Mathematical function
In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation
formula_1
where a, q are real-valued parameters. Since we may add π/2 to x to change the sign of q, it is a usual convention to set "q" ≥ 0.
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" 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_4- or formula_5-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_10 and formula_11, for formula_12. The corresponding functions are denoted formula_13 and formula_14, 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_13 and formula_14 can be further classified by parity and periodicity (both with respect to formula_18), as follows:
The indexing with the integer formula_19, besides serving to arrange the characteristic numbers in ascending order, is convenient in that formula_13 and formula_14 become proportional to formula_22 and formula_23 as formula_24. With formula_19 being an integer, this gives rise to the classification of formula_26 and formula_27 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"
formula_30
which can be related to the original Mathieu equation by taking formula_31. Accordingly, the modified Mathieu functions of the first kind of integral order, denoted by formula_32 and formula_33, are defined from
formula_34
These functions are real-valued when formula_18 is real.
Normalization.
A common normalization, which will be adopted throughout this article, is to demand
formula_36
as well as require formula_37 and formula_38 as formula_24.
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":
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Floquet's theorem — Mathieu's equation always has at least one solution formula_40 such that formula_41, where formula_42 is a constant which depends on the parameters of the equation and may be real or complex.
It is natural to associate the characteristic numbers formula_43 with those values of formula_2 which result in formula_45. Note, however, that the theorem only guarantees the existence of at least one solution satisfying formula_41, 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_4 or formula_51), and this solution is one of the formula_13, formula_14. The other solution is nonperiodic, denoted formula_54 and formula_55, respectively, and referred to as a Mathieu function of the second kind. This result can be formally stated as "Ince's theorem":
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Ince's theorem — Define a "basically periodic" function as one satisfying formula_56. Then, except in the trivial case formula_57, Mathieu's equation never possesses two (independent) basically periodic solutions for the same values of formula_2 and formula_3.
An equivalent statement of Floquet's theorem is that Mathieu's equation admits a complex-valued solution of form
 formula_60
where formula_61 is a complex number, the "Floquet exponent" (or sometimes "Mathieu exponent"), and formula_62 is a complex valued function periodic in formula_18 with period formula_4. An example formula_65 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 formula_66, where formula_67, then the regions of stability and instability are separated by the following curves:
formula_68
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_13 and formula_14, called a Mathieu function of the first kind of integral order. The nonperiodic one is denoted either formula_54 and formula_55, 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_74.
The second solutions corresponding to the modified Mathieu functions formula_32 and formula_33 are naturally defined as formula_77 and formula_78.
Fractional order.
Mathieu functions of fractional order can be defined as those solutions formula_79 and formula_80, formula_81 a non-integer, which turn into formula_82 and formula_83 as formula_24. If formula_81 is irrational, they are non-periodic; however, they remain bounded as formula_86.
An important property of the solutions formula_79 and formula_80, for formula_81 non-integer, is that they exist for the same value of formula_2. In contrast, when formula_81 is an integer, formula_79 and formula_80 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:
formula_95
The expansion coefficients formula_96 and formula_97 are functions of formula_3 but independent of formula_18. 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_100 one finds
formula_101
Being a second-order recurrence in the index formula_102, one can always find two independent solutions formula_103 and formula_104 such that the general solution can be expressed as a linear combination of the two: formula_105. Moreover, in this particular case, an asymptotic analysis shows that one possible choice of fundamental solutions has the property
formula_106
In particular, formula_103 is finite whereas formula_104 diverges. Writing formula_105, we therefore see that in order for the Fourier series representation of formula_100 to converge, formula_2 must be chosen such that formula_112 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_115. Moreover, even if the approximate value of a characteristic number is known, it cannot be used to obtain the coefficients formula_116 by numerically iterating the recurrence towards increasing formula_117. The reason is that as long as formula_2 only approximates a characteristic number, formula_119 is not identically formula_120 and the divergent solution formula_104 eventually dominates for large enough formula_117.
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_19, 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:
formula_126
where formula_127, and formula_128 and formula_129 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_19 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_132:
formula_133
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_135 and formula_136 behave similarly to formula_22 and formula_23. 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_141 and formula_142 have exactly formula_143 simple zeros in formula_144, and as formula_145 the zeros cluster about formula_146.
For formula_147 and as formula_86 the modified Mathieu functions tend to behave as damped periodic functions.
In the following, the formula_149 and formula_150 factors from the Fourier expansions for formula_135 and formula_136 may be referenced (see Explicit representation and computation). They depend on formula_3 and formula_19 but are independent of formula_18.
Reflections and translations.
Due to their parity and periodicity, formula_26 and formula_27 have simple properties under reflections and translations by multiples of formula_4:
formula_159
One can also write functions with negative formula_3 in terms of those with positive formula_3:
formula_162
Moreover,
formula_163
Orthogonality and completeness.
Like their trigonometric counterparts formula_22 and formula_23, the periodic Mathieu functions formula_13 and formula_14 satisfy orthogonality relations
formula_168
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_13 and formula_14 form a complete set, i.e. any formula_4- or formula_51-periodic function of formula_18 can be expanded as a series in formula_13 and formula_14.
Integral identities.
Solutions of Mathieu's equation satisfy a class of integral identities with respect to kernels formula_178 that are solutions of
formula_179
More precisely, if formula_180 solves Mathieu's equation with given formula_2 and formula_3, then the integral
formula_183
where formula_184 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_193 can be transformed into the wave equation and solved. For instance, one solution is formula_194. Examples of identities obtained in this way are
formula_195
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:
formula_196
valid for any complex formula_18 and real formula_3.
Asymptotic expansions.
The following asymptotic expansions hold for formula_147, formula_200, formula_201, and formula_202:
formula_203
Thus, the modified Mathieu functions decay exponentially for large real argument. Similar asymptotic expansions can be written down for formula_204 and formula_205; these also decay exponentially for large real argument.
For the even and odd periodic Mathieu functions formula_206 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_209 approximately an odd integer, i.e. formula_210
formula_211
Observe the symmetry here in replacing formula_212 and formula_209 by formula_214 and formula_215, 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_216. Here formula_209 is only approximately an odd integer because in the limit of formula_218 all minimum segments of the periodic potential formula_219 become effectively independent harmonic oscillators (hence formula_220 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_222 (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:
formula_223
Imposing these boundary conditions on the asymptotic periodic Mathieu functions associated with the above expansion for formula_2 one obtains
formula_225
The corresponding characteristic numbers or eigenvalues then follow by expansion, i.e.
formula_226
Insertion of the appropriate expressions above yields the result
formula_227
For formula_228 these are the eigenvalues associated with the even Mathieu eigenfunctions formula_229 or formula_230 (i.e. with upper, minus sign) and odd Mathieu eigenfunctions formula_231 or
formula_232 (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,