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 .
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In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory. The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function.
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.
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