Mathieu function: Difference between revisions

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==Periodic solutions==

For countably many special values of a, called ''characteristic values'', the Mathieu equation admits solutions which are periodic with period <math>2\pi</math>. The characteristic values of the MathiueMathieu cosine, sine functions respectively are written <math>a_n(q), \, b_n(q)</math>, where n in '''N'''. HereThe areperiodic thespecial firstcases fewof periodicthe Mathieu cosine and sine functions: are often written <math>CE(n,q,x), \, SE(n,q,x)</math> respectively. A very special case worthy of mention is

:<math> C(a,0,x) = CE(0,0,x) = \cos(\sqrt(a) x), \; S(a,0,x) = SE(0,0,x) = \frac{\sin(\sqrt(a) x)}{\sqrt{a}}</math>

Here are the first few periodic Mathieu cosine functions:

[[Image:MathieuCE.gif|center]]

Here, <math>CE(1,0,x)</math> (red) is sinusoidal, but none of the others are. For example, <math>CE(1,1,x)</math> (green) is too flat near the origin to be sinusoidal.