relation to Legendre elliptic integrals

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31: Bibliography H

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Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

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External Links:

MathReview (M. Lawrence Glasser)

Referenced by:

§7.14(iii).

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A fast FFT-based discrete Legendre transform. IMA J. Numer. Anal. 36 (4), pp. 1670–1684.

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ISSN 0272-4979, Document, Link, MathReview (Denis N. Sidorov)

Referenced by:

§18.16(iii).

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Elliptic Integrals. Dover Publications Inc., New York.

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Notes:

Unaltered reprint of original edition published by Wiley, New York, 1917.

External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§19.8(ii).

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Algorithm 55: Complete elliptic integral of the first kind. Comm. ACM 4 (4), pp. 180.

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Notes:

Includes Algol program, maximum accuracy 8D. For certification see same journal 6 (1966), pp. 166–167

External Links:

ISSN 0001-0782, Document

Referenced by:

§19.39(ii).

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Algorithm 56: Complete elliptic integral of the second kind. Comm. ACM 4 (4), pp. 180–181.

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Notes:

Includes Algol program, maximum accuracy 8D. For certification see same journal 9 (1966), p. 12

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ISSN 0001-0782, Document

Referenced by:

§19.39(ii).

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32: Bibliography B

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Orthogonality relations for the associated Legendre functions of imaginary order. Integral Transforms Spec. Funct. 24 (4), pp. 331–337.

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ISSN 1065-2469, Document, FullText, MathReview (Praveen Agarwal), ZentralBlatt

Referenced by:

§14.17(iii), §14.17(iii).

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An extension of the Bartky-transformation to incomplete elliptic integrals of the third kind. Numer. Math. 13 (3), pp. 266–284.

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ISSN 0029-599X, Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§19.1, §19.2(iii), §19.36(ii), §19.36(ii).

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Numerical calculation of elliptic integrals and elliptic functions. III. Numer. Math. 13 (4), pp. 305–315.

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Notes:

Algol 60 programs for the incomplete elliptic integral of the third kind, and for a general complete elliptic integral, are included.

External Links:

ISSN 0029-599X, Document, MathReview, ZentralBlatt

Referenced by:

§19.2(iii), §19.36(ii), §19.36(ii), §19.39(ii), §19.39(iii), R. Bulirsch (1965a).

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Numerical calculation of elliptic integrals and elliptic functions. Numer. Math. 7 (1), pp. 78–90.

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Notes:

Algol 60 programs are included for real and complex elliptic integrals, and for Jacobian elliptic functions of real argument.

External Links:

ISSN 0029-599X, Document, MathReview (M. Feliciano), ZentralBlatt

Referenced by:

§19.1, §19.2(iii), §19.36(ii), §19.39(iii), §22.22(ii).

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On a generalization of Barton’s integral and related integrals of complete elliptic integrals. Math. Proc. Cambridge Philos. Soc. 101 (1), pp. 1–5.

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ISSN 0305-0041, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§19.13(i).

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33: Bibliography R

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Elliptic Integrals of the First and Second Kind – Comparison of Bulirsch’s and Carlson’s Algorithms for Numerical Calculation. In Special Functions (Hong Kong, 1999), C. Dunkl, M. Ismail, and R. Wong (Eds.), pp. 293–308.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(i), §19.36(ii).

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Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome I. Gauthier-Villars, Paris.

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Notes:

Préface de H. Villat

External Links:

MathReview (R. Campbell), ZentralBlatt

Referenced by:

Ch.14.

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Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome II. Gauthier-Villars, Paris.

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MathReview (R. Campbell), ZentralBlatt

Referenced by:

Ch.14.

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The similarity solution for the Korteweg-de Vries equation and the related Painlevé transcendent. Proc. Roy. Soc. London Ser. A 361, pp. 265–275.

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FullText, MathReview (A. D. Brjuno), ZentralBlatt

Referenced by:

§32.11(ii), §32.17, §32.3(ii).

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Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.

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ISBN 978-1-107-15938-9, Document, Link, MathReview (Paul M. Jenkins), ZentralBlatt

Referenced by:

Profile Ranjan Roy.

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34: 29.6 Fourier Series

… ►With $\varphi =\frac{1}{2}\pi -\mathrm{am}(z,k)$, as in (29.2.5), we have … ►In addition, if $H$ satisfies (29.6.2), then (29.6.3) applies. … ►Consequently, ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$ reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i). … ►Here $\mathrm{dn}(z,k)$ is as in §22.2, and … ►

29.6.15 $$\frac{1}{2}{A}_0{C}_0+\sum _{p=1}^{\mathrm{\infty }}{A}_{2p}{C}_{2p}=\frac{4}{\pi }{\int }_0^K{\left({\mathrm{𝐸𝑐}}_{\nu }^{2m}(x,k^2)\right)}^{2}dx.$$

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Symbols:

${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$ , $\int$: integral, $m$: nonnegative integer, $p$: nonnegative integer, $x$: real variable, $k$: real parameter, $\nu$: real parameter, $A_{2p}$: coefficients and $C_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.6.E15

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pMML, png, TeX

See also:

Annotations for §29.6(i), §29.6 and Ch.29

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35: Bibliography

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Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem. Comm. Pure Appl. Math. 30 (1), pp. 95–148.

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External Links:

MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§23.21(ii).

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Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172 (2), pp. 289–312.

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ISSN 0377-0427, Document, MathReview (Živorad Tomovski), ZentralBlatt

Referenced by:

§19.9(i).

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Inequalities for elliptic integrals. Publ. Inst. Math. (Beograd) (N.S.) 37(51), pp. 61–63.

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ISSN 0350-1302, MathReview, ZentralBlatt

Referenced by:

§19.9(i).

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

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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.

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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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Document, MathReview (I. Marx), ZentralBlatt

Referenced by:

§29.8.

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36: 29.12 Definitions

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§29.12(i) Elliptic-Function Form

… ►There are eight types of Lamé polynomials, defined as follows: …These functions are polynomials in $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, and $\mathrm{dn}(z,k)$. … ►The superscript $m$ on the left-hand sides of (29.12.1)–(29.12.8) agrees with the number of $z$-zeros of each Lamé polynomial in the interval $(0,K)$, while $n-m$ is the number of $z$-zeros in the open line segment from $K$ to $K+\mathrm{i}{K}^{\prime }$. … ►In the fourth column the variable $z$ and modulus $k$ of the Jacobian elliptic functions have been suppressed, and $P({\mathrm{sn}}^2)$ denotes a polynomial of degree $n$ in ${\mathrm{sn}}^2(z,k)$ (different for each type). …

37: Bibliography Z

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On the Computation of Zeros of Bessel and Bessel-related Functions. In Proceedings of the Sixth International Colloquium on Differential Equations (Plovdiv, Bulgaria, 1995), D. Bainov (Ed.), Utrecht, pp. 409–416.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§10.74(vi).

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The Dilogarithm Function in Geometry and Number Theory. In Number Theory and Related Topics (Bombay, 1988), R. Askey and others (Eds.), Tata Inst. Fund. Res. Stud. Math., Vol. 12, pp. 231–249.

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External Links:

MathReview (J. Browkin), ZentralBlatt

Referenced by:

§25.12(i).

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Symmetric elliptic integrals of the third kind. Math. Comp. 24 (109), pp. 199–214.

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External Links:

FullText, Document, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.21(i), §19.21(iii), §19.22(iii), §19.25(i), §19.26(i).

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Tables of the Legendre functions $P_{-1/2+i\tau }(x)$. Part I. Translated by D. E. Brown. Mathematical Tables Series, Vol. 22, Pergamon Press, Oxford.

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Notes:

Translation of Žurina and Karmazina, Tablitsy funktsii Lezhandra $P_{-1/2+i\tau }(x)$. Tom I, Izdat. Akad. Nauk SSSR, Moscow, 1960.

External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

4th item.

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Tables of the Legendre functions $P_{-1/2+i\tau }(x)$. Part II. Translated by Prasenjit Basu. Mathematical Tables Series, Vol. 38. A Pergamon Press Book, The Macmillan Co., New York.

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Notes:

Translation of Žurina and Karmazina, Tablitsy funktsii Lezhandra $P_{-1/2+i\tau }(x)$. Tom II, Izdat. Akad. Nauk SSSR, Moscow, 1962.

External Links:

MathReview, ZentralBlatt

Referenced by:

4th item.

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38: 15.9 Relations to Other Functions

§15.9 Relations to Other Functions

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Legendre

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Meixner

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§15.9(iv) Associated Legendre Functions; Ferrers Functions

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§15.9(v) Complete Elliptic Integrals

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39: Bibliography L

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Reduction of Elliptic Integrals to Legendre Normal Form. Technical report Technical Report 97-21, Department of Computer Science, University of Waterloo, Waterloo, Ontario.

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Referenced by:

§19.14(ii).

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Traité des fonctions elliptiques et des intégrales Eulériennes. Huzard-Courcier, Paris.

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Notes:

vol.1, 1825; vol.2, 1826; three supplements, 1828–1832.

Referenced by:

§19.14(ii), Ch.19.

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Uniform asymptotic expansions of symmetric elliptic integrals. Constr. Approx. 17 (4), pp. 535–559.

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ISSN 0176-4276, Document, MathReview (Valdir A. Menegatto), ZentralBlatt

Referenced by:

§19.27(vi).

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Approximations for elliptic integrals. Math. Comp. 22 (103), pp. 627–634.

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FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

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Further approximations for elliptic integrals. Math. Comp. 24 (109), pp. 191–198.

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FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

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40: Bibliography F

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On the reciprocal modulus relation for elliptic integrals. SIAM J. Math. Anal. 1 (4), pp. 524–526.

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External Links:

Document, MathReview (B. C. Carlson), ZentralBlatt

Referenced by:

§19.7(i).

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Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.

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Notes:

Fortran, minimum precision 6D.

External Links:

ISSN 0898-1221, Document, MathReview, ZentralBlatt

Referenced by:

§10.77(ix).

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Guide to tables of elliptic functions. Math. Tables and Other Aids to Computation 3 (24), pp. 229–281.

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External Links:

FullText, MathReview (S. C. van Veen)

Referenced by:

§19.37(i).

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Numerical evaluation of the elliptic integral of the third kind. Math. Comp. 19 (91), pp. 494–496.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.36(iv).

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Series expansions of symmetric elliptic integrals. Math. Comp. 81 (278), pp. 957–990.

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External Links:

Document, ZentralBlatt

Referenced by:

§20.7(ii), §20.7(ii).

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