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as well as require <math>\text{ce}_n(x, q) \rightarrow +\cos nx</math> and <math>\text{se}_n(x, q) \rightarrow +\sin nx</math> as <math>q \rightarrow 0</math>.
== Stability ==
The Mathieu equation has two parameters. For almost all choices of parameter, by Floquet theory (see next section), any solution either converges to zero or diverges to infinity.
Parametrize Mathieu equation as <math>\ddot x + k(1-m \cos(t))x = 0</math>, where <math>k \in \R, m \geq 0</math>. The regions of stability and instability are separated by curves <ref>{{Cite journal |last=Butikov |first=Eugene I. |date=2018-04 |title=Analytical expressions for stability regions in the Ince–Strutt diagram of Mathieu equation |url=https://pubs.aip.org/aapt/ajp/article/86/4/257-267/1057663 |journal=American Journal of Physics |language=en |volume=86 |issue=4 |pages=257–267 |doi=10.1119/1.5021895 |issn=0002-9505}}</ref>
<math>\begin{aligned}
& m(k)=2 \sqrt{\frac{k(k-1)(k-4)}{3 k-8}} \quad \text { for } \quad k<0 ; \\
& m(k)=\frac{1}{4}(\sqrt{(9-4 k)(13-20 k)}-(9-4 k)), \quad \text { for } \quad k<\frac{1}{4} ; \\
& m(k)=\frac{1}{4}(9-4 k \mp \sqrt{(9-4 k)(13-20 k)}), \quad \text { for } \quad \frac{1}{4}<k<\frac{13}{20} ; \\
& m(k)=\sqrt{\frac{2(k-1)(k-4)(k-9)}{k-5}}, \quad \text { for } \quad \frac{13}{20}<k<1 ; \\
& m(k)=2 \sqrt{\frac{k(k-1)(k-4)}{3 k-8}}, \quad \text { for } \quad k>1 .
\end{aligned}</math>
== Floquet theory ==
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