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In mathematics, Weierstrass's elliptic functions are elliptic functions that take a particularly simple form; they are named for Karl Weierstrass. This class of functions are also referred to as P-functions and generally written using the symbol ℘ (or ), and known as "Weierstrass P"). The ℘ functions constitute branched double coverings of the Riemann sphere by the torus, ramified at four points. They can be used to parametrize elliptic curves over the complex numbers, thus establishing an equivalence to complex tori. They also yield solutions of the Korteweg–de Vries equation. Symbol for Weierstrass P function ==Definitions== The Weierstrass elliptic function can be defined in three closely related ways, each of which possesses certain advantages. One is as a function of a complex variable ''z'' and a lattice Λ in the complex plane. Another is in terms of ''z'' and two complex numbers ω1 and ω2 defining a pair of generators, or periods, for the lattice. The third is in terms of ''z'' and a modulus in the upper half-plane. This is related to the previous definition by = ω2/ω1, which by the conventional choice on the pair of periods is in the upper half-plane. Using this approach, for fixed ''z'' the Weierstrass functions become modular functions of . In terms of the two periods, Weierstrass's elliptic function is an elliptic function with periods ω1 and ω2 defined as : Then are the points of the period lattice, so that : for any pair of generators of the lattice defines the Weierstrass function as a function of a complex variable and a lattice. If is a complex number in the upper half-plane, then : The above sum is homogeneous of degree minus two, from which we may define the Weierstrass ℘ function for any pair of periods, as : We may compute ℘ very rapidly in terms of theta functions; because these converge so quickly, this is a more expeditious way of computing ℘ than the series we used to define it. The formula here is : There is a second-order pole at each point of the period lattice (including the origin). With these definitions, is an even function and its derivative with respect to ''z'', ℘′, is an odd function. Further development of the theory of elliptic functions shows that the condition on Weierstrass's function is determined up to addition of a constant and multiplication by a non-zero constant by the condition on the poles alone, amongst all meromorphic functions with the given period lattice. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Weierstrass's elliptic functions」の詳細全文を読む スポンサード リンク
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