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Mathieu Differential Equation

(d^2V)/(dv^2)+[a-2qcos(2v)]V=0
(1)

(Abramowitz and Stegun 1972; Zwillinger 1997, p. 125), having solution

y=C_1C(a,q,v)+C_2S(a,q,v),
(2)

where C(a,q,v) and S(a,q,v) are Mathieu functions. The equation arises in separation of variables of the Helmholtz differential equation in elliptic cylindrical coordinates. Whittaker and Watson (1990) use a slightly different form to define the Mathieu functions.

The modified Mathieu differential equation

(d^2U)/(du^2)-[a-2qcosh(2u)]U=0
(3)

(Iyanaga and Kawada 1980, p. 847; Zwillinger 1997, p. 125) arises in separation of variables of the Helmholtz differential equation in elliptic cylindrical coordinates, and has solutions

y=C_1C(a,q,-iu)+C_2S(a,q,-iu).
(4)

The associated Mathieu differential equation is given by

y^('')+[(1-2r)cotx]y^'+(a+k^2cos^2x)y=0
(5)

(Ince 1956, p. 403; Zwillinger 1997, p. 125).

See also

Hill's Differential Equation, Mathieu Function, Whittaker-Hill Differential Equation

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 722, 1972.Campbell, R. Théorie générale de l'équation de Mathieu et de quelques autres équations différentielles de la mécanique. Paris: Masson, 1955.Ince, E. L. Ordinary Differential Equations. New York: Dover, 1956.Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 847, 1980.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 556-557, 1953.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 125, 1997.

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Mathieu Differential Equation

Cite this as:

Weisstein, Eric W. "Mathieu Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MathieuDifferentialEquation.html

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