# Mathieu equation

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##### 1: 28.2 Definitions and Basic Properties
###### §28.2(iii) Floquet’s Theorem and the Characteristic Exponents
This is the characteristic equation of Mathieu’s equation (28.2.1). …
##### 2: 28.20 Definitions and Basic Properties
###### §28.20(i) Modified Mathieu’s Equation
When $z$ is replaced by $\pm\mathrm{i}z$, (28.2.1) becomes the modified Mathieu’s equation:
28.20.2 ${(\zeta^{2}-1)w^{\prime\prime}+\zeta w^{\prime}+\left(4q\zeta^{2}-2q-a\right)w% =0},$ $\zeta=\cosh z$.
28.20.6 $\operatorname{Fe}_{n}\left(z,q\right)=\mp\mathrm{i}\operatorname{fe}_{n}\left(% \pm\mathrm{i}z,q\right),$ $n=0,1,\dots$,
##### 4: 28.6 Expansions for Small $q$
###### §28.6(i) Eigenvalues
28.6.14 $\rselection{a_{m}\left(q\right)\\ b_{m}\left(q\right)}=m^{2}+\frac{1}{2(m^{2}-1)}q^{2}+\frac{5m^{2}+7}{32(m^{2}-% 1)^{3}(m^{2}-4)}q^{4}+\frac{9m^{4}+58m^{2}+29}{64(m^{2}-1)^{5}(m^{2}-4)(m^{2}-% 9)}q^{6}+\cdots.$
28.6.15 $a_{m}\left(q\right)-b_{m}\left(q\right)=\frac{2q^{m}}{\left(2^{m-1}(m-1)!% \right)^{2}}\left(1+O\left(q^{2}\right)\right).$
Higher coefficients in the foregoing series can be found by equating coefficients in the following continued-fraction equations: …
##### 5: 28.7 Analytic Continuation of Eigenvalues
###### §28.7 Analytic Continuation of Eigenvalues
The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). …
28.7.1 $\sum_{n=0}^{\infty}\left(a_{2n}\left(q\right)-(2n)^{2}\right)=0,$
28.7.4 $\sum_{n=0}^{\infty}\left(b_{2n+2}\left(q\right)-(2n+2)^{2}\right)=0.$
##### 8: 28.12 Definitions and Basic Properties
###### §28.12(i) Eigenvalues $\lambda_{\nu+2n}\left(q\right)$
For given $\nu$ (or $\cos\left(\nu\pi\right)$) and $q$, equation (28.2.16) determines an infinite discrete set of values of $a$, denoted by $\lambda_{\nu+2n}\left(q\right)$, $n=0,\pm 1,\pm 2,\dots$. … …
28.12.3 $\lambda_{m}\left(q\right)=\begin{cases}a_{m}\left(q\right),&m=0,1,\dots,\\ b_{-m}\left(q\right),&m=-1,-2,\dots.\end{cases}$
They have the following pseudoperiodic and orthogonality properties: …
##### 9: 28.16 Asymptotic Expansions for Large $q$
###### §28.16 Asymptotic Expansions for Large $q$
28.16.1 $\lambda_{\nu}\left(h^{2}\right)\sim-2h^{2}+2sh-\dfrac{1}{8}(s^{2}+1)-\dfrac{1}% {2^{7}h}(s^{3}+3s)-\dfrac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}+9)-\dfrac{1}{2^{17}h^% {3}}(33s^{5}+410s^{3}+405s)-\dfrac{1}{2^{20}h^{4}}(63s^{6}+1260s^{4}+2943s^{2}% +486)-\dfrac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+41607s)+\cdots.$