symmetric elliptic integrals

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41: 19.3 Graphics

§19.3 Graphics

… ►See Figures 19.3.1–19.3.6 for complete and incomplete Legendre’s elliptic integrals. ►

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Figure 19.3.1: $K\left(k\right)$ and $E\left(k\right)$ as functions of $k^2$ for $-2\le k^2\le 1$. Graphs of $K^{\prime }\left(k\right)$ and $E^{\prime }\left(k\right)$ are the mirror images in the vertical line $k^2=\frac{1}{2}$. Magnify

… ►In Figures 19.3.7 and 19.3.8 for complete Legendre’s elliptic integrals with complex arguments, height corresponds to the absolute value of the function and color to the phase. … ►

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Figure 19.3.12: $\mathrm{\Im }\left(E\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. … Magnify 3D Help

42: 36.5 Stokes Sets

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§36.5(ii) Cuspoids

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§36.5(iii) Umbilics

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Elliptic Umbilic Stokes Set (Codimension three)

… ►One of the sheets is symmetrical under reflection in the plane $y=0$, and is given by … ►

§36.5(iv) Visualizations

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43: 15.17 Mathematical Applications

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§15.17(ii) Conformal Mappings

… ►Hypergeometric functions, especially complete elliptic integrals, also play an important role in quasiconformal mapping. … ►First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL$(2,ℝ)$, and spherical functions on certain nonsymmetric Gelfand pairs. … ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). …

44: Bibliography T

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Some definite integrals and Fourier series for Jacobian elliptic functions. Z. Angew. Math. Mech. 49, pp. 95–96.

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External Links:

ISSN 0044-2267, Document, MathReview (B. C. Carlson), ZentralBlatt

Referenced by:

§22.11.

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Harmonic Analysis on Symmetric Spaces and Applications. II. Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-96663-3, MathReview (Stephen Gelbart), ZentralBlatt

Referenced by:

§35.1, §35.5(ii), Notations K.

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Algorithm 165: Complete elliptic integrals. Comm. ACM 6 (4), pp. 163–164.

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Notes:

Includes Algol program, maximum accuracy 10D. For certification see same journal 6 (1963), pp. 166–167

External Links:

ISSN 0001-0782, Document

Referenced by:

§19.39(ii).

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Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.

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Document

Referenced by:

§31.17(ii).

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Eigenfrequencies of an elliptic membrane. Math. Comp. 27 (124), pp. 755–765.

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FullText, Document, MathReview (Radu Badescu), ZentralBlatt

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4th item.

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45: 32.2 Differential Equations

… ►The fifty equations can be reduced to linear equations, solved in terms of elliptic functions (Chapters 22 and 23), or reduced to one of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$. … ►

§32.2(iv) Elliptic Form

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32.2.13 $$z(1-z)𝐼\left({\int }_{\mathrm{\infty }}^w\frac{dt}{\sqrt{t(t-1)(t-z)}}\right)=\sqrt{w(w-1)(w-z)}\left(\alpha +\frac{\beta z}{w^2}+\frac{\gamma (z-1)}{{(w-1)}^2}+(\delta -\frac{1}{2})\frac{z(z-1)}{{(w-z)}^2}\right),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: real, $\alpha$: arbitrary constant, $𝐼$: operator, $\beta$: arbitrary constant, $\gamma$: arbitrary constant and $\delta$: arbitrary constant

Permalink:

http://dlmf.nist.gov/32.2.E13

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pMML, png, TeX

See also:

Annotations for §32.2(iv), §32.2 and Ch.32

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§32.2(v) Symmetric Forms

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46: Bibliography I

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The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of $J_0(z)-i{J}_1(z)$ and of Bessel functions $J_m(z)$ of any real order $m$ . Linear Algebra Appl. 194, pp. 35–70.

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External Links:

ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt

Referenced by:

§10.21(ix).

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Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines. Z. Angew. Math. Mech. 75 (12), pp. 917–926.

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ISSN 0044-2267, MathReview (John P. Coleman), ZentralBlatt

Referenced by:

§10.74(vii).

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Tables of the elliptic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 52, pp. 355–433.

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External Links:

ZentralBlatt

Referenced by:

4th item, 2nd item.

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An integral which occurs in statistics. Proceedings of the Cambridge Philosophical Society 29, pp. 271–276.

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External Links:

Document, ZentralBlatt

Referenced by:

§35.3(i), §35.3(ii).

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$q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.

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External Links:

ISSN 0002-9947, FullText, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.28(vii).

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BibTeX

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47: Bibliography S

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The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. 2nd edition, Graduate Texts in Mathematics, Vol. 203, Springer-Verlag, New York.

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Notes:

First edition published by Wadsworth & Brooks/Cole Advanced Books & Software, 1991.

External Links:

ISBN 0-387-95067-2, MathReview, ZentralBlatt

Referenced by:

§26.19.

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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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Parabolic Cylinder Functions and their Application in Symmetric Two-centre Shell Model. In Proceedings of the Conference on Mathematical Analysis and its Applications (Inst. Engrs., Mysore, 1977), Matscience Rep., Vol. 91, Aarhus, pp. P81–P89.

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External Links:

MathReview

Referenced by:

§12.17.

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Some combinatorial properties of Jack symmetric functions. Adv. Math. 77 (1), pp. 76–115.

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ISSN 0001-8708, Document, MathReview (Dennis White), ZentralBlatt

Referenced by:

§18.37(iii).

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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

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3rd item.

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48: 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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External Links:

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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External Links:

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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External Links:

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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External Links:

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

Referenced by:

§29.8, §29.8.

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BibTeX

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49: Bibliography R

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Plane wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric. IEEE Trans. Antennas and Propagation 39 (2), pp. 218–223.

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External Links:

Document

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6th item.

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Normal limit theorems for symmetric random matrices. Probab. Theory Related Fields 112 (3), pp. 411–423.

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External Links:

ISSN 0178-8051, Document, MathReview (Alexei Khorunzhy), ZentralBlatt

Referenced by:

§35.9.

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Elliptic Functions, Theta Functions, and Riemann Surfaces. The Williams & Wilkins Co., Baltimore, MD.

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External Links:

MathReview (H. H. Martens), ZentralBlatt

Referenced by:

§20.11(iv), §20.12(ii).

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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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Elliptic hypergeometric series on root systems. Adv. Math. 181 (2), pp. 417–447.

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External Links:

ISSN 0001-8708, Document, MathReview, ZentralBlatt

Referenced by:

§20.11(iii).

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BibTeX

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50: Bibliography D

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The principal frequencies of vibrating systems with elliptic boundaries. Quart. J. Mech. Appl. Math. 8 (3), pp. 361–372.

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Document, MathReview (M. A. Hyman), ZentralBlatt

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3rd item.

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Pole dynamics for elliptic solutions of the Korteweg-de Vries equation. Math. Phys. Anal. Geom. 3 (1), pp. 49–74.

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ISSN 1385-0172, Document, MathReview (Giulio Soliani), ZentralBlatt

Referenced by:

§23.21(ii).

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Integral representations for elliptic functions. J. Math. Anal. Appl. 316 (1), pp. 142–160.

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External Links:

ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§23.11.

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Ramanujan’s master theorem for symmetric cones. Pacific J. Math. 175 (2), pp. 447–490.

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External Links:

ISSN 0030-8730, MathReview (Jacques Faraut), ZentralBlatt

Referenced by:

§35.7(ii), §35.8(iv), §35.8(v).

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The Legendre Relation for Elliptic Integrals. In Paul Halmos: Celebrating 50 Years of Mathematics, J. H. Ewing and F. W. Gehring (Eds.), pp. 305–315.

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External Links:

ISBN 0-387-97509-8, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§19.7(i).

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