Handbook of Elliptic Integrals for Engineers and Scientists

Byrd, Paul F.; Friedman, Morris D.

doi:10.1007/978-3-642-65138-0

Cited by 1,654 publications

(1,325 citation statements)

References 0 publications

Supporting

Mentioning

1,291

Contrasting

Unclassified

Order By: Relevance

“…Explicit calculation gives [35] K 1 = iX (−α 6 + 2α 4 + α 2 k 2 − 2α 4 k 2 )E(k) − (k 2 − α 2 )(α 4 − 2α 2 + k 2 )K(k) + (α 8 − 4α 6 k 2 + 6α 4 k 2 − 4α 2 k 2 + k 4 )Π(α 2 , k 2 ) K 2 = iX −α 2 (α 4 − 2α 2 − 2α 2 k 2 + 3k 2 )E(k) − (k 2 − α 2 )(α 4 − 2α 2 + 4α 2 k 2 − 3k 2 )K(k) + (α 8 − 6α 4 k 2 + 4α 2 k 4 + 4α 2 k 2 − 3k 4 )Π(α 2 , k 2 )…”

Section: Comparison With Instanton Computationsunclassified

Gravitational corrections in supersymmetric gauge theory and matrix models

Klemm¹,

Mariño²,

Theisen³

2003

J. High Energy Phys.

143

208

View full text Add to dashboard Cite

Gravitational corrections in N = 1 and N = 2 supersymmetric gauge theories are obtained from topological string amplitudes. We show how they are recovered in matrix model computations. This provides a test of the proposal by Dijkgraaf and Vafa beyond the planar limit. Both, matrix model and topological string theory, are used to check a conjecture of Nekrasov concerning these gravitational couplings in Seiberg-Witten theory. Our analysis is performed for those gauge theories which are related to the cubic matrix model, i.e. pure SU (2) SeibergWitten theory and N = 2 U (N ) SYM broken to N = 1 via a cubic superpotential. We outline the computation of the topological amplitudes for the local Calabi-Yau manifolds which are relevant for these two cases.

show abstract

Section: Comparison With Instanton Computationsunclassified

Gravitational corrections in supersymmetric gauge theory and matrix models

Klemm¹,

Mariño²,

Theisen³

2003

J. High Energy Phys.

143

208

View full text Add to dashboard Cite

show abstract

“…All integrals are evaluated using the tables in [38]. Suppose E 0 < E 1 < E 2 < E 3 are given and R 1/2 4 (z) is defined as usual.…”

Section: The Elliptic Case Genus Onementioning

confidence: 99%

“…For the explicit calculations in the genus one case, I have used the tables for elliptic integrals in [38]. In addition, you might also find [10], [123], and [127] useful.…”

Section: Notes On Literaturementioning

confidence: 99%

See 1 more Smart Citation

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

324

349

View full text Add to dashboard Cite

Abstract. This book is intended to serve both as an introduction and a reference to spectral and inverse spectral theory of Jacobi operators (i.e., second order symmetric difference operators) and applications of these theories to the Toda and Kac-van Moerbeke hierarchy.Starting from second order difference equations we move on to self-adjoint operators and develop discrete Weyl-Titchmarsh-Kodaira theory, covering all classical aspects like Weyl m-functions, spectral functions, the moment problem, inverse spectral theory, and uniqueness results. Next, we investigate some more advanced topics like locating the essential, absolutely continuous, and discrete spectrum, subordinacy, oscillation theory, trace formulas, random operators, almost periodic operators, (quasi-)periodic operators, scattering theory, and spectral deformations.Then, the Lax approach is used to introduce the Toda hierarchy and its modified counterpart, the Kac-van Moerbeke hierarchy. Uniqueness and existence theorems for the initial value problem, solutions in terms of Riemann theta functions, the inverse scattering transform, Bäcklund transformations, and soliton solutions are discussed.Corrections and complements to this book are available from:

show abstract

“…The length of the arc of a meridian s m between the equator u = 0 and the parallel u = u i on a triaxial ellipsoid described by equation (1) If a > b > c then n m 2 > 0 and according to (Byrd and Friedmann, 1954) the integral I 1m can be presented in the form of: If a > b then n p 2 > 0 and the integral I 1p , similarly as for a meridian can be presented in the form of (Byrd and Friedmann, 1954 Finally, the length of the parallel's arc can be expressed using the equation:…”

Section: Derivation Of Equations For the Lengths Of Meridians Betweenmentioning

confidence: 99%

Equidistant map projections of a triaxial ellipsoid with the use of reduced coordinates

Pędzich¹

2017

Geodesy and Cartography

View full text Add to dashboard Cite

Abstract:The paper presents a new method of constructing equidistant map projections of a triaxial ellipsoid as a function of reduced coordinates. Equations for x and y coordinates are expressed with the use of the normal elliptic integral of the second kind and Jacobian elliptic functions. This solution allows to use common known and widely described in literature methods of solving such integrals and functions. The main advantage of this method is the fact that the calculations of x and y coordinates are practically based on a single algorithm that is required to solve the elliptic integral of the second kind. Equations are provided for three types of map projections: cylindrical, azimuthal and pseudocylindrical. These types of projections are often used in planetary cartography for presentation of entire and polar regions of extraterrestrial objects. The paper also contains equations for the calculation of the length of a meridian and a parallel of a triaxial ellipsoid in reduced coordinates. Moreover, graticules of three coordinates systems (planetographic, planetocentric and reduced) in developed map projections are presented. The basic properties of developed map projections are also described. The obtained map projections may be applied in planetary cartography in order to create maps of extraterrestrial objects.

show abstract

Handbook of Elliptic Integrals for Engineers and Scientists

Cited by 1,654 publications

References 0 publications

Gravitational corrections in supersymmetric gauge theory and matrix models

Gravitational corrections in supersymmetric gauge theory and matrix models

Jacobi Operators and Completely Integrable Nonlinear Lattices

Equidistant map projections of a triaxial ellipsoid with the use of reduced coordinates

Contact Info

Product

Resources

About