integral equation for Lamé functions

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1: 29.8 Integral Equations

§29.8 Integral Equations

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29.8.2 $$\mu w(z_1)w(z_2)w(z_3)={\int }_{-2K}^{2K}{𝖯}_{\nu }\left(x\right)w(z)dz,$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $x$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $w(z)$: solution and $\mu$

Referenced by:

§29.8

Permalink:

http://dlmf.nist.gov/29.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

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2: Bibliography

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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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3: Bibliography S

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Integral equations and relations for Lamé functions and ellipsoidal wave functions. Proc. Cambridge Philos. Soc. 64, pp. 113–126.

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External Links:

Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§29.8.

Encodings:

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4: 29.2 Differential Equations

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§29.2(i) Lamé’s Equation

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§29.2(ii) Other Forms

… ►we have …For the Weierstrass function $\mathrm{\wp }$ see §23.2(ii). … ►

5: 29.3 Definitions and Basic Properties

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§29.3(i) Eigenvalues

… ►satisfies the continued-fraction equation … ►

§29.3(iv) Lamé Functions

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§29.3(v) Normalization

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§29.3(vi) Orthogonality

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6: Bibliography E

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On Lamé functions. Philos. Mag. (7) 31, pp. 123–130.

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External Links:

MathReview (G. Szegő), ZentralBlatt

Referenced by:

§29.3(ii).

Encodings:

BibTeX

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On algebraic Lamé functions. Philos. Mag. (7) 32, pp. 348–350.

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External Links:

MathReview (G. Szegő), ZentralBlatt

Referenced by:

§29.17(ii).

Encodings:

BibTeX

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Integral equations for Heun functions. Quart. J. Math., Oxford Ser. 13, pp. 107–112.

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External Links:

Document, MathReview (E. Rothe), ZentralBlatt

Referenced by:

§31.10(i).

Encodings:

BibTeX

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The Fuchsian equation of second order with four singularities. Duke Math. J. 9 (1), pp. 48–58.

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External Links:

Document, MathReview (R. E. Langer), ZentralBlatt

Referenced by:

§31.11(i).

Encodings:

BibTeX

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A catalogue of Sturm-Liouville differential equations. In Sturm-Liouville theory, pp. 271–331.

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External Links:

Document, Link, MathReview Entry

Referenced by:

§1.18(ix), §1.18(x).

Encodings:

BibTeX

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7: Bibliography L

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Some differential equations and associated integral equations. Quart. J. Math. (Oxford) 5, pp. 81–97.

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External Links:

Document, ZentralBlatt

Referenced by:

§31.10(i), §31.9(i).

Encodings:

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Lamé-Wangerin functions. Quart. J. Math., Oxford Ser. (2) 3, pp. 107–114.

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External Links:

Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§29.17(ii).

Encodings:

BibTeX

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Some Differential Equations Satisfied by Hypergeometric Functions. In Approximation and Computation (West Lafayette, IN, 1993), Internat. Ser. Numer. Math., Vol. 119, pp. 371–381.

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External Links:

MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§16.8(ii).

Encodings:

BibTeX

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Evaluation of Bessel function integrals with algebraic singularities. J. Comput. Appl. Math. 37 (1-3), pp. 101–112.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Integrals of Bessel Functions. McGraw-Hill Book Co., Inc., New York.

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External Links:

MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§10.22(ii), §10.22(iv), §10.22(vi), §10.43(iii), §10.43(vi), §11.7(v).

Encodings:

BibTeX

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8: 29.1 Special Notation

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$${(s_{\nu }^m(k^2))}^2=\frac{4}{\pi }{\int }_0^K{\left({\mathrm{𝐸𝑠}}_{\nu }^m(x,k^2)\right)}^{2}dx.$$

9: Bibliography V

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An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

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External Links:

ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt

Referenced by:

§31.10(i).

Encodings:

BibTeX

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Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-Like Functions. CWI Tract, Vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam.

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Notes:

A reprint was published in 1986.

External Links:

ISBN 90-6196-277-3, MathReview (John P. Coleman), ZentralBlatt

Referenced by:

§5.21, §6.18(iv), §7.22(v), §7.6(i), §7.7(i), §7.7(ii).

Encodings:

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Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.

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External Links:

ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8, §29.8.

Encodings:

BibTeX

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Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.

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External Links:

ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8.

Encodings:

BibTeX

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Four remarks on eigenvalues of Lamé’s equation. Anal. Appl. (Singap.) 2 (2), pp. 161–175.

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External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§29.3(ii), §29.3(vii), §29.5, §29.7(i), §29.7(i).

Encodings:

BibTeX

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10: 29.17 Other Solutions

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§29.17(i) Second Solution

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§29.17(ii) Algebraic Lamé Functions

►Algebraic Lamé functions are solutions of (29.2.1) when $\nu$ is half an odd integer. … ►

§29.17(iii) Lamé–Wangerin Functions

►Lamé–Wangerin functions are solutions of (29.2.1) with the property that ${(\mathrm{sn}(z,k))}^{1/2}w(z)$ is bounded on the line segment from $\mathrm{i}{K}^{\prime }$ to $2K+\mathrm{i}{K}^{\prime }$. …