Weierstrass functions information

necessitated infinitely jagged functions (nowadays known as fractal curves). In Weierstrass's original paper, the function was defined as a Fourier series:...

mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are named...

mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class...

the Bolzano–Weierstrass theorem, and used the latter to study the properties of continuous functions on closed bounded intervals. Weierstrass was born into...

be evaluated in terms of the gamma function as well. Due to the Weierstrass factorization theorem, analytic functions can be written as infinite products...

continuous function on a closed and bounded set obtains its extreme values The Weierstrass–Casorati theorem describes the behavior of holomorphic functions near...

ellipse. Important elliptic functions are Jacobi elliptic functions and the Weierstrass ℘ {\displaystyle \wp } -function. Further development of this...

plane follows naturally from a curious property of Weierstrass's elliptic functions. These functions and their first derivative are related by the formula...

analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions...

Lindemann–Weierstrass theorem, the hyperbolic functions have a transcendental value for every non-zero algebraic value of the argument. Hyperbolic functions were...

In mathematics, the Weierstrass transform of a function f : R → R, named after Karl Weierstrass, is a "smoothed" version of f(x) obtained by averaging...

particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite)...

\sin } . The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions...

functions: The inverses of elliptic integrals; used to model double-periodic phenomena. Jacobi's elliptic functions Weierstrass's elliptic functions Lemniscate...

the Weierstrass sigma function, which is quasiperiodic in two independent quasiperiods, the periods of the corresponding Weierstrass ℘ function. Functions...

theorem Weierstrass coordinates Weierstrass's elliptic functions Weierstrass equation Weierstrass factorization theorem Weierstrass function Weierstrass functions...

meromorphic function), then for entire functions there is a generalization of the factorization — the Weierstrass theorem on entire functions. Every entire...

traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is...

where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between...

is continuous everywhere but differentiable nowhere is the Weierstrass function. A function f {\textstyle f} is said to be continuously differentiable...

by sigma function one can mean one of the following: The sum-of-divisors function σa(n), an arithmetic function Weierstrass sigma function, related to...

In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point P. It...

of the Weierstrass factorization theorem. Following from the infinite product definitions for the gamma function, due to Euler and Weierstrass respectively...

to define the Weierstrass transform. Gaussian functions arise by composing the exponential function with a concave quadratic function: f ( x ) = exp...

holomorphic function is constant, while there always exist non-constant meromorphic functions. Cousin problems Mittag-Leffler's theorem Weierstrass factorization...

In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces...