Mathieu function: Difference between revisions

Revision as of 00:15, 4 October 2005

Revision as of 00:18, 4 October 2005

Line 1:

In [[mathematics]], the '''Mathieu functions''' are solutions to thecertain [[Mathieuspecial differential equationfunctions]], whichuseful for treating a variety of interesting problems in applied mathematics, isincluding

*problems [[parametric resonance]],

* vibrating elliptical drumheads,

:<math> \frac{d^2y}{dx^2}+[a-2q\cos (2x) ]y=0 </math>

*gravitational waves in [[general relativity]].

The Mathieu functions are used in treating problems involving [[parametric resonance]], vibrating elliptical drumheads, and gravitational waves in [[general relativity]], among other applications. They were introduced by [[Emile Mathieu]] in [[1868]] in the context of the second problem.

==Definition==

Mathieu functions are, by definition, solutions to the [[Mathieu equation]]

:<math> \frac{d^2y}{dx^2}+[a-2q\cos (2x) ]y=0 </math>

The substitution <math>x \rightarrow \arccos(x)</math> transforms Matheiu's equation to the ''rational form''

:<math> y^{\prime \prime} = \frac{x}{1-x^2} \, y^\prime + \frac{( 4x^2-2) \, q -a}{1-x^2} \, y</math>

This has two regular singularities at x = -1,1 and one irregularity singularity at infinity, whic impleis that in general (unlike many other [[special functions]]), the solutions of Mathieu's equation ''cannot'' be expressed in terms of [[hypergeometric function]]s.

=Floquet solution==