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Wikipedia / Mathieu function

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

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. 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.

Definition

Mathieu functions

In some usages, Mathieu function., called characteristic numbers, respectively. They are sometimes also referred to as cosine-elliptic and sine-elliptic, or Mathieu functions of the first kind.

is real, both the characteristic numbers and associated functions are real-valued.

can be further classified by parity), as follows:

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being an integer, 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

, are defined from

is real.

Normalization

A common normalization, which will be adopted throughout this article, is to demand

.

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:

., respectively, and referred to as a Mathieu function of the second kind. 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

is a complex number, the Floquet exponent (or sometimes Mathieu exponent 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.

, then the regions of stability and instability are separated by the following curves:

Other types of Mathieu functions

Second kind

Since Mathieu's equation is a second order differential equation, one can construct two linearly independent.

.

Fractional order

..

. (See Ince's Theorem above.)

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

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|+Classification of Mathieu functions |}

Explicit representation and computation

First kind

Mathieu functions of the first kind can be represented as Fourier series:

. By substitution into the Mathieu equation, they can be shown to obey three-term recurrence relations one finds

such that the general solution can be expressed as a linear combination. Moreover, in this particular case, an asymptotic analysis shows that one possible choice of fundamental solutions has the property

correspond to the characteristic numbers.

.

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

In practice, Mathieu functions and the corresponding characteristic numbers can be calculated using pre-packaged software, such as Mathematica, Maple, MATLAB, and SciPy, they can also be expressed perturbatively as power series, which can be useful in physical applications.

Second kind

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

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., the form of the series must be chosen carefully to avoid subtraction errors.

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:

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

Qualitative behavior

simple zeros.

the modified Mathieu functions tend to behave as damped periodic functions.

may be referenced (see Explicit representation and computation.

Reflections and translations

:

:

Moreover,

Orthogonality and completeness

satisfy orthogonality relations

treated as the eigenvalue, the Mathieu equation is of Sturm–Liouville.

Integral identities

Solutions of Mathieu's equation satisfy a class of integral identities with respect to kernels that are solutions of

, then the integral

is a path in the complex plane, provided the following conditions are met:

can be transformed into the wave equation. Examples of identities obtained in this way are

Identities of the latter type are useful for studying asymptotic properties of the modified Mathieu functions.

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

.

Asymptotic expansions

:

; these also decay exponentially for large real argument.

.

. (in quantum mechanics called eigenvalues) corresponding to even and odd periodic Mathieu functions. This splitting is obtained with boundary conditions (in quantum mechanics this provides the splitting of the eigenvalues into energy bands). The boundary conditions are:

one obtains

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

Insertion of the appropriate expressions above yields the result

(i.e. with lower, plus sign). The explicit and normalised expansions of the eigenfunctions can be found in or.

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.

The asymptotic nature of the asymptotic solutions of the Mathieu equation has been investigated in detail by investigating the large order behaviour of the expansions.

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. 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.

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

Elliptic coordinates are defined by

is a positive constant. The Helmholtz equation in these coordinates is

curves are confocal ellipses with focal length yields the Mathieu equations

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:

, which in turn requires

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. A classic example along these lines is the inverted pendulum. Other examples are

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 radial Schrödinger equation

can be converted into the equation

The transformation is achieved with the following substitutions

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.

Originally the Schrödinger equation with cosine function was solved in 1928 by Strutt.

See also

Notes

References

  • Note: Reprinted lithographically in Great Britain at the University Press, Oxford, 1951 from corrected sheets of the (1947) first edition.

  • (free online access to the appendix on Mathieu functions)

  • Ho-Chul Shin, "Numerical Review of Mathieu Function Programs for Integer Orders and Real Parameters", SIAM Review, Vol.67, No.4 (2025)