relation to Mathieu functions

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1: 31.12 Confluent Forms of Heun’s Equation

… ►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty }$. …

2: 28.20 Definitions and Basic Properties

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

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

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3: Bibliography N

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Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview, ZentralBlatt

Referenced by:

§28.1, §28.26(i), 6th item, Notations S, Notations S.

Encodings:

BibTeX

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4: 28.2 Definitions and Basic Properties

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§28.2(vi) Eigenfunctions

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5: Bibliography B

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Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

1st item.

Encodings:

BibTeX

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Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

2nd item, 1st item.

Encodings:

BibTeX

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6: 28.32 Mathematical Applications

§28.32 Mathematical Applications

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§28.32(i) Elliptical Coordinates and an Integral Relationship

… ► ►This leads to integral equations and an integral relation between the solutions of Mathieu’s equation (setting $\zeta =\mathrm{i}\xi$, $z=\eta$ in (28.32.3)). … ► …

7: Bibliography

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Algorithm 804: Subroutines for the computation of Mathieu functions of integer orders. ACM Trans. Math. Software 26 (3), pp. 408–414.

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Notes:

Double-precision C++, minimum accuracy 14D

External Links:

ISSN 0098-3500, Document, GAMS (module 804; package TOMS)

Referenced by:

1st item, 1st item.

Encodings:

BibTeX

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Scattering by singular potentials with a perturbation – Theoretical introduction to Mathieu functions. J. Mathematical Phys. 16, pp. 961–970.

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Notes:

Erratum: same journal 17 (1975), no. 7, p. 1361

External Links:

Document, MathReview (K. Veselic), ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

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Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

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External Links:

ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt

Referenced by:

(25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.

Encodings:

BibTeX

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Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.

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External Links:

Document, MathReview (I. Marx), ZentralBlatt

Referenced by:

§29.8.

Encodings:

BibTeX

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Periodic Differential Equations. An Introduction to Mathieu, Lamé, and Allied Functions. International Series of Monographs in Pure and Applied Mathematics, Vol. 66, Pergamon Press, The Macmillan Co., New York.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§28.1, §28.1, §28.10(i), §28.11, §28.12(ii), §28.17, §28.2(ii), §28.2(iii), §28.2(iv), §28.28(i), §28.28(ii), §28.29(i), §28.32(i), §28.32(ii), §28.4(i), §28.5(i), Ch.28, §29.1, §29.11, §29.12(i), §29.12(ii), §29.12(iii), §29.14, §29.15(ii), §29.15(ii), §29.17(i), §29.18(iii), Ch.29, §30.1, §30.10, §30.11(i), §30.11(vi), §30.2(ii), §30.4(iii), §30.6, §30.8(i), §30.8(ii), Ch.30, §31.4, §31.5, §31.9(ii), Notations F, Notations G, Notations M, Notations N.

Encodings:

BibTeX

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8: 28.8 Asymptotic Expansions for Large $q$

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§28.8(ii) Sips’ Expansions

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§28.8(iii) Goldstein’s Expansions

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Barrett’s Expansions

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Dunster’s Approximations

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9: 28.14 Fourier Series

§28.14 Fourier Series

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28.14.2 $${\mathrm{ce}}_{\nu }(z,q)=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_{2m}^{\nu }(q)\cos (\nu +2m)z,$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $m$: integer, $q=h^2$: parameter, $z$: complex variable, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

Permalink:

http://dlmf.nist.gov/28.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

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28.14.4 $$q{c}_{2m+2}-\left(a-{(\nu +2m)}^2\right)c_{2m}+q{c}_{2m-2}=0,$$ $a={\lambda }_{\nu }\left(q\right),c_{2m}=c_{2m}^{\nu }(q)$,

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Symbols:

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

Referenced by:

item (d), item (d)

Permalink:

http://dlmf.nist.gov/28.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

►and the normalization relation … ►When $q\to 0$ with $m$ ($\ge 1$) and $\nu$ fixed, …

10: 28.10 Integral Equations

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§28.10(i) Equations with Elementary Kernels

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§28.10(ii) Equations with Bessel-Function Kernels

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28.10.10 $${\int }_0^{\pi }{J}_0\left(2\sqrt{q}(\cos \tau +\cos \zeta )\right){\mathrm{ce}}_n(\tau ,q)d\tau =w_{\text{II}}(\pi ;a_n\left(q\right),q){\mathrm{ce}}_n(\zeta ,q).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\mathrm{ce}}_n(z,q)$: Mathieu function, $a_n\left(q\right)$: eigenvalues of Mathieu equation, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $q=h^2$: parameter, $n$: integer and $w_{\text{II}}(z;a,q)$: fundamental solution

Permalink:

http://dlmf.nist.gov/28.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(ii), §28.10 and Ch.28

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§28.10(iii) Further Equations

… ►For relations with variable boundaries see Volkmer (1983).