Mathieu function: Difference between revisions

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Many properties of the Mathieu differential equation can be deduced from the general theory of ordinary differential equations with periodic coefficients, called [[Floquet theory]]. The central result is ''Floquet's theorem'':

{{boxmath theorem|widthname=600px|padding=10px|border size=1px|'''Floquet's theorem<ref>Arscott (1964), p. 29.</ref> —'''| math_statement = Mathieu's equation always has at least one solution <math>y(x)</math> such that <math>y(x + \pi) = \sigma y(x)</math>, where <math>\sigma</math> is a constant which depends on the parameters of the equation and may be real or complex.}}

It is natural to associate the characteristic numbers <math>a(q)</math> with those values of <math>a</math> which result in <math>\sigma = \pm 1</math>.<ref>It is not true, in general, that a <math>2 \pi</math> periodic function has the property <math>y(x + \pi) = -y(x)</math>. However, this turns out to be true for functions which are solutions of Mathieu's equation.</ref> Note, however, that the theorem only guarantees the existence of at least one solution satisfying <math>y(x + \pi) = \sigma y(x)</math>, when Mathieu's equation in fact has two independent solutions for any given <math>a</math>, <math>q</math>. Indeed, it turns out that with <math>a</math> equal to one of the characteristic numbers, Mathieu's equation has only one periodic solution (that is, with period <math>\pi</math> or <math>2 \pi</math>), and this solution is one of the <math>\text{ce}_n(x, q)</math>, <math>\text{se}_n(x, q)</math>. The other solution is nonperiodic, denoted <math>\text{fe}_n(x, q)</math> and <math>\text{ge}_n(x, q)</math>, respectively, and referred to as a '''Mathieu function of the second kind'''.<ref>McLachlan (1951), pp. 141-157, 372</ref> This result can be formally stated as ''Ince's theorem'':

{{boxmath theorem|width name =600px|padding=10px|border size=1px|'''Ince's theorem<ref>Arscott (1964), p. 34</ref> —'''| math_statement = Define a ''basically periodic'' function as one satisfying <math>y(x + \pi) = \pm y(x)</math>. Then, except in the trivial case <math>q = 0</math>, Mathieu's equation never possesses two (independent) basically periodic solutions for the same values of <math>a</math> and <math>q</math>.}}

[[Image:MathieuFloquet.gif|thumb|250px|An example <math>P(a, q, x)</math> from Floquet's theorem, with <math>a = 1</math>, <math>q = 1/5</math>, <math>\mu \approx 1 + 0.0995 i</math> (real part, red; imaginary part, green)]]