Mathieu function
In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation where formula_2 and formula_3 are parameters. 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 boundary value problems possessing elliptic symmetry. Definition. Mathieu functions. In some usages, "Mathieu function" refers to solutions of the Mathieu differential equation for arbitrary values  of formula_2 and formula_3. When no confusion can arise, other authors use the term to refer specifically to formula_6- or formula_7-periodic solutions, which exist only for special values of formula_2 and formula_3. More precisely, for given (real) formula_3 such periodic solutions exist for an infinite number of values of formula_2, called "characteristic numbers", conventionally indexed as two separate sequences formula_12 and formula_13, for formula_14. The corresponding functions are denoted formula_15 and formula_16, 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 formula_3  is real, both the characteristic numbers and associated functions are real-valued. formula_15 and formula_16 can be further classified by parity and periodicity (both with respect to formula_20), as follows: The indexing with the integer formula_21, besides serving to arrange the characteristic numbers in ascending order, is convenient in that formula_15 and formula_16 become proportional to formula_24 and formula_25 as formula_26. With formula_21 being an integer, this gives rise to the classification of formula_28 and formula_29 as Mathieu functions (of the first kind) of integral order. For general formula_2 and formula_3, 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" which can be related to the original Mathieu equation by taking formula_33. Accordingly, the modified Mathieu functions of the first kind of integral order, denoted by formula_34 and formula_35, are defined from These functions are real-valued when formula_20 is real. Normalization. A common normalization, which will be adopted throughout this article, is to demand as well as require formula_39 and formula_40 as formula_26. 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 formula_42 with those values of formula_2 which result in formula_44. Note, however, that the theorem only guarantees the existence of at least one solution satisfying formula_45, when Mathieu's equation in fact has two independent solutions for any given formula_2, formula_3. Indeed, it turns out that with formula_2 equal to one of the characteristic numbers, Mathieu's equation has only one  periodic solution (that is, with period formula_6 or formula_50), and this solution is one of the formula_15, formula_16. The other solution is nonperiodic, denoted formula_53 and formula_54, 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 where formula_56 is a complex number, the "Floquet exponent" (or sometimes "Mathieu exponent"), and formula_57 is a complex valued function periodic in formula_20 with period formula_6. An example formula_60 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 formula_2 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 formula_15 and formula_16, called a Mathieu function of the first kind of integral order. The nonperiodic one is denoted either formula_53 and formula_54, respectively, and is called a Mathieu function of the second kind (of integral order). The nonperiodic solutions  are unstable, that is, they diverge as formula_66. The second solutions corresponding to the modified Mathieu functions formula_34 and formula_35 are naturally defined as formula_69 and formula_70. Fractional order. Mathieu functions of fractional order can be defined as those solutions formula_71 and formula_72, formula_73 a non-integer, which turn into formula_74 and formula_75 as formula_26. If formula_73 is irrational, they are non-periodic; however, they remain bounded as formula_78. An important property of the solutions formula_71 and formula_72, for formula_73 non-integer, is that they exist for the same value of formula_2. In contrast, when formula_73 is an integer, formula_71  and formula_72 never occur for the same value of formula_2. (See Ince's Theorem above.) These classifications are summarized in the table below. The modified Mathieu function counterparts are defined similarly. Explicit representation and computation. First kind. Mathieu functions of the first kind can be represented as Fourier series: The expansion coefficients formula_88 and formula_89 are functions of formula_3 but independent of formula_20. By substitution into the Mathieu equation, they can be shown to obey three-term recurrence relations in the lower index. For instance, for each formula_92 one finds Being a second-order recurrence in the index formula_94, one  can always find two independent solutions formula_95 and formula_96 such that the general solution can be expressed as a linear combination of the two: formula_97. Moreover, in this particular case, an asymptotic analysis shows that one possible choice of fundamental solutions has the property In particular, formula_95 is finite whereas formula_96 diverges. Writing formula_97, we therefore see that in order for the Fourier series representation of formula_92 to converge, formula_2 must be chosen such that formula_104. These choices of formula_2 correspond to the characteristic numbers. In general, however, the solution of a three-term recurrence with variable coefficients cannot  be represented in a simple manner, and hence there is no simple way to determine formula_2 from the condition formula_104. Moreover, even if the approximate value of a characteristic number is known, it cannot be used to obtain the coefficients formula_108 by numerically iterating the recurrence towards increasing formula_109. The reason is that as long as formula_2 only approximates a characteristic number, formula_111 is not identically formula_112 and the divergent solution formula_96 eventually dominates for large enough formula_109. 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. For small values of formula_3 and low order formula_21, they can also be expressed perturbatively as power series of formula_3, 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: where formula_119, and formula_120 and formula_121 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. For large formula_21 and formula_3, 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. This can be seen by transformation  of Mathieu's equation to algebraic form, using the change of variable formula_124: Since this equation has an irregular singular point at infinity, it cannot be transformed into an equation of the hypergeometric type. Qualitative behavior. For small formula_3, formula_127 and formula_128 behave similarly to formula_24 and formula_25. For arbitrary formula_3, they may deviate significantly from their trigonometric counterparts; however, they remain periodic in general. Moreover, for any real formula_3, formula_133 and formula_134 have exactly formula_135 simple zeros in formula_136, and as formula_137 the zeros cluster about formula_138. For formula_139 and as formula_78 the modified Mathieu functions tend to  behave as damped periodic functions. In the following, the formula_141 and formula_142 factors from the Fourier expansions for formula_127 and formula_128 may be referenced (see Explicit representation and computation). They depend on formula_3 and formula_21 but are independent of formula_20. Reflections and translations. Due to their parity and periodicity, formula_28 and formula_29 have simple properties under reflections and translations by multiples of formula_6: One can also write functions with negative formula_3 in terms of those with positive formula_3: Moreover, Orthogonality and completeness. Like their trigonometric counterparts formula_24 and formula_25, the periodic Mathieu functions formula_15 and formula_16 satisfy orthogonality relations Moreover , with formula_3 fixed and formula_2 treated as the eigenvalue, the Mathieu equation is of Sturm-Liouville form. This implies that the eigenfunctions formula_15 and formula_16 form a complete set, i.e. any formula_6- or formula_50-periodic function of formula_20 can be expanded as a series in formula_15 and formula_16. Integral identities. Solutions of Mathieu's equation satisfy a class of integral identities with respect to kernels formula_170 that are solutions of More precisely, if formula_172 solves Mathieu's equation with given formula_2 and formula_3, then the integral where formula_176 is a path in the complex plane, also solves Mathieu's equation  with the same formula_2 and formula_3, provided the following conditions are met: Using an appropriate change of variables, the equation for formula_185 can be transformed into the wave equation and solved. For instance, one solution is formula_186. 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: valid for any complex formula_20 and real formula_3. Asymptotic expansions. The following asymptotic expansions hold for formula_139, formula_192, formula_193, and formula_194: Thus, the modified  Mathieu functions decay exponentially for large real argument. Similar asymptotic expansions can be written down for formula_196 and formula_197; these also decay exponentially for large real argument. For the even and odd periodic Mathieu functions formula_198 and the associated characteristic numbers formula_2 one can also derive asymptotic expansions for large formula_3. For the characteristic numbers in particular, one has with formula_201 approximately an odd integer, i.e. formula_202 Observe the symmetry here in replacing formula_205 and formula_201 by formula_207 and formula_208, which is a significant feature of the expansion. Terms of this expansion have been obtained explicitly up to and including  the term of order formula_209. Here formula_201 is only approximately an odd integer because in the limit of formula_211 all minimum segments of the periodic potential formula_212 become effectively independent harmonic oscillators (hence formula_213 an odd integer). By decreasing formula_3, tunneling through the barriers becomes possible (in physical language), leading to a splitting of the characteristic numbers formula_215 (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: Imposing these boundary conditions on  the asymptotic periodic Mathieu functions associated with the above expansion for formula_2 one obtains The corresponding characteristic numbers or eigenvalues then follow by expansion, i.e. Insertion of the appropriate expressions above yields the result For formula_222 these are the eigenvalues associated with the even Mathieu eigenfunctions formula_223 or formula_224 (i.e. with upper, minus sign) and odd Mathieu eigenfunctions formula_225 or formula_226 (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. 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 where formula_229, formula_230, and formula_231 is a positive constant. The Helmholtz equation in these coordinates is The constant formula_56 curves are confocal ellipses with focal length formula_231; hence, these coordinates are convenient for solving the Helmholtz equation on domains with elliptic boundaries. Separation of variables via formula_235 yields the Mathieu equations where formula_2 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: For given formula_242, this restricts the solutions to those of the form formula_243 and formula_244, where formula_245. This is the same as restricting allowable values of formula_2, for given formula_242. Restrictions on formula_242 then arise due to imposition of physical conditions on some bounding surface, such as an elliptic boundary defined by formula_249. For instance, clamping the membrane at formula_250 imposes formula_251, 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 modified Mathieu equation also arises when describing the quantum mechanics of singular potentials. For the particular singular potential formula_253 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.