Floquet solutions

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1: 28.34 Methods of Computation

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§28.34(iii) Floquet Solutions

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2: 28.29 Definitions and Basic Properties

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

… ►A solution satisfying (28.29.7) is called a Floquet solution with respect to $\nu$ (or Floquet solution). … ►

28.29.10 $$F_{\nu }(z)={\mathrm{e}}^{\mathrm{i}\nu z}{P}_{\nu }(z),$$

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Defines:

$F_{\nu }(z)$: Floquet solution (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.29.E10

Encodings:

pMML, png, TeX

See also:

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

… ► … ►A nontrivial solution $w(z)$ is either a Floquet solution with respect to $\nu$, or $w(z+\pi )-{\mathrm{e}}^{\mathrm{i}\nu \pi }w(z)$ is a Floquet solution with respect to $-\nu$. …

3: 28.2 Definitions and Basic Properties

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§28.2(iv) Floquet Solutions

►A solution with the pseudoperiodic property (28.2.14) is called a Floquet solution with respect to $\nu$. … ► ►The Fourier series of a Floquet solution …leads to a Floquet solution. …

4: 28.12 Definitions and Basic Properties

… ►In consequence, for the Floquet solutions $w(z)$ the factor ${\mathrm{e}}^{\pi \mathrm{i}\nu }$ in (28.2.14) is no longer $\pm 1$. … ►The Floquet solution with respect to $\nu$ is denoted by ${\mathrm{me}}_{\nu }(z,q)$. …The other eigenfunction is ${\mathrm{me}}_{\nu }(-z,q)$, a Floquet solution with respect to $-\nu$ with $a={\lambda }_{\nu }\left(q\right)$. …

5: 28.8 Asymptotic Expansions for Large $q$

… ►Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions ${\mathrm{me}}_{\nu }(z,q)$ (§28.12(ii)) and modified Mathieu functions ${\mathrm{M}}_{\nu }^{(j)}(z,h)$ (§28.20(iii)). …

6: Bibliography D

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Unification of one-dimensional Fokker-Planck equations beyond hypergeometrics: Factorizer solution method and eigenvalue schemes. Phys. Rev. E (3) 57 (1), pp. 252–275.

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External Links:

ISSN 1063-651X, Document, MathReview (Victor I. Stepanov), ZentralBlatt

Referenced by:

§31.17(ii).

Encodings:

BibTeX

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Computing spectra of linear operators using the Floquet-Fourier-Hill method. J. Comput. Phys. 219 (1), pp. 296–321.

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External Links:

ISSN 0021-9991, MathReview, ZentralBlatt

Referenced by:

§22.11.

Encodings:

BibTeX

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Pole dynamics for elliptic solutions of the Korteweg-de Vries equation. Math. Phys. Anal. Geom. 3 (1), pp. 49–74.

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External Links:

ISSN 1385-0172, Document, MathReview (Giulio Soliani), ZentralBlatt

Referenced by:

§23.21(ii).

Encodings:

BibTeX

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Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one. Methods Appl. Anal. 3 (1), pp. 109–134.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

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Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions. Stud. Appl. Math. 107 (3), pp. 293–323.

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External Links:

ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§10.20(ii), §2.8(i).

Encodings:

BibTeX

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