Weierstrass theorem
Encyclopedia
Several theorems are named after Karl Weierstrass
. These include:
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....
. These include:
- The Weierstrass approximation theorem, of which one well known generalization is the Stone–Weierstrass theorem
- The Bolzano–Weierstrass theoremBolzano–Weierstrass theoremIn real analysis, the Bolzano–Weierstrass theorem is a fundamental result about convergence in a finite-dimensional Euclidean space Rn. The theorem states thateach bounded sequence in Rn has a convergent subsequence...
, which ensures compactness of closed and bounded sets in Rn - The Weierstrass extreme value theoremExtreme value theoremIn calculus, the extreme value theorem states that if a real-valued function f is continuous in the closed and bounded interval [a,b], then f must attain its maximum and minimum value, each at least once...
, which states that a continuous function on a closed and bounded set obtains its extreme values - The Weierstrass–Casorati theoremWeierstrass–Casorati theoremIn complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of meromorphic functions near essential singularities...
describes the behavior of holomorphic functions near essential singularities - The Weierstrass preparation theoremWeierstrass preparation theoremIn mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point P...
describes the behavior of analytic functions near a specified point - The Lindemann–Weierstrass theoremLindemann–Weierstrass theoremIn mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states that if 1, ..., are algebraic numbers which are linearly independent over the rational numbers ', then 1, ..., are algebraically...
concerning the transcendental numbers - The Weierstrass factorization theoremWeierstrass factorization theoremIn mathematics, the Weierstrass factorization theorem in complex analysis, named after Karl Weierstrass, asserts that entire functions can be represented by a product involving their zeroes...
asserts that entire functions can be represented by a product involving their zeroes - The Sokhatsky–Weierstrass theorem which helps evaluate certain Cauchy-type integrals