Mathieu Function

The form given by Whittaker and Watson (1990, p. 405) defines the Mathieu function based on the equation

$${d^2u\over dz^2}+[a+16q\cos(2z)]u=0.$$ (1)

This equation is closely related to Hill's Differential Equation. For an Even Mathieu function,

$$G(\eta)=\lambda \int_{-\pi}^\pi e^{k\cos\eta\cos\theta} G(\theta)\,d\theta,$$ (2)

where $k\equiv \sqrt{32q}$. For an Odd Mathieu function,

$$G(\eta)=\lambda \int_{-\pi}^\pi \sin(k\sin\eta\sin\theta)G(\theta)\,d\theta.$$ (3)

Both Even and Odd functions satisfy

$$G(\eta)=\lambda\int_{-\pi}^\pi e^{ik\sin\eta\sin\theta}G(\theta)\,d\theta.$$ (4)

Letting $\zeta\equiv \cos^2 z$ transforms the Mathieu Differential Equation to

$$4\zeta(1-\zeta){d^2u\over d\zeta^2}+2(1-2\zeta){du\over d\zeta}+(a-16q+32q\zeta)u=0.$$ (5)

See also Mathieu Differential Equation

References

Abramowitz, M. and Stegun, C. 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.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 562-568 and 633-642, 1953.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.