These functions appear in physical problems involving elliptical shapes or periodic potentials, and were first introduced by Mathieu (1868) when analyzing the motion of elliptical membranes. Unfortunately, the analytic determination of Mathieu functions "presents great difficulties" (Whittaker 1914, Frenkel and Portugal 2001), and they are difficult to employ, "mainly because of the impossibility of analytically representing them in a simple and handy way" (Sips 1949, Frenkel and Portugal 2001).
The Mathieu functions have the special values
(2)
(3)
For nonzero ,
the Mathieu functions are only periodic in for certain values of . Such characteristic values are given by the Wolfram
Language functions MathieuCharacteristicA[r,
q] and MathieuCharacteristicB[r,
q] with
an integer or rational number. These values are often denoted and . In general, both and are multivalued functions with very complicated branch cut
structures. Unfortunately, there is no general agreement on how to define the branch
cuts. As a result, the Wolfram Language's
implementation simply picks a convenient sheet.
For integer ,
the even and odd Mathieu functions with characteristic values and are often denoted and , known as the elliptic
cosine and elliptic sine functions, respectively
(Abramowitz and Stegun 1972, p. 725; Frenkel and Portugal 2001). The left plot
above shows
for ,
1, ..., 4 and the right plot shows for , ..., 4.
Whittaker and Watson (1990, p. 405) define the Mathieu function based on the equation
Abramowitz, M. and Stegun, I. A. (Eds.). "Mathieu Functions." Ch. 20 in Handbook
of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 721-746, 1972.Blanch, G. "Asymptotic
Expansion for the Odd Periodic Mathieu Functions." Trans. Amer. Math. Soc.97,
357-366, 1960.Dingle, R. B. and Müller, H. J. W.
"Asymptotic Expansions of Mathieu Functions and Their Characteristic Numbers."
J. reine angew. Math.211, 11-32, 1962.Frenkel, D. and
Portugal, R. "Algebraic Methods to Compute Mathieu Functions." J. Phys.
A: Math. Gen.34, 3541-3551, 2001.Gradshteyn, I. S.
and Ryzhik, I. M. "Mathieu Functions." §6.9 and 8.6 in Tables
of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press,
pp. 800-804 and 1006-1013, 2000.Humbert, P. Fonctions de Lamé
et Fonctions de Mathieu. Paris: Gauthier-Villars, 1926.Mathieu,
É. "Mémoire sur le mouvement vibratoire d'une membrane de forme
elliptique." J. math. pure appl.13, 137-203, 1868.McLachlan,
N. W. Theory
and Applications of Mathieu Functions. New York: Dover, 1964.Mechel,
F. P. Mathieu
Functions: Formulas, Generation, Use. Stuttgart, Germany: Hirzel, 1997.Meixner,
J. and Schäfke, F. W. Mathieusche Funktionen und Sphäroidfunktionen.
Berlin: Springer-Verlag, 1954.Morse, P. M. and Feshbach, H. Methods
of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 562-568 and
633-642, 1953.Rubin, H. "Anecdote on Power Series Expansions of
Mathieu Functions." J. Math. Phys.43, 339-341, 1964.Sips,
R. "Représentation asymptotique des fonctions de Mathieu et des fonctions
d'onde sphéroidales." Trans. Amer. Math. Soc.66, 93-134,
1949.Whittaker, E. T. "On the General Solution of Mathieu's
Equation." Proc. Edinburgh Math. Soc.32, 75-80, 1914.Whittaker,
E. T. and Watson, G. N. A
Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University
Press, 1990.
![(d^2V)/(dv^2)+[a-2qcos(2v)]V=0.](/images/equations/MathieuFunction/NumberedEquation1.svg)
and odd solutions by
. These are returned by the Wolfram Language functions MathieuC[a,
q, z] and MathieuS[a,
q, z], respectively. Similarly, their derivatives are implemented as
MathieuCPrime[a,
q, z] and MathieuSPrime[a,
q, z].
,
the Mathieu functions are only periodic in 
for certain values of 
. Such characteristic values are given by the Wolfram
Language functions MathieuCharacteristicA[r,
q] and MathieuCharacteristicB[r,
q] with 
an integer or rational number. These values are often denoted 
and 
. In general, both 
and 
are multivalued functions with very complicated branch cut
structures. Unfortunately, there is no general agreement on how to define the branch
cuts. As a result, the Wolfram Language's
implementation simply picks a convenient sheet.
,
the even and odd Mathieu functions with characteristic values 
and 
are often denoted 
and 
, known as the elliptic
cosine and elliptic sine functions, respectively
(Abramowitz and Stegun 1972, p. 725; Frenkel and Portugal 2001). The left plot
above shows 
for 
,
1, ..., 4 and the right plot shows 
for 
, ..., 4.
![(d^2u)/(dz^2)+[a+16qcos(2z)]u=0.](/images/equations/MathieuFunction/NumberedEquation2.svg)
.
For an odd Mathieu function,
transforms the Mathieu differential equation
to
