In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation for . The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by Carl Gustav Jakob Jacobi (1829). Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular,[1] but his work was published much later.
^Armitage, J. V.; Eberlein, W. F. (2006). Elliptic Functions (First ed.). Cambridge University Press. ISBN 978-0-521-78078-0. p. 48
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In mathematics, the Jacobiellipticfunctions are a set of basic ellipticfunctions. They are found in the description of the motion of a pendulum (see...
to ellipticfunctions, dynamics, differential equations, determinants, and number theory. His name is sometimes given as Karl Gustav Jakob. Jacobi was...
Weierstrass ellipticfunctions are ellipticfunctions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also...
work on ellipticfunctions that was actually published. Abel's work on ellipticfunctions also influenced Jacobi's studies of ellipticfunctions, whose...
the approximation problem given by Zolotarev was in terms of Jacobiellipticfunctions. Zolotarev gave the general solution where the number of zeroes...
trigonometric functions sine and cosine and the Jacobiellipticfunctions sn and cn; Göran Dillner described them earlier in 1873. The functions sm and cm...
equations Jacobi eigenvalue algorithm, a method for calculating the eigenvalues and eigenvectors of a real symmetric matrix Jacobiellipticfunctions, a set...
mathematics the elliptic rational functions are a sequence of rational functions with real coefficients. Elliptic rational functions are extensively used...
(w,1/\xi )} where cd() is the Jacobielliptic cosine function and using the definition of the elliptic rational functions yields: 1 + ϵ 2 c d 2 ( n w K...
_{3}^{2}(0|\tau )} . The Neville theta functions are related to the Jacobiellipticfunctions. If pq(u,m) is a Jacobiellipticfunction (p and q are one of s,c,n,d)...
Ellipticfunctions: The inverses of elliptic integrals; used to model double-periodic phenomena. Jacobi'sellipticfunctions Weierstrass's elliptic functions...
Foundations of the Theory of EllipticFunctions) is a treatise on ellipticfunctions by German mathematician Carl Gustav Jacob Jacobi. The book was first published...
field theory. The most common form of theta function is that occurring in the theory of ellipticfunctions. With respect to one of the complex variables...
function with just one zero. Ellipticfunction Abel ellipticfunctionsJacobiellipticfunctions Weierstrass ellipticfunctions Lemniscate elliptic functions...
Protocol Symmetric group or Sn n-sphere or Sn sn (ellipticfunction), one of Jacobi'sellipticfunctions SN, METAR code for snow Spotter Network, a system...
Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product...
Families of Exact Sums of Squares Formulas, JacobiEllipticFunctions, Continued Fractions, and Schur Functions. Springer Science & Business Media. p. 9...
called elliptic modular forms to emphasize the point) are related to elliptic curves. Jacobi forms are a mixture of modular forms and ellipticfunctions. Examples...
Korteweg–de Vries equation. These solutions are in terms of the Jacobiellipticfunction cn, which is why they are coined cnoidal waves. They are used to...
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