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Concept
Weierstrass elliptic function
Summary
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 are 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. 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.
Symbol for Weierstrass -function
Let be two complex numbers that are linearly independent over and let be the lattice generated by those numbers. Then the -function is defined as follows:
This series converges locally uniformly absolutely in . Oftentimes instead of only is written.
The Weierstrass -function is constructed exactly in such a way that it has a pole of the order two at each lattice point.
Because the sum alone would not converge it is necessary to add the term .
It is common to use and in the upper half-plane as generators of the lattice. Dividing by maps the lattice isomorphically onto the lattice with . Because can be substituted for , without loss of generality we can assume , and then define .
A cubic of the form , where are complex numbers with , cannot be rationally parameterized. Yet one still wants to find a way to parameterize it.
For the quadric , the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function:
Because of the periodicity of the sine and cosine is chosen to be the domain, so the function is bijective.
In a similar way one can get a parameterization of by means of the doubly periodic -function (see in the section "Relation to elliptic curves"). This parameterization has the domain , which is topologically equivalent to a torus.
There is another analogy to the trigonometric functions. Consider the integral function
It can be simplified by substituting and :
That means .
Official source
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