In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. 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 C g 2 , g 3 C = { ( x , y ) ∈ C 2 : y 2 = 4 x 3 − g 2 x − g 3 } {\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }=\{(x,y)\in \mathbb {C} ^{2}:y^{2}=4x^{3}-g_{2}x-g_{3}\}} , where g 2 , g 3 ∈ C {\displaystyle g_{2},g_{3}\in \mathbb {C} } are complex numbers with g 2 3 − 27 g 3 2 ≠ 0 {\displaystyle g_{2}^{3}-27g_{3}^{2}\neq 0} , cannot be rationally parameterized. Yet one still wants to find a way to parameterize it. For the quadric K = { ( x , y ) ∈ R 2 : x 2 + y 2 = 1 } {\displaystyle K=\left\{(x,y)\in \mathbb {R} ^{2}:x^{2}+y^{2}=1\right\}} ; the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function:
ψ : R / 2 π Z → K , t ↦ ( sin t , cos t ) . {\displaystyle \psi :\mathbb {R} /2\pi \mathbb {Z} \to K,\quad t\mapsto (\sin t,\cos t).}
Because of the periodicity of the sine and cosine R / 2 π Z {\displaystyle \mathbb {R} /2\pi \mathbb {Z} } is chosen to be the domain, so the function is bijective. In a similar way one can get a parameterization of C g 2 , g 3 C {\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }} by means of the doubly periodic ℘ {\displaystyle \wp } -function and its derivative, namely via ( x , y ) = ( ℘ ( z ) , ℘ ′ ( z ) ) {\displaystyle (x,y)=(\wp (z),\wp '(z))} . This parameterization has the domain C / Λ {\displaystyle \mathbb {C} /\Lambda } , which is topologically equivalent to a torus. There is another analogy to the trigonometric functions. Consider the integral function
a ( x ) = ∫ 0 x d y 1 − y 2 . {\displaystyle a(x)=\int _{0}^{x}{\frac {dy}{\sqrt {1-y^{2}}}}.}
It can be simplified by substituting y = sin t {\displaystyle y=\sin t} and s = arcsin x {\displaystyle s=\arcsin x} :
a ( x ) = ∫ 0 s d t = s = arcsin x . {\displaystyle a(x)=\int _{0}^{s}dt=s=\arcsin x.}
That means a − 1 ( x ) = sin x {\displaystyle a^{-1}(x)=\sin x} . So the sine function is an inverse function of an integral function. Elliptic functions are the inverse functions of elliptic integrals. In particular, let:
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