relation to symmetric elliptic integrals

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

Referenced by:

§31.10(i).

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On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.

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FullText, MathReview, ZentralBlatt

Referenced by:

§19.12, §19.5.

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Modular hypergeometric residue sums of elliptic Selberg integrals. Lett. Math. Phys. 58 (3), pp. 223–238.

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ISSN 0377-9017, Document, MathReview (Laurent Habsieger), ZentralBlatt

Referenced by:

§19.35(i).

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Construction of a crossing-symmetric, Regge-behaved amplitude for linearly rising trajectories. Il Nuovo Cimento A 57 (1), pp. 190–197.

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FullText

Referenced by:

§5.20.

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

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

Referenced by:

§29.8, §29.8.

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22: 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.

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

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Elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 99–102.

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Notes:

Includes Algol program.

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MathReview

Referenced by:

§19.39(iii).

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On the calculation of elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 91–94.

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MathReview, ZentralBlatt

Referenced by:

§19.36(ii).

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Bounds for symmetric elliptic integrals. J. Approx. Theory 122 (2), pp. 249–259.

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ISSN 0021-9045, Document, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§19.24(ii), §19.24(i), §19.9(i).

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23: 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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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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MathReview (J. Browkin), ZentralBlatt

Referenced by:

§25.12(i).

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On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval. J. Approx. Theory 94 (1), pp. 73–106.

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ISSN 0021-9045, Link, MathReview (T. S. Chihara), ZentralBlatt

Referenced by:

§18.33(iii).

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Generalized Watson Transforms and Applications to Group Representations. Ph.D. Thesis, University of Vermont, Burlington,VT.

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Referenced by:

§35.7(ii).

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

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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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24: 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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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.

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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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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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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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Document, ZentralBlatt

Referenced by:

§20.7(ii), §20.7(ii).

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25: 19.7 Connection Formulas

§19.7 Connection Formulas

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Reciprocal-Modulus Transformation

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Imaginary-Modulus Transformation

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§19.7(iii) Change of Parameter of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$

►There are three relations connecting $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ and $\mathrm{\Pi }(\varphi ,{\omega }^2,k)$, where ${\omega }^2$ is a rational function of ${\alpha }^2$. …

26: Bibliography M

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Connection between quantum systems involving the fourth Painlevé transcendent and $k$-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57 (5), pp. Paper 052101, 15 pp..

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ISSN 0022-2488, Document, Link, MathReview Entry

Referenced by:

§18.38(iii).

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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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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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27: 2.6 Distributional Methods

… ►This leads to integrals of the form … ►To assign a distribution to the function $f_n(t)$, we first let $f_{n,n}(t)$ denote the $n$th repeated integral (§1.4(v)) of $f_n$: … ►An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). … ►The replacement of $f(t)$ by its asymptotic expansion (2.6.9), followed by term-by-term integration leads to convolution integrals of the form … ►The method of distributions can be further extended to derive asymptotic expansions for convolution integrals: …