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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 .
In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is also an example of a fractal curve .
In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function.
They are named elliptic functions because they come from elliptic integrals. Those integrals are in turn named elliptic because they first were encountered for the calculation of the arc length of an ellipse. Important elliptic functions are Jacobi elliptic functions and the Weierstrass ℘-function.
with integral coefficients , reducing the coefficients modulo p defines an elliptic curve over the finite field F p (except for a finite number of primes p, where the reduced curve has a singularity and thus fails to be elliptic, in which case E is said to be of bad reduction at p). The zeta function of an elliptic curve over a finite field F p ...
(The Weierstrass gap theorem or Lückensatz is the statement that there must be gaps.) For hyperelliptic curves , for example, we may have a function F {\displaystyle F} with a double pole at P {\displaystyle P} only.
The Stone–Weierstrass theorem can be used to prove the following two statements, which go beyond Weierstrass's result. If f is a continuous real-valued function defined on the set [a, b] × [c, d] and ε > 0, then there exists a polynomial function p in two variables such that | f (x, y) − p(x, y) | < ε for all x in [a, b] and y in [c, d].
In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point P.It states that such a function is, up to multiplication by a function not zero at P, a polynomial in one fixed variable z, which is monic, and whose coefficients of lower degree terms are analytic functions in the remaining variables and zero at P.