characteristic equation

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1: 28.34 Methods of Computation

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§28.34(i) Characteristic Exponents

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§28.34(ii) Eigenvalues

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Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

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2: 2.9 Difference Equations

… ►where ${\rho }_1,{\rho }_2$ are the roots of the characteristic equation … ►

§2.9(ii) Coincident Characteristic Values

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3: 28.29 Definitions and Basic Properties

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§28.29(ii) Floquet’s Theorem and the Characteristic Exponent

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28.29.9 $$2\cos \left(\pi \nu \right)=w_{\text{I}}(\pi ,\lambda )+w_{\text{II}}^{\prime }(\pi ,\lambda ).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\nu$: complex parameter, $w_{\text{I}}(z;a,q)$: fundamental solution, $w_{\text{II}}(z;a,q)$: fundamental solution and $\lambda$: parameter

Referenced by:

§28.29(ii), §28.29(ii), §28.29(ii)

Permalink:

http://dlmf.nist.gov/28.29.E9

Encodings:

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See also:

Annotations for §28.29(ii), §28.29 and Ch.28

►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 $△(\lambda )-2\cos \left(\pi \nu \right)=0$ 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 $$\sum _{n=0}^{\mathrm{\infty }}\left(b_{2n+2}\left(q\right)-{(2n+2)}^2\right)=0.$$

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Symbols:

$b_n\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $n$: integer

Permalink:

http://dlmf.nist.gov/28.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.7 and Ch.28

5: 2.7 Differential Equations

… ►where ${\lambda }_1$, ${\lambda }_2$ are the roots of the characteristic equation … ►See §2.11(v) for other examples. … ►The transformed differential equation either has a regular singularity at $t=\mathrm{\infty }$, or its characteristic equation has unequal roots. …

6: 28.2 Definitions and Basic Properties

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28.2.16 $$\cos \left(\pi \nu \right)=w_{\text{I}}(\pi ;a,q)=w_{\text{I}}(\pi ;a,-q).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $q=h^2$: parameter, $\nu$: complex parameter, $a$: parameter and $w_{\text{I}}(z;a,q)$: fundamental solution

A&S Ref:

20.3.10 (in slightly different form)

Referenced by:

§28.12(i), §28.12(i), §28.12(ii), §28.2(iii), §28.2(iv), §28.2(v), item (c), §28.34(i)

Permalink:

http://dlmf.nist.gov/28.2.E16

Encodings:

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See also:

Annotations for §28.2(iii), §28.2 and Ch.28

►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. … ►

Distribution

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Change of Sign of $q$

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7: 28.15 Expansions for Small $q$

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§28.15(i) Eigenvalues ${\lambda }_{\nu }\left(q\right)$

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28.15.2 $$a-{\nu }^2-\frac{q^2}{a-{(\nu +2)}^2-}\frac{q^2}{a-{(\nu +4)}^2-}\mathrm{\cdots }=\frac{q^2}{a-{(\nu -2)}^2-}\frac{q^2}{a-{(\nu -4)}^2-}\mathrm{\cdots }.$$

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Symbols:

$q=h^2$: parameter, $\nu$: complex parameter and $a$: parameter

Referenced by:

item (e)

Permalink:

http://dlmf.nist.gov/28.15.E2

Encodings:

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See also:

Annotations for §28.15(i), §28.15 and Ch.28

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8: 28.16 Asymptotic Expansions for Large $q$

§28.16 Asymptotic Expansions for Large $q$

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9: 28.17 Stability as $x\to \pm \mathrm{\infty }$

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10: 28.12 Definitions and Basic Properties

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§28.12(i) Eigenvalues ${\lambda }_{\nu +2n}\left(q\right)$

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28.12.2 $${\lambda }_{\nu }\left(-q\right)={\lambda }_{\nu }\left(q\right)={\lambda }_{-\nu }\left(q\right).$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.12(i), §28.12 and Ch.28

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