# characteristic equation

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##### 1: 28.34 Methods of Computation
###### §28.34(ii) Eigenvalues
• (f)

Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

• ##### 2: 2.9 Difference Equations
where $\rho_{1},\rho_{2}$ are the roots of the characteristic equation
##### 3: 28.29 Definitions and Basic Properties
###### §28.29(ii) Floquet’s Theorem and the Characteristic Exponent
28.29.9 $2\cos\left(\pi\nu\right)=w_{\mbox{\tiny I}}(\pi,\lambda)+w_{\mbox{\tiny II}}^{% \prime}(\pi,\lambda).$
This is the characteristic equation of (28.29.1), and $\cos\left(\pi\nu\right)$ is an entire function of $\lambda$. … For a given $\nu$, the characteristic equation $\bigtriangleup(\lambda)-2\cos\left(\pi\nu\right)=0$ has infinitely many roots $\lambda$. …
##### 4: 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). …
28.7.4 $\sum_{n=0}^{\infty}\left(b_{2n+2}\left(q\right)-(2n+2)^{2}\right)=0.$
##### 5: 2.7 Differential Equations
where $\lambda_{1}$, $\lambda_{2}$ are the roots of the characteristic equationSee §2.11(v) for other examples. … The transformed differential equation either has a regular singularity at $t=\infty$, or its characteristic equation has unequal roots. …
##### 6: 28.2 Definitions and Basic Properties
This is the characteristic equation of Mathieu’s equation (28.2.1). …If $\widehat{\nu}=0$ or $1$, or equivalently, $\nu=n$, then $\nu$ is a double root of the characteristic equation, otherwise it is a simple root. …
##### 7: 28.15 Expansions for Small $q$
###### §28.15(i) Eigenvalues $\lambda_{\nu}\left(q\right)$
28.15.2 $a-\nu^{2}-\cfrac{q^{2}}{a-(\nu+2)^{2}-\cfrac{q^{2}}{a-(\nu+4)^{2}-\cdots}}=% \cfrac{q^{2}}{a-(\nu-2)^{2}-\cfrac{q^{2}}{a-(\nu-4)^{2}-\cdots}}.$
##### 10: 28.12 Definitions and Basic Properties
###### §28.12(i) Eigenvalues $\lambda_{\nu+2n}\left(q\right)$
28.12.2 $\lambda_{\nu}\left(-q\right)=\lambda_{\nu}\left(q\right)=\lambda_{-\nu}\left(q% \right).$