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つづき
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Weierstrass elliptic function
This class of functions is also referred to as ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy script p.
They play an important role in the theory of elliptic functions, i.e., meromorphic functions that are doubly periodic. A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice.
Motivation
A cubic of the form
(Cg2,g3)^C={(x,y)∈C2:y2=4x3－g2x－g3}, where
g2,g3∈C are complex numbers with
(g2)^3－27(g3)^2≠0, cannot be rationally parameterized.[1]
Yet one still wants to find a way to parameterize it.
For the quadric
K={(x,y)∈R2:x2+y2=1}; the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function:
ψ:R/2πZ→K,t↦(sin t,cos t).
Because of the periodicity of the sine and cosine
R/2πZ is chosen to be the domain, so the function is bijective.
In a similar way one can get a parameterization of
(Cg2,g3)^C
by means of the doubly periodic
℘-function (see in the section "Relation to elliptic curves"). This parameterization has the domain
C/Λ, which is topologically equivalent to a torus.[2]
There is another analogy to the trigonometric functions. Consider the integral function
略
It can be simplified by substituting
y=sin t and s=arcsin x:
略
That means a^－1(x)=sin x. So the sine function is an inverse function of an integral function.[3]
Elliptic functions are the inverse functions of elliptic integrals. In particular, let:
略
Then the extension of
u^－1 to the complex plane equals the ℘-function.[4]
This invertibility is used in complex analysis to provide a solution to certain nonlinear differential equations satisfying the Painlevé property, i.e., those equations that admit poles as their only movable singularities.[5]
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