Legendre%20elliptic%20integrals

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1: 19.36 Methods of Computation

… ►Because of cancellations in (19.26.21) it is advisable to compute $R_G$ from $R_F$ and $R_D$ by (19.21.10) or else to use §19.36(ii). … ►Computation of Legendre’s integrals of all three kinds by quadratic transformation is described by Cazenave (1969, pp. 128–159, 208–230). … ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). … ►For series expansions of Legendre’s integrals see §19.5. … ►

2: 22.3 Graphics

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§22.3(i) Real Variables: Line Graphs

►Line graphs of the functions $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, $\mathrm{dn}(x,k)$, $\mathrm{cd}(x,k)$, $\mathrm{sd}(x,k)$, $\mathrm{nd}(x,k)$, $\mathrm{dc}(x,k)$, $\mathrm{nc}(x,k)$, $\mathrm{sc}(x,k)$, $\mathrm{ns}(x,k)$, $\mathrm{ds}(x,k)$, and $\mathrm{cs}(x,k)$ for representative values of real $x$ and real $k$ illustrating the near trigonometric ($k=0$), and near hyperbolic ($k=1$) limits. … ► $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, and $\mathrm{dn}(x,k)$ as functions of real arguments $x$ and $k$. … ►

§22.3(iii) Complex $z$; Real $k$

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§22.3(iv) Complex $k$

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3: Software Index

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	Open Source													With Book						Commercial
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14 Legendre and Related Functions
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19 Elliptic Integrals
19.39(ii) $K\left(k\right)$, $E\left(k\right)$, $0\le k^2\le 1$	✓		✓		✓	✓		✓	✓			✓	✓							✓	✓	✓	✓
19.39(iii) $F(\varphi ,k)$, $E(\varphi ,k)$, $\varphi ,k\in ℝ$	✓		✓		✓	✓		✓	✓			✓	✓			✓					✓	✓	a	✓
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20 Theta Functions
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4: 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).

Encodings:

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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.

Encodings:

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

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External Links:

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

Referenced by:

§19.27(vi).

Encodings:

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

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

Encodings:

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

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

Encodings:

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5: Bibliography B

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Numerical calculation of a generalized complete elliptic integral. Rev. Mod. Phys. 10, pp. 264–269.

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External Links:

Document, ZentralBlatt

Referenced by:

§19.2(iii).

Encodings:

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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).

Encodings:

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Numerical calculation of elliptic integrals and elliptic functions. II. Numer. Math. 7 (4), pp. 353–354.

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Notes:

An improved Algol 60 program for the complete elliptic integral of the third kind is included, but superseded by a program in Bulirsch (1969b).

External Links:

ISSN 0029-599X, Document, MathReview (M. P. S. Madan), ZentralBlatt

Referenced by:

§19.1.

Encodings:

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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).

Encodings:

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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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External Links:

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

Referenced by:

§19.13(i).

Encodings:

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6: Bibliography F

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Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

Encodings:

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Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:

Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.

Referenced by:

§19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).

Encodings:

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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).

Encodings:

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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).

Encodings:

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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).

Encodings:

BibTeX

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7: Bibliography

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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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External Links:

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

Referenced by:

§19.9(i).

Encodings:

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Repeated integrals and derivatives of $K$ Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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External Links:

ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt

Referenced by:

§10.43(iii).

Encodings:

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Generalized elliptic integrals and modular equations. Pacific J. Math. 192 (1), pp. 1–37.

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External Links:

ISSN 0030-8730, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.35(i).

Encodings:

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Functional inequalities for complete elliptic integrals and their ratios. SIAM J. Math. Anal. 21 (2), pp. 536–549.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

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

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External Links:

ISSN 0350-1302, MathReview, ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

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8: Bibliography S

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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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External Links:

ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

Encodings:

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A Table of the Incomplete Elliptic Integral of the Third Kind. Dover Publications Inc., New York.

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External Links:

MathReview (C. J. Bouwkamp), ZentralBlatt

Referenced by:

§19.37(iii).

Encodings:

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

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Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. National Bureau of Standards Applied Mathematics Series, No. 19, U. S. Government Printing Office, Washington, D.C..

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External Links:

MathReview, ZentralBlatt

Referenced by:

§13.23(iv), Ch.14, §31.3(iii).

Encodings:

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A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

External Links:

Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

Referenced by:

3rd item.

Encodings:

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9: Bibliography P

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Tables of the Incomplete Elliptic Integrals of the First and Third Kind. Technical report Curtiss-Wright Corp., Research Division, Quehanna, PA.

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Notes:

Reviewed in Math. Comp. 14(1960)209-210.

Referenced by:

§19.37(iii).

Encodings:

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Numerical calculation of the generalized Fermi-Dirac integrals. Comput. Phys. Comm. 55 (2), pp. 127–136.

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External Links:

Document

Referenced by:

§25.18(i).

Encodings:

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Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:

Double-precision Fortran, maximum accuracy 20S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

6th item.

Encodings:

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Sur une généralisation des polynomes de Legendre. C. R. Acad. Sci. Paris 228, pp. 1363–1365.

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External Links:

ISSN 0001-4036, MathReview (G. Szegö)

Referenced by:

§18.35(i).

Encodings:

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Elliptic integrals. Computers in Physics 4 (1), pp. 92–96.

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Referenced by:

2nd item.

Encodings:

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10: Bibliography C

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On computing elliptic integrals and functions. J. Math. and Phys. 44, pp. 36–51.

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External Links:

MathReview (R. Nicolovius), ZentralBlatt

Referenced by:

§19.36(ii), §19.36(ii).

Encodings:

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Inequalities for a symmetric elliptic integral. Proc. Amer. Math. Soc. 25 (3), pp. 698–703.

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External Links:

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

Referenced by:

§19.24(ii), §19.9(ii).

Encodings:

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Elliptic integrals of the first kind. SIAM J. Math. Anal. 8 (2), pp. 231–242.

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External Links:

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

Referenced by:

§19.29(iii).

Encodings:

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Short proofs of three theorems on elliptic integrals. SIAM J. Math. Anal. 9 (3), pp. 524–528.

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External Links:

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

Referenced by:

§19.26(i).

Encodings:

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Integrals involving complete elliptic integrals. J. Comput. Appl. Math. 106 (1), pp. 169–175.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§19.13(i).

Encodings:

BibTeX

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