Weierstrass

Weierstrass is a German surname most closely associated with the 19th‑century mathematician Karl Theodor Wilhelm Weierstrass (1815–1897). A foundational figure in analysis, he helped establish rigorous approaches to real and complex functions and influenced the modern theory of mathematical analysis. His work emphasized precise definitions, proofs, and the systematic study of functions, culminating in a number of results and concepts that bear his name.

Several key contributions are named after Weierstrass. The Weierstrass approximation theorem states that every continuous function on a closed interval can be uniformly approximated by polynomials, a central result in approximation theory. The Weierstrass factorization (or Weierstrass product) theorem provides canonical infinite product representations for entire functions based on their zeros, enabling a structured factorization theory for holomorphic functions. The Weierstrass preparation theorem, important in several complex variables and analytic geometry, describes how certain holomorphic functions can locally factor into a unit and a polynomial.

In complex analysis and the theory of elliptic functions, Weierstrass introduced the Weierstrass elliptic functions, notably the p-function, which are doubly periodic meromorphic functions used to construct and analyze elliptic curves. The general Weierstrass form of an elliptic curve, often written as y^2 = x^3 + ax + b, is named after him and remains a standard representation in algebraic geometry and number theory. Weierstrass’s influence extends through the rigorous foundation of analysis, shaping modern approaches to both real and complex function theory.