Mathieu function: Difference between revisions

Revision as of 00:36, 4 October 2005

Revision as of 00:37, 4 October 2005

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:<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\frac{d^2y}{\prime \primedx^2} = \frac{x}{1-x^2} \, y^\primefrac{d y}{dx} + \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.