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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.
Graphs of curves y 2 = x 3 − x and y 2 = x 3 − x + 1. Although the formal definition of an elliptic curve requires some background in algebraic geometry, it is possible to describe some features of elliptic curves over the real numbers using only introductory algebra and geometry.
Elliptic curve; Watt's curve; Curves with genus > 1. Bolza surface (genus 2) Klein quartic (genus 3) ... Cardiac function curve; Dose–response curve; Growth curve ...
Fig. 1: The graph of the hyperelliptic curve : = where () = + + = (+) (+) ().. In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus g > 1, given by an equation of the form + = where f(x) is a polynomial of degree n = 2g + 1 > 4 or n = 2g + 2 > 4 with n distinct roots, and h(x) is a polynomial of degree < g + 2 (if the characteristic of the ground field is not 2, one can ...
In a nutshell, an elliptic curve is a special kind of function. They take the unthreatening-looking form y²=x³+ax+b. It turns out functions like this have certain properties that cast insight ...
For an elliptic curve defined over the complex numbers the group is isomorphic to the additive group of the complex plane modulo the period lattice of the corresponding elliptic functions. The intersection of two quadric surfaces is, in general, a nonsingular curve of genus one and degree four, and thus an elliptic curve, if it has a rational ...
Pages in category "Elliptic curves" The following 52 pages are in this category, out of 52 total. ... Hasse's theorem on elliptic curves; Heegner point; Height function;
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.