Mathieu Differential Equation
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
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25-6-2019
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Mathieu Differential Equation
![(d^2V)/(dv^2)+[a-2qcos(2v)]V=0](http://mathworld.wolfram.com/images/equations/MathieuDifferentialEquation/NumberedEquation1.gif) |
(1)
|
(Abramowitz and Stegun 1972; Zwillinger 1997, p. 125), having solution
 |
(2)
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where 
and 
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](http://mathworld.wolfram.com/images/equations/MathieuDifferentialEquation/NumberedEquation3.gif) |
(3)
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(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
 |
(4)
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The associated Mathieu differential equation is given by
 |
(5)
|
(Ince 1956, p. 403; Zwillinger 1997, p. 125).
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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