elliptic curves

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11: Bibliography C

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Algorithms for Modular Elliptic Curves. 2nd edition, Cambridge University Press, Cambridge.

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ISBN 0-521-59820-6, MathReview (Joseph H. Silverman), ZentralBlatt

Referenced by:

§23.20(ii).

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12: Bibliography S

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Rational Points on Elliptic Curves. Undergraduate Texts in Mathematics, Springer-Verlag, New York.

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ISBN 0-387-97825-9, MathReview (Andrew Bremner), ZentralBlatt

Referenced by:

§22.18(iv), §23.1, §23.20(ii), §23.20(ii), §23.20(v).

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13: 22.3 Graphics

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Figure 22.3.2: $k=0.7$, $-3K\le x\le 3K$, $K=1.8456\mathrm{\dots }$. For $\mathrm{cn}(x,k)$ the curve for $k=1/\sqrt{2}=0.70710\mathrm{\dots }$ is a boundary between the curves that have an inflection point in the interval $0\le x\le 2K\left(k\right)$, and its translates, and those that do not; see Walker (1996, p. 146). Magnify

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14: 19.25 Relations to Other Functions

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§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals

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§19.25(v) Jacobian Elliptic Functions

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§19.25(vi) Weierstrass Elliptic Functions

… ►The sign on the right-hand side of (19.25.40) will change whenever one crosses a curve on which , for some $j$. …

15: 19.17 Graphics

§19.17 Graphics

►See Figures 19.17.1–19.17.8 for symmetric elliptic integrals with real arguments. … ►

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Figure 19.17.7: Cauchy principal value of $R_J(0.5,y,1,p)$ for $y=0,\mathrm{\hspace{0.17em}0.01},\mathrm{\hspace{0.17em}0.05},\mathrm{\hspace{0.17em}0.2},\mathrm{\hspace{0.17em}1}$, . …As $p\to 0$ the curve for $y=0$ has the finite limit $-8.10386\mathrm{\dots }$; see (19.20.10). Magnify

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Figure 19.17.8: $R_J(0,y,1,p)$, $0\le y\le 1$, $-1\le p\le 2$. … Magnify 3D Help

16: 28.32 Mathematical Applications

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§28.32(i) Elliptical Coordinates and an Integral Relationship

►If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient. … ►Also let $ℒ$ be a curve (possibly improper) such that the quantity …defines a solution of Mathieu’s equation, provided that (in the case of an improper curve) the integral converges with respect to $z$ uniformly on compact subsets of $ℂ$. … ► …

17: 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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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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Transverse cusp diffraction catastrophes produced by the reflection of ultrasonic tone bursts from a curved surface in water. J. Acoust. Soc. Amer. 92 (5), pp. 2869–2877.

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Document

Referenced by:

§36.14(iv).

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Travel time surface of a transverse cusp caustic produced by reflection of acoustical transients from a curved metal surface. J. Acoust. Soc. Amer. 95 (2), pp. 650–660.

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Document

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§36.14(iv).

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18: 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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Computing Riemann matrices of algebraic curves. Phys. D 152/153, pp. 28–46.

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ISSN 0167-2789, Document, MathReview (John B. Little), ZentralBlatt

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3rd item.

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

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

Referenced by:

§23.11.

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

Referenced by:

§19.7(i).

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19: Bibliography H

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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.

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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

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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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Inverse virial symmetry of diatomic potential curves. J. Chem. Phys. 109 (1), pp. 11–19.

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Document

Referenced by:

§15.18.

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20: Bibliography W

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Evaluating elliptic functions and their inverses. Comput. Math. Appl. 39 (3-4), pp. 131–136.

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ISSN 0898-1221, Document, MathReview, ZentralBlatt

Referenced by:

§22.20(ii), §22.20(v).

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Caustics: Mathematical curves generated by light shined through rippled plastic. Scientific American 249, pp. 146–153.

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Referenced by:

§36.14(ii).

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The distribution of the zeros of Jacobian elliptic functions with respect to the parameter $k$ . Comput. Methods Funct. Theory 9 (2), pp. 579–591.

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ISSN 1617-9447, MathReview (M. K. Jain), ZentralBlatt

Referenced by:

§22.4(i).

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Elliptic Functions According to Eisenstein and Kronecker. Classics in Mathematics, Springer-Verlag, Berlin.

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Notes:

Reprint of the 1976 original

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ISBN 3-540-65036-9, MathReview, ZentralBlatt

Referenced by:

§22.12, §23.8(ii).

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Elliptic genera and quantum field theory. Comm. Math. Phys. 109 (4), pp. 525–536.

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ISSN 0010-3616, Document, MathReview (J. Stasheff), ZentralBlatt

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1st item.

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