Mathieu function: Difference between revisions

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Revision as of 23:23, 3 October 2005

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The Mathieu functions are used in treating [[parametric resonance]], and were introduced by [[Emile Mathieu]].

==Floquet solution==

According to [[Floquet's theorem]], for fixed values of a,q, Mathieu's equation admits a ''complex valued'' solution of form

:<math>F(a,q,x) = \exp(i \mu) \, P(a,q,x)</math>

where <math>\mu</math> is a complex number, the ''Mathieu exponent'', and P is a complex valued function which is ''periodic'' with period <math>2 \pi</math>. However, P is in general ''not'' sinusoidal.

[[Image:MatheiuFloquet.gif|frame|right|A Floquet solution F(1,0.2, x) to Mathieu's equation (real part, red; imaginary part, green).]]

==Mathieu sine and cosine==

[[Image:MatheiuCosine.gif|frame|right|Mathieu cosine MCC(1,0.2,x), in red, and cos(x) for comparison, in green.]]

For fixed a,q, the '''Mathieu cosine''' <math>MCC(a,q,\xi)</math> is a function of <math>\xi</math> defined as the unique solution of the Mathieu equation which

#takes the value <math>MCC(a,q,0)=1</math>.,

#is an [[even function]], or equivalently <math>C^\prime(a,q,0)=0</math>.

Similarly, the '''Mathieu sine''' <math>MSS(a,q,\xi)</math> is the unique solution which

#takes the value <math>MSS(a,q,0)=0</math>.,

#is an [[odd function]], or equivalently <math>S^\prime(a,q,0)=0</math>.

These are closely related to the Floquet solution:

For fixed a,q, the '''Mathieu cosine''' <math>MC(a,q,\xi)</math> is a function of <math>\xi</math> defined as the unique solution of the Mathieu equation which

:<math> C(a,q,x) = \frac{\exp(i \mu)}{F(a,q,0} \, \frac{P(a,q,x) + P(a,q,-x)}{2}</math>

#is an [[even function]],

:<math> S(a,q,x) = \frac{\exp(i \mu)}{F(a,q,0} \, \frac{P(a,q,x) - P(a,q,-x)}{2}</math>

#takes the value <math>MC(a,q,0)=1</math>.

Similarly, the '''Mathieu sine''' <math>MS(a,q,\xi)</math> is the unique solution which

#is an [[odd function]],

#takes the value <math>MS(a,q,0)=0</math>.

==Periodic solutions==