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On Jacobi’s transformation theory of elliptic functions
Archive for History of Exact Sciences, 2013The modern theory of elliptic functions was created during a historical race between Abel and Jacobi. Whereas the publications of Abel have been studied extensively (see, e.g. [\textit{C. Houzel}, in: The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, 2002. Berlin: Springer.
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1989
The elliptic functions sn u, cn u, and dn u are defined as ratios of theta functions as below: $$sn\;u = \frac{{{\theta _3}(0)}}{{{\theta _2}(0)}}\cdot \frac{{{\theta _1}(z)}}{{{\theta _4}(z)}},$$ (2.1.1) $$cn\,u = \frac{{{\theta _4}(0)}}{{{\theta _2}(0)}}\cdot \frac{{{\theta _2}(z)}}{{{\theta _4}(z)}},$$ (2.1.2) $$dn\;u = \frac{
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The elliptic functions sn u, cn u, and dn u are defined as ratios of theta functions as below: $$sn\;u = \frac{{{\theta _3}(0)}}{{{\theta _2}(0)}}\cdot \frac{{{\theta _1}(z)}}{{{\theta _4}(z)}},$$ (2.1.1) $$cn\,u = \frac{{{\theta _4}(0)}}{{{\theta _2}(0)}}\cdot \frac{{{\theta _2}(z)}}{{{\theta _4}(z)}},$$ (2.1.2) $$dn\;u = \frac{
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A note on the Jacobi elliptic function expansion method
Physics Letters, Section A: General, Atomic and Solid State Physics, 2003zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Shou-Feng Shen
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JACOBI ELLIPTIC FUNCTIONS, NONLINEAR EVOLUTION EQUATIONS AND RECURSION
International Journal of Modern Physics C, 2000Jacobi elliptic functions play a central role in nonlinear evolution equations. We show how with one recursive call we can calculate the Jacobi elliptic functions sn, cn, and dn. We give an implementation using C++.
Hardy, Yorick, Steeb, W.-H., Stoop, R.
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The Rayleigh method with Jacobi elliptic functions
Journal of Sound and Vibration, 1989Exemples d'utilisation de fonctions elliptiques de Jacobi en tant que fonctions d'essai pour la ...
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Jacobi's elliptic functions and Lagrangian immersions
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1996First, we establish a sharp inequality between the squared mean curvature and the scalar curvature for a Lagrangian submanifold in a nonflat complex-space-form. Then, by utilising the Jacobi's elliptic functions en and dn, we introduce three families of Lagrangian submanifolds and two exceptional Lagrangian submanifolds Fn, Ln in nonflat complex-space-
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Elliptic Functions With A View Toward Jacobi Hyperbolic Functions
Global Journal For Research Analysis, 2012In Complex Analysis an elliptic function is a doubly periodic meromorphic function which is determined by its value on a fundamental parallelogram which then repeat in a lattice. Elliptic functions must have at least two poles in a fundamental parallelogram. Jacobi elliptic functions can be expressed in various forms namely as a Trigonometric functions,
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A Final Application in Elasticity with Jacobi Elliptic Functions
2021Last of all, we want to return to the curva elastica, but this time from the engineering point of view. This is after all a problem which arose in statics of deformable bodies, where deflection of beams is of central interest and already caught Galileo’s attention as early as 1638.
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Unveiling the solitons mystery: The Jacobi elliptic functions
American Journal of Physics, 1986Several examples are given in which the soliton solutions belonging to well-known nonlinear differential equations arise as simple limits of the Jacobian elliptic functions. The knowledge of the solutions of a single ordinary nonlinear differential equation is sufficient for the subsequent derivation of soliton solutions.
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On the So-Called “Jacobi Elliptic Cliffordian Functions”
Advances in Applied Clifford Algebras, 2009The aim of this talk is to introduce the analogous of the well-known classical elliptic functions sn, cn and dn in the framework of Clifford Analysis. All the results which will be given below were obtained by G. Laville and the speaker [1].
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