^{2 \pi} _{0} ce _{n} se _{m} dx = 0 \end{align}

Moreover, with

q fixed and a treated as the eigenvalue, the Mathieu equation is of Sturm–Liouville form. This implies that the eigenfunctions ce _{n(x,} q) and se _{n(x,} q) form a complete set, i.e. any \pi - or 2 \pi -periodic function of x can be expanded as a series in ce _{n(x,} q) and se _{n(x,} q) .^[4]

Integral identities

\chi(x, x') that are solutions of(\partial ^{2} \chi)/(\partial x ^{2}) - (\partial ^{2} \chi)/(\partial x' ^{2}) = 2 q \left(\cos 2 x - \cos 2 x' \right) \chi More precisely, if \phi(x) solves Mathieu's equation with given a and q , then the integral\psi(x) \equiv \int _{C} \chi(x, x') \phi(x') dx' where C is a path in the complex plane, also solves Mathieu's equation with the same a and q , provided the following conditions are met:^[28] \chi(x, x') solves (\partial ^{2} \chi)/(\partial x ^{2}) - (\partial ^{2} \chi)/(\partial x' ^{2}) = 2 q \left(\cos 2 x - \cos 2 x' \right) \chi

• In the regions under consideration,

\psi(x) exists and \chi(x, x') is analytic\left( \phi (\partial \chi)/(\partial x') - (\partial \phi)/(\partial x') \chi \right) has the same value at the endpoints of C

Using an appropriate change of variables, the equation for

\chi can be transformed into the wave equation and solved. For instance, one solution is \chi(x, x') = \sinh ( 2 q ^{1/2} \sin x \sin x' ) . Examples of identities obtained in this way are^[29] \begin{align} se _{2n+1} (x, q) & = (se ' _{2n+1} ( 0 , q))/(\pi q ^{1/2} B ^{( 2 n +1 )} _{1}) \int ^{\pi} _{0} \sinh ( 2 q ^{1/2} \sin x \sin x' ) se _{2n+1} (x', q) dx' (q > 0 ) \\ Ce _{2n} (x, q) & = (ce _{2n} (\pi / 2 , q))/(\pi A ^{( 2 n)} _{0}) \int ^{\pi} _{0} \cos ( 2 q ^{1/2} \cosh x \cos x' ) ce _{2n} (x', q) dx' (q > 0 ) \end{align}

Identities of the latter type are useful for studying asymptotic properties of the modified Mathieu functions.^[30]

There also exist integral relations between functions of the first and second kind, for instance:^[21]

fe _{2n} (x, q) = 2 n \int ^{x} _{0} ce _{2n} (\tau, - q) J _{0} \left( \sqrt{ 2 q(\cos 2 x - \cos 2 \tau)} \right) d \tau, n \geq 1 valid for any complex x and real q .

Asymptotic expansions

The following asymptotic expansions hold for

q > 0 , Im (x) = 0 , Re (x) → inf ty , and 2 q ^{1/2} \cosh x \simeq q ^{1/2} e ^{x} :^[31] \begin{align} Ce _{2n} (x,q) &\sim \left( (2)/(\pi q ^{1/2}) \right) ^{1/2} (ce _{2n} ( 0 , q) ce _{2n} (\pi/ 2 , q))/(A ^{( 2 n)} _{0}) ⋅ e ^{-x/2} \sin \left( q ^{1/2} e ^{x} + (\pi)/(4) \right) \\ Ce _{2n+1} (x,q) &\sim \left( (2)/(\pi q ^{3/2}) \right) ^{1/2} (ce _{2n+1} ( 0 , q) ce ' _{2n+1} (\pi/ 2 , q))/(A ^{( 2 n +1 )} _{1}) ⋅ e ^{-x/2} \cos \left( q ^{1/2} e ^{x} + (\pi)/(4) \right) \\ Se _{2n+1} (x,q) &\sim - \left( (2)/(\pi q ^{3/2}) \right) ^{1/2} (se ' _{2n+1} ( 0 , q) se _{2n+1} (\pi/ 2 , q))/(B ^{( 2 n +1 )} _{1}) ⋅ e ^{-x/2} \cos \left( q ^{1/2} e ^{x} + (\pi)/(4) \right) \\ Se _{2n+2} (x,q) &\sim \left( (2)/(\pi q ^{5/2}) \right) ^{1/2} (se ' _{2n+2} ( 0 , q) se ' _{2n+2} (\pi/ 2 , q))/(B ^{( 2 n +2 )} _{2}) ⋅ e ^{-x/2} \sin \left( q ^{1/2} e ^{x} + (\pi)/(4) \right) \end{align}

Thus, the modified Mathieu functions decay exponentially for large real argument. Similar asymptotic expansions can be written down for

Fe _{n} and Ge _{n} ; these also decay exponentially for large real argument.

For the even and odd periodic Mathieu functions

ce, se and the associated characteristic numbers a one can also derive asymptotic expansions for large q .^[32] For the characteristic numbers in particular, one has with N approximately an odd integer, i.e. N ≈ N _{0=} 2 n +1 , n =1 , 2 , 3 ,..., \begin{align} a(N) = {}& -2 q + 2 q ^{1/2} N - (1)/(2 ^{3}) (N ^{2+1)} - (1)/(2 ^{7q} ^{1/2}) N(N ^{2} +3 ) - (1)/(2 ^{12} q) ( 5 N ^{4} + 34 N ^{2} +9 ) \\ & - (1)/(2 ^{17} q ^{3/2}) N( 33 N ^{4} +410 N ^{2} +405 ) - (1)/(2 ^{20} q ^{2}) ( 63 N ^{6} + 1260 N ^{4} + 2943 N ^{2} +41807 ) + l{O}(q ^{-5/2} ) \end{align} Observe the symmetry here in replacing q ^{1/2} and N by - q ^{1/2} and - N , which is a significant feature of the expansion. Terms of this expansion have been obtained explicitly up to and including the term of order |q| ^{-7/2} .^[33] Here N is only approximately an odd integer because in the limit of q\to inf ty all minimum segments of the periodic potential \cos 2 x become effectively independent harmonic oscillators (hence N _{0} an odd integer). By decreasing q , tunneling through the barriers becomes possible (in physical language), leading to a splitting of the characteristic numbers a \to a _{\mp} (in quantum mechanics called eigenvalues) corresponding to even and odd periodic Mathieu functions. This splitting is obtained with boundary conditions^[33] (in quantum mechanics this provides the splitting of the eigenvalues into energy bands).^[34] The boundary conditions are:\left( (dce _{N _{0-1}})/(dx) \right) _{\pi/2} =0 , ce _{N _{0}} (\pi/ 2 ) =0 , \left( (dse _{N _{0}})/(dx) \right) _{\pi/2} =0 , se _{N _{0+1}} (\pi/ 2 ) =0 . Imposing these boundary conditions on the asymptotic periodic Mathieu functions associated with the above expansion for a one obtainsN - N _{0} = \mp 2 \left( (2)/(\pi) \right) ^{1/2} (( 16 q ^{1 / 2} ) ^{N _{0/2}} e ^{-4 q ^{1/2}})/(\left[ 1- [ (1)/((N _{0-1)]!} 2)) (3 (N ^{2+1 )} _{0})/(2 ^{6q} ^{1/2}) + (1)/(2 ^{13} q) ( 9 N ^{4} _{0} -40 N ^{3} _{0} +18 N ^{2} _{0} -136 N _{0} + 9 ) + ... \right]. The corresponding characteristic numbers or eigenvalues then follow by expansion, i.e.a(N) = a(N _{0)} + (N - N _{0)} \left( (\partial a)/(\partial N) \right) _{N _{0}} + … . Insertion of the appropriate expressions above yields the result\begin{align} a(N)\to a _{\mp} (N _{0)} = {} & -2 q + 2 q ^{1/2} N _{0} - (1)/(2 ^{3}) (N ^{2} _{0+1 )} - (1)/(2 ^{7q} ^{1/2}) N _{0(N} - ^{2+3 )} _{0} (1)/(2 ^{12} q) ( 5 N ^{2} _{0} +9 ) - … \\ & \mp (( 16 q ^{1 / 2} ) ^{N _{0/2+1}} e ^{-4 q ^{1/2}})/([ 1- ( 8 \pi) ^{1 / 2} [ (1)/(2) (N _{0-1)]!}) (N _{0})/(2 ^{6q} ^{1/2}) ( 3 N ^{2+8 N} _{0+3 )} + … ] . \end{align} For N _{0=1,3,5,...} these are the eigenvalues associated with the even Mathieu eigenfunctions ce _{N _{0}} or ce _{N _{0-1}} (i.e. with upper, minus sign) and odd Mathieu eigenfunctions se _{N _{0+1}} orse _{N _{0}} (i.e. with lower, plus sign). The explicit and normalised expansions of the eigenfunctions can be found in ^[33] or.^[34]

Similar asymptotic expansions can be obtained for the solutions of other periodic differential equations, as for Lamé functions and prolate and oblate spheroidal wave functions.

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

Mathieu functions arise when separation of variables in elliptic coordinates is applied to 1) the Laplace equation in 3 dimensions, and 2) the Helmholtz equation in either 2 or 3 dimensions. Since the Helmholtz equation is a prototypical equation for modeling the spatial variation of classical waves, Mathieu functions can be used to describe a variety of wave phenomena. For instance, in computational electromagnetics they can be used to analyze the scattering of electromagnetic waves off elliptic cylinders, and wave propagation in elliptic waveguides.^[35] In general relativity, an exact plane wave solution to the Einstein field equation can be given in terms of Mathieu functions.

More recently, Mathieu functions have been used to solve a special case of the Smoluchowski equation, describing the steady-state statistics of self-propelled particles.^[36]

The remainder of this section details the analysis for the two-dimensional Helmholtz equation.^[37] In rectangular coordinates, the Helmholtz equation is

\left( (\partial ^{2})/(+ \partial x ^{2}) (\partial ^{2})/(\partial y ^{2}) \right) \psi + k ^{2} \psi = 0 , Elliptic coordinates are defined by\begin{align} x & = c \cosh \mu \cos \nu \\ y & = c \sinh \mu \sin \nu \end{align} where 0 \leq \mu < inf ty , 0 \leq \nu < 2 \pi , and c is a positive constant. The Helmholtz equation in these coordinates is(1)/(\left( c ^{2} (\sinh ^{2} \mu + \sin ^{2} \nu)) (\partial ^{2})/(+ \partial \mu ^{2}) (\partial ^{2})/(\partial \nu ^{2}) \right) \psi + k ^{2} \psi = 0 The constant \mu curves are confocal ellipses with focal length c ; hence, these coordinates are convenient for solving the Helmholtz equation on domains with elliptic boundaries. Separation of variables via \psi(\mu, \nu) = F(\mu) G(\nu) yields the Mathieu equations\begin{align} & (d ^{2F})/(d \mu ^{2}) - \left(a - (c ^{2} k ^{2})/(2) \cosh 2 \mu \right) F = 0 \\ & (d ^{2G})/(d \nu ^{2}) + \left(a - (c ^{2} k ^{2})/(2) \cos 2 \nu \right) G = 0 \\ \end{align} where a is a separation constant.

As a specific physical example, the Helmholtz equation can be interpreted as describing normal modes of an elastic membrane under uniform tension. In this case, the following physical conditions are imposed:^[38]

• Periodicity with respect to

\nu , i.e. \psi(\mu, \nu) = \psi(\mu, \nu + 2 \pi)

• Continuity of displacement across the interfocal line:

\psi( 0 , \nu) = \psi( 0 , - \nu)

• Continuity of derivative across the interfocal line:

\psi _{\mu} ( 0 , \nu) = - \psi _{\mu} ( 0 , - \nu)

For given

k , this restricts the solutions to those of the form Ce _{n} (\mu, q) ce _{n(\nu,} q) and Se _{n} (\mu, q) se _{n(\nu,} q) , where q = c ^{2} k ^{2} / 4 . This is the same as restricting allowable values of a , for given k . Restrictions on k then arise due to imposition of physical conditions on some bounding surface, such as an elliptic boundary defined by \mu = \mu _{0} > 0 . For instance, clamping the membrane at \mu = \mu _{0} imposes \psi(\mu _{0,} \nu) = 0 , which in turn requires\begin{align} Ce _{n} (\mu _{0,} q) = 0 \\ Se _{n} (\mu _{0,} q) = 0 \end{align} These conditions define the normal modes of the system.

Dynamical problems

In dynamical problems with periodically varying forces, the equation of motion sometimes takes the form of Mathieu's equation. In such cases, knowledge of the general properties of Mathieu's equation— particularly with regard to stability of the solutions—can be essential for understanding qualitative features of the physical dynamics.^[39] A classic example along these lines is the inverted pendulum.^[40] Other examples are

• vibrations of a string with periodically varying tension^[39]
• stability of railroad rails as trains drive over them
• seasonally forced population dynamics
• the phenomenon of parametric resonance in forced oscillators
• motion of ions in a quadrupole ion trap^[41]
• the Stark effect for a rotating electric dipole
• the Floquet theory of the stability of limit cycles

Quantum mechanics

Mathieu functions play a role in certain quantum mechanical systems, particularly those with spatially periodic potentials such as the quantum pendulum and crystalline lattices.

The modified Mathieu equation also arises when describing the quantum mechanics of singular potentials. For the particular singular potential

V(r) = g ^{2/r} ^{4} the radial Schrödinger equation(d ^{2y})/(dr ^{2}) + \left[ k ^{2} - (\ell(\ell +1 ))/(- r ^{2}) (g ^{2})/(r ^{4}) \right] y = 0 can be converted into the equation(d ^{2\varphi})/(dz ^{2}) + \left[ 2 h ^{2} \cosh 2 z - \left(\ell + (1)/(2) \right) ^{2\right]} \varphi = 0 . The transformation is achieved with the following substitutionsy = r ^{1/2} \varphi, r = \gamma e ^{z,} \gamma = (ig)/(h) , h ^{2} = ikg, h = e ^{I\pi/4} (kg) ^{1/2} . 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 absorptivity can be obtained.^[42]

See also

• Almost Mathieu operator
• Bessel function
• Hill differential equation
• Inverted pendulum
• Lamé function
• List of mathematical functions
• Monochromatic electromagnetic plane wave

References

• Book: Arscott. Felix. Felix Arscott. Periodic differential equations: an introduction to Mathieu, Lamé, and allied functions. Pergamon Press. 1964. 9781483164885.
  • Book: Malcolm M.. Bibby. Andrew F.. Peterson. 2014. Accurate Computation of Mathieu Functions. Morgan & Claypool. 9781627050852. 10.2200/S00526ED1V01Y201307CEM032. 28354918 .
  • Robert B.. Dingle. Harald J.W.. Müller. The Form of the Coefficients of the Late Terms in Asymptotic Expansions of the Characteristic Numbers of Mathieu and Spheroidal-Wave Functions. . 216. 1964. 123–133. 0075-4102.
• Book: Table of Integrals, Series, and Products . Izrail Solomonovich . Gradshteyn . Izrail Solomonovich Gradshteyn . Iosif Moiseevich . Ryzhik . Iosif Moiseevich Ryzhik . Yuri Veniaminovich . Geronimus . Yuri Veniaminovich Geronimus . Michail Yulyevich . Tseytlin . Michail Yulyevich Tseytlin . Alan . Jeffrey . Daniel . Zwillinger . ((Scripta Technica, Inc.)) . February 2007 . 7 . . 978-0-12-373637-6 . 2360010. 1.
  • Book: Iyanaga . Shōkichi . Kawada . Yukiyosi . Encyclopedic Dictionary of Mathematics, Volume I . 1977 . . 1st . Translated from the 2nd Japanese edition, paperback version of the 1977 edition . 978-0-262-59010-5 . 591028 . 1980 . registration .
• Book: Jin. J.M.. Zhang. Shan Jjie. Computation of special functions. Wiley. New York. 1996 . 9780471119630.
  • • March. Raymond E.. An Introduction to Quadrupole Ion Trap Mass Spectrometry. Journal of Mass Spectrometry. April 1997. 32. 4. 351–369. 10.1002/(SICI)1096-9888(199704)32:4<351::AID-JMS512>3.0.CO;2-Y. 1997JMSp...32..351M . 16506573 . free.
  • Book: McLachlan, N. W. . Theory and application of Mathieu functions. Oxford University Press . 1951. Note: Reprinted lithographically in Great Britain at the University Press, Oxford, 1951 from corrected sheets of the (1947) first edition.
• Book: Mathieusche Funktionen und Sphäroidfunktionen. Meixner. Josef. Schäfke. Friedrich Wilhelm. 1954. Springer-Verlag. Berlin. de. 10.1007/978-3-662-00941-3. 978-3-540-01806-3.
• Book: Methods of Theoretical Physics: Pt. 1. Morse. Philip McCord. Feshbach. Herman. 1953-01-01. McGraw-Hill Inc., US. 9780070433168. Reprint. Boston, Mass. en.
• Book: Müller-Kirsten, Harald J.W.. Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral. World Scientific. 2nd. 2012. 978-981-4397--73-5.
• R.B.. Dingle. H.J.W.. Müller. Asymptotic Expansions of Mathieu Functions and their Characteristic Numbers. Journal für die reine und angewandte Mathematik. 1962. 1962. 211. 11–32. 0075-4102. 10.1515/crll.1962.211.11. 117516747 .
• H.J.W.. Müller. On Asymptotic Expansions of Mathieu Functions. Journal für die reine und angewandte Mathematik. 1962. 1962. 211. 179–190. 0075-4102. 10.1515/crll.1962.211.179. 118909645 .
  • • • Van Buren. Arnie L.. Boisvert. Jeffrey E.. Accurate calculation of the modified Mathieu functions of integer order. Quarterly of Applied Mathematics. 65. 1. 2007. 1–23. 0033-569X. 10.1090/S0033-569X-07-01039-5. free.
• Book: Lew Yan Voon LC, Willatzen M . 2011. Separable Boundary-Value Problems in Physics . Wiley-VCH. 978-3-527-41020-0. 10.1002/9783527634927. (free online access to the appendix on Mathieu functions)
• Book: Wimp. Jet . 1984. Computation with Recurrence Relations . Pitman Publishing. 0-273-08508-5. 83–84.
  • Brimacombe. Chris. Corless. Robert M.. Zamir. Mair. 2021. Computation and Applications of Mathieu Functions: A Historical Perspective. SIAM Review. en. 63. 4. 653–720. 10.1137/20M135786X. 220969117 . 0036-1445. free. 2008.01812.

External links

• • List of equations and identities for Mathieu Functions functions.wolfram.com
  • Timothy Jones, Mathieu's Equations and the Ideal rf-Paul Trap (2006)
• Mathieu equation, EqWorld
• NIST Digital Library of Mathematical Functions: Mathieu Functions and Hill's Equation

Notes and References

1. Mathieu (1868).
2. Morse and Feshbach (1953).
3. Brimacombe, Corless and Zamir (2021)
4. Gutiérrez-Vega (2015).
5. Arscott (1964), chapter III
6. Arscott (1964) 43–44
7. McLachlan (1947), chapter II.
8. Arscott (1964); Iyanaga (1980); Gradshteyn (2007); This is also the normalization used by the computer algebra system Maple.
9. Butikov . Eugene I. . April 2018 . Analytical expressions for stability regions in the Ince–Strutt diagram of Mathieu equation . American Journal of Physics . en . 86 . 4 . 257–267 . 10.1119/1.5021895 . 2018AmJPh..86..257B . 0002-9505. free .
10. It is not true, in general, that a 2 \pi periodic function has the property y(x + \pi) = - y(x) . However, this turns out to be true for functions which are solutions of Mathieu's equation.
11. McLachlan (1951), pp. 141-157, 372
12. McLachlan (1947), p. 144
13. McLachlan (1947), p. 372
14. McLachlan (1947), p. 28
15. Wimp (1984), pp. 83-84
16. McLachlan (1947)
17. Chaos-Cador and Ley-Koo (2001)
18. Temme (2015), p. 234
19. Müller-Kirsten (2012), pp. 420-428
20. Meixner and Schäfke (1954); McLachlan (1947)
21. Malits (2010)
22. Jin and Zhang (1996)
23. Van Buren and Boisvert (2007)
24. Bibby and Peterson (2013)
25. Meixner and Schäfke (1954), p.134
26. McLachlan (1947), pp. 234–235
27. Gradshteyn (2007), p. 953
28. Arscott (1964), pp. 40-41
29. Gradshteyn (2007), pp. 763–765
30. Arscott (1964), p. 86
31. McLachlan (1947), chapter XI
32. McLachlan (1947), p. 237; Dingle and Müller (1962); Müller (1962); Dingle and Müller(1964)
33. Dingle and Müller (1962)
34. Müller-Kirsten (2012)
35. Bibby and Peterson (2013); Barakat (1963); Sebak and Shafai (1991); Kretzschmar (1970)
36. Solon et al (2015)
37. see Willatzen and Voon (2011), pp. 61–65
38. McLachlan (1947), pp. 294–297
39. Meixner and Schäfke (1954), pp. 324–343
40. Ruby (1996)
41. March (1997)
42. Müller-Kirsten (2006)