Mathieu function: Difference between revisions

Revision as of 14:44, 19 November 2020

Revision as of 13:37, 12 December 2020

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For <math>q > 0</math> and as <math>x \rightarrow \infty</math> the modified Mathieu functions tend to behave as damped periodic functions.

In the following, the <math>A</math> and <math>B</math> factors from the Fourier expansions for <math>\text{ce}_{n}</math> and <math>\text{se}_{n}</math> may be referenced (see [[Mathieu function#Explicit representation and computation|Explicit representation and computation]]). They depend on <math>q</math> and <math>n</math> but are independent of <math>x</math>.

=== Reflections and translations ===

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</math>

For <math>N_0=1,3,5,...</math> these are the eigenvalues associated with the even Mathieu eigenfunctions <math> ce_{N_0} </math> or <math>ce_{N_0-1}</math> (i.e. with upper, minus sign) and odd Mathieu eigenfunctions <math>se_{N_0+1}</math> or

<math>se_{N_0}</math> (i.e. with lower, plus sign). The explicit and normalised expansions of the eigenfunctions can be found in <ref>Dingle and Müller (1962)</ref> or .<ref>Müller-Kirsten (2012)</ref>.

Similar asymptotic expansions can be obtained for the solutions of other periodic differential equations, as for [[Lamé function]]s and prolate and oblate [[spheroidal wave function]]s.

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*seasonally forced [[population dynamics]]

*the phenomenon of [[parametric oscillator#Parametric resonance|parametric resonance]] in forced [[oscillator]]s

*motion of ions in a [[Quadrupole ion trap|quadrupole ion trap]]<ref>March (1997)</ref>

*the [[Stark effect]] for a rotating [[electric dipole]]

* the [[Floquet theory]] of the stability of [[limit cycles]]