basic elliptic integrals

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1: 19.1 Special Notation

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$l,m,n$	nonnegative integers.
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2: 19.29 Reduction of General Elliptic Integrals

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§19.29(ii) Reduction to Basic Integrals

… ►All other cases are integrals of the second kind. …

3: 29.3 Definitions and Basic Properties

§29.3 Definitions and Basic Properties

… ►For each pair of values of $\nu$ and $k$ there are four infinite unbounded sets of real eigenvalues $h$ for which equation (29.2.1) has even or odd solutions with periods $2K$ or $4K$. … ►In this table the nonnegative integer $m$ corresponds to the number of zeros of each Lamé function in $(0,K)$, whereas the superscripts $2m$, $2m+1$, or $2m+2$ correspond to the number of zeros in $[0,2K)$. … ►For $\mathrm{dn}(z,k)$ see §22.2. ►To complete the definitions, ${\mathrm{𝐸𝑐}}_{\nu }^m(K,k^2)$ is positive and ${d{\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)/dz|}_{z=K}$ is negative. …

4: Bibliography G

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Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

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ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt

Referenced by:

§17.7(iii).

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Definite integrals of the complete elliptic integral $K$ . J. Res. Nat. Bur. Standards Sect. B 80B (2), pp. 313–323.

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MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.13(i).

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Automatic reduction of elliptic integrals using Carlson’s relations. Math. Comp. 71 (237), pp. 311–318.

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ISSN 0025-5718, Document, MathReview (David A. Richter), ZentralBlatt

Referenced by:

§19.29(ii).

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Multilateral summation theorems for ordinary and basic hypergeometric series in $\mathrm{U}(n)$ . SIAM J. Math. Anal. 18 (6), pp. 1576–1596.

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ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§17.15.

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Staircase polygons, elliptic integrals, Heun functions, and lattice Green functions. Phys. Rev. E 47 (4), pp. R2233–R2236.

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Document

Referenced by:

§19.35(ii).

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

Referenced by:

§19.9(i).

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

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

Referenced by:

§19.35(i).

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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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ISSN 0036-1410, Document, MathReview (K. C. Gupta), 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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Some basic hypergeometric extensions of integrals of Selberg and Andrews. SIAM J. Math. Anal. 11 (6), pp. 938–951.

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ISSN 0036-1410, Document, MathReview (W. A. Al-Salam), ZentralBlatt

Referenced by:

§17.13, §17.13.

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6: Bibliography F

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Calculation of elliptic integrals of the third kind by means of Gauss’ transformation. Math. Comp. 19 (89), pp. 97–104.

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FullText, MathReview (J. E. L. Peck), ZentralBlatt

Referenced by:

§19.36(ii).

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

Referenced by:

§19.7(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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FullText, MathReview, ZentralBlatt

Referenced by:

§19.36(iv).

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Fast computation of incomplete elliptic integral of first kind by half argument transformation. Numer. Math. 116 (4), pp. 687–719.

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ISSN 0029-599X, Document, FullText, MathReview (Mehdi Hassani), ZentralBlatt

Referenced by:

§19.36(iv), §19.36(iv).

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

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Document, ZentralBlatt

Referenced by:

§20.7(ii), §20.7(ii).

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7: 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).

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A note on the computation of the incomplete beta function. Adv. Eng. Software 12 (1), pp. 39–44.

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Notes:

Includes Basic code.

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Document

Referenced by:

§8.28(iv).

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

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Document

Referenced by:

§25.18(i).

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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.

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ISSN 0010-4655, Document, Code

Referenced by:

6th item.

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

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Referenced by:

2nd item.

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8: Bibliography M

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Algorithm 149: Complete elliptic integral. Comm. ACM 5 (12), pp. 605.

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Notes:

Includes Algol program, maximum accuracy 10D. For remark see ACM Trans. Math. Software 4 (1978), p. 95

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ISSN 0001-0782, Document

Referenced by:

§19.39(ii).

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An improved calculation of the general elliptic integral of the second kind in the neighbourhood of $x=0$ . Numer. Math. 25 (1), pp. 99–101.

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ISSN 0029-599X, Document, MathReview (Roland Bulirsch), ZentralBlatt

Referenced by:

§19.36(ii).

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A new symmetry related to $\mathrm{𝑆𝑈}(n)$ for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

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ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.15.

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Balanced ${}_3{}^{}\Theta _2^{}$ summation theorems for $U(n)$ basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.

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ISSN 0001-8708, Document, MathReview (Jiang Zeng), ZentralBlatt

Referenced by:

§17.15.

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Calculation of the complete elliptic integrals with complex modulus. Numer. Math. 29 (2), pp. 233–236.

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Notes:

An Algol procedure for calculating the complete elliptic integrals of the first and the second kind with complex modulus $k$ is included.

External Links:

ISSN 0029-599X, Document, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.36(ii), §19.39(ii).

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9: 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).

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

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

Referenced by:

§19.24(ii), §19.9(ii).

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

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

Referenced by:

§19.29(iii).

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

Referenced by:

§19.26(i).

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

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§19.13(i).

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10: Bibliography S

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Very high accuracy Chebyshev expansions for the basic trigonometric functions. Math. Comp. 34 (149), pp. 237–244.

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ISSN 0025-5718, FullText, Document, MathReview (Claude Carasso), ZentralBlatt

Referenced by:

§4.47(i).

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Spline Functions: Basic Theory. John Wiley & Sons Inc., New York.

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Pure and Applied Mathematics, A Wiley-Interscience Publication. Reprinted by Robert E. Krieger Publishing Co. Inc., 1993.

External Links:

ISBN 0-471-76475-2, MathReview (D. D. Stancu), ZentralBlatt

Referenced by:

§24.17(ii), §3.11(vi).

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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).

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The elliptical microstrip antenna with circular polarization. IEEE Trans. Antennas and Propagation 29 (1), pp. 90–94.

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FullText

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5th item.

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An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.

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ISBN 1-4020-1221-7, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§17.3(i).

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