We demonstrate and elaborate upon the very intimate relationship between the elliptic functions sm and cm of Dixon and the ternary elliptic functions of Du Val. (c) 2025 Elsevier Inc. All rights are reserved, includin...
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We demonstrate and elaborate upon the very intimate relationship between the elliptic functions sm and cm of Dixon and the ternary elliptic functions of Du Val. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
This paper offers a simple description of the elementary properties of the Jacobian elliptic functions and of the Landen transformation, which connects them with the circular and hyperbolic functions and thereby provi...
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This paper offers a simple description of the elementary properties of the Jacobian elliptic functions and of the Landen transformation, which connects them with the circular and hyperbolic functions and thereby provides one of the most accurate methods of evaluating elliptic functions, The use of elliptic functions in creating equal-ripple lowpass filters is explained and their numerical evaluation is illustrated by means of an example, A Fortran program for effecting the design is included and a faster and more accurate replacement for the Matlab program ELLIPAP is given.
We study two elliptic functions to the quintic base and find two nonlinear second order differential equations satisfied by them. We then derive two recurrence relations involving certain Eisenstein series associated ...
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We study two elliptic functions to the quintic base and find two nonlinear second order differential equations satisfied by them. We then derive two recurrence relations involving certain Eisenstein series associated with the group Gamma(0)(5). These recurrence relations allow us to derive infinite families of identities involving the Eisenstein series and Dedekind eta-products. An imaginary transformation for one of the elliptic functions is also derived.
Finite temperature boson and fermion free field theories on the space-time manifolds RxS(d) are discussed with one eye on the questions of temperature inversion symmetry and modular invariance. For conformally invaria...
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Finite temperature boson and fermion free field theories on the space-time manifolds RxS(d) are discussed with one eye on the questions of temperature inversion symmetry and modular invariance. For conformally invariant theories it is shown that the total energy at any temperature for any odd dimension, d, is given as a power series in the d = 3 and d = 5 energies, for scalars, and the d = 1 and d = 3 energies for spinors. Further, these energies can be given in finite terms at specific temperatures associated with singular moduli of elliptic function theory. Some examples are listed and numbers given. (C) 2002 Elsevier Science B.V. All rights reserved.
Scattering amplitudes at loop level can be reduced to a basis of linearly independent Feynman integrals. The integral coefficients are extracted from generalized unitarity cuts which define algebraic varieties. The to...
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Scattering amplitudes at loop level can be reduced to a basis of linearly independent Feynman integrals. The integral coefficients are extracted from generalized unitarity cuts which define algebraic varieties. The topology of an algebraic variety characterizes the difficulty of applying maximal cuts. In this work, we analyze a novel class of integrals of which the maximal cuts give rise to an algebraic variety with irrational irreducible components. As a phenomenologically relevant example, we examine the two-loop planar double-box contribution with internal massive lines. We derive unique projectors for all four master integrals in terms of multivariate residues along with Weierstrass’ elliptic functions. We also show how to generate the leading-topology part of otherwise infeasible integration-by-parts identities analytically from exact meromorphic differential forms.
We consider the class of elliptic functions whose critical points in the Julia set are eventually mapped onto infinity. This paper is a continuation of our previous papers, namely [11] and [12]. We study the geometry ...
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We consider the class of elliptic functions whose critical points in the Julia set are eventually mapped onto infinity. This paper is a continuation of our previous papers, namely [11] and [12]. We study the geometry and ergodic properties of this class of elliptic functions. In particular, we obtain a lower bound on the Hausdorff dimension of the Julia set that is bigger than the estimate proved in [11]. Let h be the Hausdorff dimension of the Julia set of f. We construct an atomless h-conformal measure m and prove the existence of a (unique up to a multiplicative constant) sigma-finite f-invariant measure mu equivalent to m. The measure mu is ergodic and conservative.
Abstract: It is proved that any complete minimal surfaces in ${{\mathbf {R}}^n}(n \geqslant 3)$ with total curvature $- 4\pi$ is conformally equivalent to the complex plane or the punctured plane, just like th...
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Abstract: It is proved that any complete minimal surfaces in ${{\mathbf {R}}^n}(n \geqslant 3)$ with total curvature $- 4\pi$ is conformally equivalent to the complex plane or the punctured plane, just like the case in ${{\mathbf {R}}^3}$.
In this article, we derive the quintuple, Hirschhorn and Winquist product identities using the theory of elliptic functions. Our method can be used to establish generalizations of these identities due to the second au...
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In this article, we derive the quintuple, Hirschhorn and Winquist product identities using the theory of elliptic functions. Our method can be used to establish generalizations of these identities due to the second author.
With the help of the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to obtain the Jacobi doubly periodic wave solutions...
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With the help of the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to obtain the Jacobi doubly periodic wave solutions of the (2+1)-dimensionai B-type Kadomtsev-Petviashvili (BKP) equation and the generalized Klein-Gordon equation. The method is also valid for other (1+1)-dimensional and higher dimensional systems.
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