basic solutions of Mathieu equation

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1: 28.2 Definitions and Basic Properties

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§28.2(ii) Basic Solutions $w_{\text{I}}$, $w_{\text{II}}$

… ►(28.2.1) possesses a fundamental pair of solutions $w_{\text{I}}(z;a,q),w_{\text{II}}(z;a,q)$ called basic solutions with … ►

§28.2(vi) Eigenfunctions

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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

►The basic solutions $w_{\text{I}}(z,\lambda )$, $w_{\text{II}}(z,\lambda )$ are defined in the same way as in §28.2(ii) (compare (28.2.5), (28.2.6)). … ► … ► … ► …

3: 28.4 Fourier Series

§28.4 Fourier Series

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§28.4(ii) Recurrence Relations

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§28.4(iii) Normalization

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§28.4(v) Change of Sign of $q$

… ►For the basic solutions $w_{\text{I}}$ and $w_{\text{II}}$ see §28.2(ii).

4: 28.20 Definitions and Basic Properties

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§28.20(i) Modified Mathieu’s Equation

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28.20.1 $$w^{\prime \prime }-\left(a-2q\cosh \left(2z\right)\right)w=0,$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

A&S Ref:

20.1.2 (in slightly different form) 20.8.6

Referenced by:

§28.20(iii), §28.32(i), 1st item, 2nd item, item (b), §28.8(iv), §28.8(iv)

Permalink:

http://dlmf.nist.gov/28.20.E1

Encodings:

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See also:

Annotations for §28.20(i), §28.20 and Ch.28

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§28.20(ii) Solutions ${\mathrm{Ce}}_{\nu }$, ${\mathrm{Se}}_{\nu }$, ${\mathrm{Me}}_{\nu }$, ${\mathrm{Fe}}_n$, ${\mathrm{Ge}}_n$

… ►It follows that (28.20.1) has independent and unique solutions ${\mathrm{M}}_{\nu }^{(3)}(z,h)$, ${\mathrm{M}}_{\nu }^{(4)}(z,h)$ such that … ►

§28.20(iv) Radial Mathieu Functions ${\mathrm{Mc}}_n^{(j)}$, ${\mathrm{Ms}}_n^{(j)}$

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5: 28.12 Definitions and Basic Properties

§28.12 Definitions and Basic Properties

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§28.12(ii) Eigenfunctions ${\mathrm{me}}_{\nu }(z,q)$

… ►They have the following pseudoperiodic and orthogonality properties: … ►When $\nu =s/m$ is a rational number, but not an integer, all solutions of Mathieu’s equation are periodic with period $2m\pi$. …

6: Bibliography

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Asymptotics of solutions of the generalized sine-Gordon equation, the third Painlevé equation and the d’Alembert equation. Dokl. Akad. Nauk SSSR 280 (2), pp. 265–268 (Russian).

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Notes:

English translation: Soviet Math. Dokl. 31(1985), no. 1, pp. 45–47

External Links:

ISSN 0002-3264, MathReview (M. S. Agranovich), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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Rational solutions of Painlevé equations. Stud. Appl. Math. 61 (1), pp. 31–53.

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External Links:

ISSN 0022-2526, MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§32.10(ii), §32.8(i), §32.8(v).

Encodings:

BibTeX

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Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.

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External Links:

MathReview (R. N. Jain), ZentralBlatt

Referenced by:

§17.11.

Encodings:

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Applications of basic hypergeometric functions. SIAM Rev. 16 (4), pp. 441–484.

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External Links:

Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§17.16, Ch.17.

Encodings:

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Perturbation solutions of the ellipsoidal wave equation. Quart. J. Math. Oxford Ser. (2) 7, pp. 161–174.

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External Links:

ISSN 0033-5606, Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§29.11.

Encodings:

BibTeX

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7: Bibliography G

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Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge.

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Notes:

A revised and updated edition, with three new chapters, was published in 2004 by Cambridge University Press (Encyclopedia of Mathematics and its Applications, vol. 96).

External Links:

ISBN 0-521-35049-2, MathReview (Shaun Cooper), ZentralBlatt

Referenced by:

(17.9.3), Erratum (V1.0.23) for Equation (17.9.3).

Encodings:

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Basic Hypergeometric Series. Second edition, Encyclopedia of Mathematics and its Applications, Vol. 96, Cambridge University Press, Cambridge.

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Notes:

With a foreword by Richard Askey

External Links:

ISBN 0-521-83357-4, MathReview (Shaun Cooper), ZentralBlatt

Referenced by:

§17.1, §17.1, §17.10, (17.11.1), (17.11.2), (17.11.3), §17.13, §17.16, (17.4.5), (17.4.6), (17.4.7), (17.4.8), §17.4(i), §17.5, §17.6(i), §17.6(ii), §17.6(iii), §17.6(iv), §17.6(v), §17.7(i), §17.7(ii), §17.7(iii), §17.8, (17.9.3), §17.9(i), §17.9(ii), §17.9(iii), Ch.17, §18.27(i), §18.27(i), §18.27(ii), §18.27(iii), §18.27(iv), (18.28.24), (18.28.25), §18.28(i), §18.28(ii), §18.28(v), §18.28(viii), §26.19, Erratum (V1.0.10) for Section 17.1, Erratum (V1.0.23) for Equation (17.9.3).

Encodings:

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Special classes of solutions of Painlevé equations. Differ. Uravn. 18 (3), pp. 419–429 (Russian).

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Notes:

In Russian. English translation: Differential Equations 18(3), pp. 317–326.

External Links:

ISSN 0374-0641, MathReview, ZentralBlatt

Referenced by:

§32.10(iv), §32.10(v), §32.8(iii), §32.8(v), §32.9(ii).

Encodings:

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The solutions of Painlevé’s fifth equation. Differ. Uravn. 12 (4), pp. 740–742 (Russian).

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Notes:

In Russian. English translation: Differential Equations 12(4), pp. 519–521 (1977)

External Links:

MathReview (Z. Nehari), ZentralBlatt

Referenced by:

§32.7(v).

Encodings:

BibTeX

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One-parameter systems of solutions of Painlevé equations. Differ. Uravn. 14 (12), pp. 2131–2135 (Russian).

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Notes:

In Russian. English translation: Differential Equations 14(12), pp. 1510–1513 (1979)

External Links:

ISSN 0374-0641, MathReview (M. Tvrdý), ZentralBlatt

Referenced by:

§32.10(i), §32.10(ii), §32.10(iii), §32.7(iv).

Encodings:

BibTeX

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8: Bibliography F

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Basic Hypergeometric Series and Applications. Mathematical Surveys and Monographs, Vol. 27, American Mathematical Society, Providence, RI.

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Notes:

With a foreword by George E. Andrews

External Links:

ISBN 0-8218-1524-5, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.1, §17.16, §17.6(ii), Ch.17, Notations F.

Encodings:

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On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

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External Links:

ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.13(i), §32.7(i), §32.7(iii).

Encodings:

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The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.

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External Links:

ISSN 0024-1318, MathReview (D. A. Lutz)

Referenced by:

§32.10(vi).

Encodings:

BibTeX

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Solution of the transcendental equation $w{e}^w=x$ . Comm. ACM 16 (2), pp. 123–124.

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Notes:

Includes Fortran code.

External Links:

ISSN 0001-0782, Document

Referenced by:

§4.48(iv).

Encodings:

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Characteristic values of the integral equation satisfied by the Mathieu functions and its application to a system with chirality-pair interaction on a one-dimensional lattice. Phys. A 190 (3-4), pp. 346–362.

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External Links:

ISSN 0378-4371, Document, MathReview

Referenced by:

6th item.

Encodings:

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