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Mathieu Functions

Mathieu functions appear frequently in physical problems involving elliptical shapes or periodic potentials. These functions were first introduced by Mathieu (1868) when analyzing the solutions to the equation $y'' + (a - 2 q \cos (2 z)) y = 0$ , which arises from the separation of the 2-D or 3-D wave equation modeling the motion of an elliptic membrane. The rational form of Mathieu's equation has two regular singularities and one irregular singularity; hence, Mathieu functions are perhaps the simplest class of special functions (Heun type), which are not essentially hypergeometric. The Maple implementation of Mathieu functions includes: MathieuC and MathieuS , representing the solution to Mathieu's equation; MathieuCE and MathieuSE representing the periodic cases; MathieuFloquet representing the Floquet type solutions, and the set of auxiliary functions MathieuA , MathieuB , and MathieuExponent relating the parameters (a,q) entering Mathieu's equation.

$restart$

The MathieuC and MathieuS Functions

The Mathieu functions MathieuC(a,q,z) and MathieuS(a,q,z) are linearly independent solutions of Mathieu's equation

$\frac{{&DifferentialD;}^{2}}{&DifferentialD; z^{2}} y (z) + (a - 2 q \cos (2 z)) y (z)$

where a and q are arbitrary parameters. MathieuC and MathieuS are even and odd functions of z. The derivatives of MathieuC and MathieuS with respect to z are MathieuCPrime and MathieuSPrime, respectively. From this definition and the differential equation, the differentiation rules for MathieuCPrime and MathieuSPrimeare:

$FunctionAdvisor (differentiation_rule, MathieuCPrime)$

$\frac{&DifferentialD;}{&DifferentialD; z} MathieuCPrime (a, b, z) = (2 b \cos (2 z) - a) MathieuC (a, b, z)$

(1.1)

$FunctionAdvisor (differentiation_rule, MathieuSPrime)$

$\frac{&DifferentialD;}{&DifferentialD; z} MathieuSPrime (a, b, z) = (2 b \cos (2 z) - a) MathieuS (a, b, z)$

(1.2)

Special values for MathieuC and MathieuS:

$FunctionAdvisor (special_values, MathieuC)$

$[MathieuC (a, 0, z) = \cos (\sqrt{a} z), MathieuC (a, b, 0) = 1]$

(1.3)

$FunctionAdvisor (special_values, MathieuS)$

$[MathieuS (a, 0, z) = \frac{\sin (\sqrt{a} z)}{\sqrt{a}}, MathieuS (a, b, 0) = 0]$

(1.4)

For $q \neq 0$ and arbitrary values of the first parameter a, the Mathieu functions are not periodic in z. There exist, however, infinitely many particular values of a, commonly known as characteristic values, such that the Mathieu functions have period $k π$ , where $k$ is a positive integer, corresponding to periodic solutions $y (z) = y (k π + z)$ to Mathieu's equation.

For general complex $q \neq 0$ , when a is a characteristic value the Mathieu functions are entire functions of z and the only linearly independent periodic solutions to Mathieu's equation. When the period is $π$ or $2 π$ , then only one periodic solution exists, otherwise both even and odd periodic solutions exist. In summary, given some (a,q), three different situations can happen:

1. Neither MathieuC(a,q,z) nor MathieuS(a,q,z) is periodic;

2. Mathieu's equation admits a solution with period $π$ or $2 π$ and only one of MathieuC(a,q,z) or MathieuS(a,q,z) is periodic - the other one is not;

3. Mathieu's equation admits solutions with period $2 n π$ , where n >= 2 is a positive integer, and both MathieuC(a,q,z) and MathieuS(a,q,z) are periodic.

The MathieuFloquet Function and the Normalization for MathieuC and MathieuS

According to Floquet's theorem, in all three cases Mathieu's equation admits a solution of the form $F (a, q, z) = {&ExponentialE;}^{I ν (a, q) z} P (z)$ , called a Floquet solution, where $ν (a, q)$ is an arbitrary complex number, commonly known as a characteristic exponent, and $P (z)$ has the same periodicity as the coefficients of Mathieu's equation, namely $π$ . From the invariance of Mathieu's equation under $z \to - z$ it follows that $F (a, q, - z)$ is also a solution, which is linearly independent of $F (a, q, z)$ if and only if $ν (a, q)$ is not an integer.

The cases 1, 2, and 3 above respectively correspond to the cases $ν (a, q)$ not a rational number, $ν (a, q)$ an integer, and $ν (a, q)$ a proper rational number, respectively. In case 2, when $ν (a, q)$ is an integer, a is called a characteristic value. Then $F (a, q, z)$ and $F (a, q, - z)$ are proportional and a second (non-periodic) independent solution is constructed as in 20.3.6 and 20.3.7 of "Handbook of Mathematical Functions." For further details, see also Chap. IV of "Theory and Applications of Mathieu Functions" (see references below). Moreover, the period is $π$ if $ν (a, q)$ is even, and $2 π$ otherwise.

The Maple functions MathieuFloquet(a,q,z) and MathieuFloquetPrime(a,q,z) represent the Floquet solution $F (a, q, z)$ and its derivative with respect to z, respectively. In Maple, MathieuFloquet is normalized so that the $L_{2}$ norm of P(z) is equal to $2 π$ , and MathieuC and MathieuS are defined in terms of MathieuFloquet, as follows.

$MathieuC (a, q, z) = \frac{MathieuFloquet (a, q, z) + MathieuFloquet (a, q, - z)}{2 MathieuFloquet (a, q, 0)} &semi;$ $MathieuS (a, q, z) = \frac{MathieuFloquet (a, q, z) - MathieuFloquet (a, q, - z)}{2 MathieuFloquetPrime (a, q, 0)}$

$MathieuC (a, q, z) = \frac{MathieuFloquet (a, q, z) + MathieuFloquet (a, q, - z)}{2 MathieuFloquet (a, q, 0)}$

$MathieuS (a, q, z) = \frac{MathieuFloquet (a, q, z) - MathieuFloquet (a, q, - z)}{2 MathieuFloquetPrime (a, q, 0)}$

(2.1)

This yields the following normalization for MathieuC and MathieuS.

$MathieuC (a, q, 0) = MathieuC (a, q, 0) &semi; MathieuCPrime (a, q, 0) = MathieuCPrime (a, q, 0) &semi; MathieuS (a, q, 0) = MathieuS (a, q, 0) &semi; MathieuSPrime (a, q, 0) = MathieuSPrime (a, q, 0)$

$MathieuC (a, q, 0) = 1$

$MathieuCPrime (a, q, 0) = 0$

$MathieuS (a, q, 0) = 0$

$MathieuSPrime (a, q, 0) = 1$

(2.2)

Note: To obtain the result shown in (2.2), after typing $MathieuC (a, q, 0) = MathieuC (a, q, 0)$ , Maple must be told not to evaluate the left hand side of the equation. This was accomplished by highlighting the first $MathieuC$ and selecting 2-D Math>Convert To>Atomic Variable from the context menu. This was done for all four equations above. This method is used throughout the worksheet. An alternative way to accomplish this is to place unevaluation quotes (' ') around each name.

You have the following special cases when q = 0.

$MathieuFloquet (a, 0, z) = MathieuFloquet (a, 0, z) &semi;$ $MathieuFloquetPrime (a, 0, z) = MathieuFloquetPrime (a, 0, z)$

$MathieuFloquet (a, 0, z) = {&ExponentialE;}^{I \sqrt{a} z}$

$MathieuFloquetPrime (a, 0, z) = I \sqrt{a} {&ExponentialE;}^{I \sqrt{a} z}$

(2.3)

The Auxiliary Functions MathieuA, MathieuB, and MathieuExponent

Given (a,q), the characteristic exponent $ν$ entering Floquet solutions is computed using MathieuExponent(a,q). Conversely, given( $ν$ ,q), the values of a entering Floquet solutions and so entering MathieuC(a,q,z) and MathieuS(a,q,z) - regardless of whether $ν$ is rational and so regardless of whether the functions are periodic - are respectively computed using MathieuA(nu,q) and MathieuB(nu,q).Therefore, the values returned by MathieuA and MathieuB are characteristic values of the corresponding Mathieu function only when $ν$ is a rational number and MathieuA and MathieuB return a different number only when $ν$ is an integer (Case 2 in the itemization of section entitled, The MathieuC and MathieuS Functions).

The Mathieu Characteristic functions can be viewed as inverses of each other in that they satisfy: nu = MathieuExponent(MathieuA(nu,q),q) and nu = MathieuExponent(MathieuB(nu,q),q).

The Periodic Solutions MathieuCE and MathieuSE

For a non-negative integer $ν$ , the even periodic solution corresponding to the $ν$ th characteristic value is given by MathieuCE(nu,q,z). Similarly, for positiveinteger $ν$ , the odd periodic solution corresponding to the $ν$ th characteristic value is MathieuSE(nu,q,z). Their first derivatives with respect to z are MathieuCEPrime(nu,q,z) and MathieuSEPrime(nu,q,z), respectively. These even and odd periodic Mathieu functions are normalized to have $L_{2}$ norm equal to $π$ if $0 < ν$ , and MathieuCE(0,q,z) is normalized to have $L_{2}$ norm equal to $2 π$ . The following equalities hold.

$MathieuCE (ν, q, z) = MathieuCE (ν, q, 0) MathieuC (MathieuA (ν, q), q, z) &semi;$ $MathieuSE (ν, q, z) = MathieuSEPrime (ν, q, 0) MathieuS (MathieuB (ν, q), q, z)$

$MathieuCE (ν, q, z) = MathieuCE (ν, q, 0) MathieuC (MathieuA (ν, q), q, z)$

$MathieuSE (ν, q, z) = MathieuSEPrime (ν, q, 0) MathieuS (MathieuB (ν, q), q, z)$

(4.1)

The even and odd periodic Mathieu functions assume the following special values for q = 0:

$MathieuCE (ν, 0, z) = MathieuCE (ν, 0, z) &semi;$ $MathieuCEPrime (ν, 0, z) = MathieuCEPrime (ν, 0, z) &semi;$ $MathieuSE (ν, 0, z) = MathieuSE (ν, 0, z) &semi;$ $MathieuSEPrime (ν, 0, z) = MathieuSEPrime (ν, 0, z)$

$MathieuCE (ν, 0, z) = \cos (ν z)$

$MathieuCEPrime (ν, 0, z) = - ν \sin (ν z)$

$MathieuSE (ν, 0, z) = \sin (ν z)$

$MathieuSEPrime (ν, 0, z) = ν \cos (ν z)$

(4.2)

Maple recognizes the symmetry and periodicity of MathieuCE and MathieuSE and their derivatives.

$MathieuCE (ν, q, - z) = MathieuCE (ν, q, - z) &semi;$ $MathieuCEPrime (ν, q, - z) = MathieuCEPrime (ν, q, - z) &semi;$ $MathieuSE (ν, q, - z) = MathieuSE (ν, q, - z) &semi;$ $MathieuSEPrime (ν, q, - z) = MathieuSEPrime (ν, q, - z)$

$MathieuCE (ν, q, - z) = MathieuCE (ν, q, - z)$

$MathieuCEPrime (ν, q, - z) = - MathieuCEPrime (ν, q, z)$

$MathieuSE (ν, q, - z) = - MathieuSE (ν, q, z)$

$MathieuSEPrime (ν, q, - z) = MathieuSEPrime (ν, q, z)$

(4.3)

$MathieuCE (ν, q, z + 2 π) = MathieuCE (ν, q, z + 2 π) &semi; MathieuSE (ν, q, z + 2 π) = MathieuSE (ν, q, z + 2 π) &semi; MathieuCE (ν, q, z + π) assuming ν ∷ even &semi; MathieuSE (ν, q, z + π) assuming ν ∷ even$

$MathieuCE (ν, q, z + 2 π) = MathieuCE (ν, q, z)$

$MathieuSE (ν, q, z + 2 π) = MathieuSE (ν, q, z)$

$MathieuCE (ν, q, z)$

$MathieuSE (ν, q, z)$

(4.4)

A Rational Form for Mathieu's Equation and the Relation with Spheroidal Wave Functions

A rational form of Mathieu's equation, showing the connection between Mathieu and spheroidal wave functions, is obtained by changing variables $z \to \arccos (z)$ .

$\frac{&DifferentialD;}{&DifferentialD; z} (\frac{&DifferentialD;}{&DifferentialD; z} y (z)) = \frac{z}{1 - z^{2}} (\frac{&DifferentialD;}{&DifferentialD; z} y (z)) + \frac{((4 z^{2} - 2) q - a)}{1 - z^{2}} \cdot y (z)$

$\frac{{&DifferentialD;}^{2}}{&DifferentialD; z^{2}} y (z) = \frac{z (\frac{&DifferentialD;}{&DifferentialD; z} y (z))}{- z^{2} + 1} + \frac{((4 z^{2} - 2) q - a) y (z)}{- z^{2} + 1}$

(5.1)

Comparing the above with the ODE satisfied by the spheroidal wave functions

$\frac{&DifferentialD;}{&DifferentialD; z} (\frac{&DifferentialD;}{&DifferentialD; z} y (z)) = \frac{2 (b + 1) z}{1 - z^{2}} (\frac{&DifferentialD;}{&DifferentialD; z} y (z)) + \frac{(4 q z^{2} - c)}{1 - z^{2}} \cdot y (z)$

$\frac{{&DifferentialD;}^{2}}{&DifferentialD; z^{2}} y (z) = \frac{(2 b + 2) z (\frac{&DifferentialD;}{&DifferentialD; z} y (z))}{- z^{2} + 1} + \frac{(4 q z^{2} - c) y (z)}{- z^{2} + 1}$

(5.2)

you see that Mathieu's equation is the special case of the above when $b = - \frac{1}{2}, c = a + 2 q$ . This rational form of Mathieu's equation has two regular singular points at $\{- 1, 1\}$ and one irregular singular point at $\infty$ . Therefore, Mathieu functions admit only hypergeometric representation for some particular values of the parameters (a,q).

Series Expansions

Series expansions with respect to the argument are Taylor expansions.

$series (MathieuC (a, q, z), z)$

$1 + (q - \frac{a}{2}) z^{2} + (\frac{1}{24} a^{2} - \frac{1}{6} a q + \frac{1}{6} q^{2} - \frac{1}{3} q) z^{4} + O (z^{6})$

(6.1)

$series (MathieuS (a, q, z), z = 1, 3)$

$MathieuS (a, q, 1) + MathieuSPrime (a, q, 1) (z - 1) + (q \cos (2) - \frac{a}{2}) MathieuS (a, q, 1) {(z - 1)}^{2} + O ({(z - 1)}^{3})$

(6.2)

$MathieuC (a, q, z) = \frac{MathieuFloquet (a, q, z) + MathieuFloquet (a, q, - z)}{2 MathieuFloquet (a, q, 0)}$

(6.3)

$map (series,, z, 3)$

$1 + (q - \frac{a}{2}) z^{2} + O (z^{4}) = 1 + (q - \frac{a}{2}) z^{2} + O (z^{3})$

(6.4)

$series (MathieuCE (a, q, z), z, 3)$

$MathieuCE (a, q, 0) + (q - \frac{MathieuA (a, q)}{2}) MathieuCE (a, q, 0) z^{2} + O (z^{3})$

(6.5)

$series (MathieuSEPrime (a, q, z), z, 3)$

$MathieuSEPrime (a, q, 0) + (q - \frac{MathieuB (a, q)}{2}) MathieuSEPrime (a, q, 0) z^{2} + O (z^{3})$

(6.6)

Series expansions in the parameter q about 0 and asymptotic expansions are implemented for MathieuCE, MathieuSE, their derivatives, and the characteristic value functions. The series expansions about 0 require the first argument to be an integer, whereas for asymptotic expansions, the first parameter can be symbolic as well.

$series (MathieuCE (1, q, z), q, 3)$

$\cos (z) - \frac{1}{8} \cos (3 z) q + (- \frac{\cos (z)}{128} - \frac{\cos (3 z)}{64} + \frac{\cos (5 z)}{192}) q^{2} + O (q^{3})$

(6.7)

$series (MathieuSEPrime (2, q, z), q, 3)$

$2 \cos (2 z) - \frac{1}{3} \cos (4 z) q + (- \frac{\cos (2 z)}{144} + \frac{\cos (6 z)}{64}) q^{2} + O (q^{3})$

(6.8)

$series (MathieuA (3, q), q)$

$9 + \frac{1}{16} q^{2} + \frac{1}{64} q^{3} + \frac{13}{20480} q^{4} - \frac{5}{16384} q^{5} + O (q^{6})$

(6.9)

$asympt (MathieuSE (1, q, z), q, 1)$

$\frac{\sin (z) {&ExponentialE;}^{- \sqrt{q} {\cos (z)}^{2}} HermiteH (0, q^{\frac{1}{4}} \cos (z) \sqrt{2}) π^{\frac{1}{4}} 2^{\frac{3}{4}}}{2 {(\frac{1}{q})}^{\frac{1}{8}}} + π^{\frac{1}{4}} \sin (z) (\frac{HermiteH (0, q^{\frac{1}{4}} \cos (z) \sqrt{2}) 2^{\frac{3}{4}}}{32} + \frac{2^{\frac{3}{4}} HermiteH (2, q^{\frac{1}{4}} \cos (z) \sqrt{2})}{64} - \frac{HermiteH (4, q^{\frac{1}{4}} \cos (z) \sqrt{2}) 2^{\frac{3}{4}}}{512}) {&ExponentialE;}^{- \sqrt{q} {\cos (z)}^{2}} {(\frac{1}{q})}^{\frac{3}{8}} + π^{\frac{1}{4}} \sin (z) (\frac{27 HermiteH (0, q^{\frac{1}{4}} \cos (z) \sqrt{2}) 2^{\frac{3}{4}}}{2048} + \frac{5 2^{\frac{3}{4}} HermiteH (2, q^{\frac{1}{4}} \cos (z) \sqrt{2})}{512} - \frac{9 HermiteH (4, q^{\frac{1}{4}} \cos (z) \sqrt{2}) 2^{\frac{3}{4}}}{8192} - \frac{2^{\frac{3}{4}} HermiteH (6, q^{\frac{1}{4}} \cos (z) \sqrt{2})}{16384} + \frac{HermiteH (8, q^{\frac{1}{4}} \cos (z) \sqrt{2}) 2^{\frac{3}{4}}}{262144}) {&ExponentialE;}^{- \sqrt{q} {\cos (z)}^{2}} {(\frac{1}{q})}^{\frac{7}{8}} + O (\frac{1}{q})$

(6.10)

$asympt (MathieuB (n, q), q, 3)$

$- 2 q + \frac{4 n - 2}{\sqrt{\frac{1}{q}}} - \frac{{(n - 1)}^{2}}{2} - \frac{n}{2} + \frac{1}{4} + (\frac{3}{32} n^{2} - \frac{3}{32} n + \frac{1}{32} - \frac{1}{16} n^{3}) \sqrt{\frac{1}{q}} + \frac{\frac{11}{256} n - \frac{3}{256} - \frac{5}{256} n^{4} + \frac{5}{128} n^{3} - \frac{1}{16} n^{2}}{q} + (- \frac{225}{8192} n + \frac{53}{8192} - \frac{33}{4096} n^{5} + \frac{165}{8192} n^{4} - \frac{185}{4096} n^{3} + \frac{195}{4096} n^{2}) {(\frac{1}{q})}^{\frac{3}{2}} + \frac{\frac{1413}{65536} n - \frac{297}{65536} + \frac{189}{16384} n^{5} - \frac{63}{16384} n^{6} - \frac{2205}{65536} n^{4} + \frac{1575}{32768} n^{3} - \frac{1431}{32768} n^{2}}{q^{2}} + (- \frac{20649}{1048576} n + \frac{3961}{1048576} - \frac{527}{262144} n^{7} - \frac{6671}{262144} n^{5} + \frac{3689}{524288} n^{6} + \frac{48265}{1048576} n^{4} - \frac{7613}{131072} n^{3} + \frac{11695}{262144} n^{2}) {(\frac{1}{q})}^{\frac{5}{2}} + O (\frac{1}{q^{3}})$

(6.11)

Plots

The following plots are from "Handbook of Mathematical Functions" (see references below). Note that the Maple normalization of MathieuCE(0,q,z) differs from that in the referenced book by a factor of $\sqrt{2}$ .

$plotsetup (inline) &semi;$ $plot ([MathieuA (5 - n, q) &dollar; (n = 0 .. 5)], q = 0 .. 15, legend = [' a || (5 - n)' &dollar; (n = 0 .. 5)])$

$plotsetup (inline) &semi;$ $plot ([MathieuB (6 - n, q) &dollar; (n = 1 .. 5)], q = 0 .. 15, legend = [' b || (6 - n)' &dollar; (n = 1 .. 5)])$

$plotsetup (inline) &semi;$ $plot ([MathieuCE (n, 1, z) &dollar; (n = 0 .. 5)], z = 0 .. \frac{π}{2}, title = Even periodic Mathieu functions, q=1, legend = [' ce || n' &dollar; (n = 0 .. 5)])$

$plotsetup (inline) &semi;$ $plot ([MathieuSE (n, 1, z) &dollar; (n = 1 .. 5)], z = 0 .. \frac{π}{2}, title = Odd periodic Mathieu functions, q=1, legend = [' se || n' &dollar; (n = 1 .. 5)])$

$plotsetup (inline) &semi;$ $plot ([MathieuCE (n, 10, z) &dollar; (n = 0 .. 5)], z = 0 .. \frac{π}{2}, title = Even periodic Mathieu functions, q=10, legend = [' ce || n' &dollar; (n = 0 .. 5)])$

$plotsetup (inline) &semi;$ $plot ([MathieuSE (n, 10, z) &dollar; (n = 1 .. 5)], z = 0 .. \frac{π}{2}, title = Odd periodic Mathieu functions, q=10, legend = [' se || n' &dollar; (n = 1 .. 5)])$

References

Abramowitz, M., and Stegun, I.A. Handbook of Mathematical Functions. Dover, 1965.

Frenkel, D., and Portugal, R. "Algebraic methods to compute Mathieu functions." Journal of Physics A: Mathematical and General, Vol. 34,(2001): 3541-3551.

McLachlan, N.W. Theory and Applications of Mathieu Functions. Oxford, 1951.

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