Mathieu Differential Equation

$${d^2V\over dv^2}+[b-2q\cos(2v)]V=0.$$

It arises in separation of variables of Laplace's Equation in Elliptic Cylindrical Coordinates. Whittaker and Watson (1990) use a slightly different form to define the Mathieu Functions.

The modified Mathieu differential equation

$${d^2U\over du^2}-[b-2q\cosh(2u)]U=0$$

arises in Separation of Variables of the Helmholtz Differential Equation in Elliptic Cylindrical Coordinates.

See also Mathieu Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 722, 1972.

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.