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The fundamental rectangle in the complex plane of . There are twelve Jacobi elliptic functions denoted by (,), where and are any of the letters , , , and . (Functions of the form (,) are trivially set to unity for notational completeness.) is the argument, and is the parameter, both of which may be complex.
Fundamenta nova theoriae functionum ellipticarum [1] (from Latin: New Foundations of the Theory of Elliptic Functions) is a treatise on elliptic functions by German mathematician Carl Gustav Jacob Jacobi. [2] The book was first published in 1829, and has been reprinted in volume 1 of his collected works and on several later occasions.
They are named elliptic functions because they come from elliptic integrals. Those integrals are in turn named elliptic because they first were encountered for the calculation of the arc length of an ellipse. Important elliptic functions are Jacobi elliptic functions and the Weierstrass ℘-function.
The incomplete elliptic integral of the first kind F is defined as (,) = = (;) = .This is Legendre's trigonometric form of the elliptic integral; substituting t = sin θ and x = sin φ, one obtains Jacobi's algebraic form:
In mathematics, specifically the theory of elliptic functions, the nome is a special function that belongs to the non-elementary functions. This function is of great importance in the description of the elliptic functions, especially in the description of the modular identity of the Jacobi theta function, the Hermite elliptic transcendents and the Weber modular functions, that are used for ...
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Created Date: 8/30/2012 4:52:52 PM
This method of inversion, and its subsequent extension by Weierstrass and Riemann to arbitrary algebraic curves, may be seen as a higher genus generalization of the relation between elliptic integrals and the Jacobi or Weierstrass elliptic functions. Carl Gustav Jacob Jacobi. Jacobi was the first to apply elliptic functions to number theory ...