# Weierstrass, Karl Theodor Wilhelm

## Weierstrass, Karl Theodor Wilhelm

Born Oct. 31, 1815, in Ostenfelde; died Feb. 19, 1897, in Berlin. German mathematician.

Weierstrass studied law in Bonn and mathematics in Münster and was a professor at the University of Berlin from 1856. His investigations were in mathematical analysis, the theory of functions, the calculus of variations, differential geometry, and linear algebra. Weierstrass developed a system for the logical foundation of mathematical analysis based on the theory of real numbers that he had constructed. He systematically used the concepts of upper and lower bounds and the limit point of number sets; he provided a rigorous proof of the fundamental properties of functions continuous over a closed interval and introduced into general use the concept of the uniform convergence of a series of functions. Weierstrass’ predecessor in these studies was the Czech mathematician B. Bolzano. Weierstrass devised an example of a continuous function without a derivative at any one point; he proved the possibility of a polynomial approximation (to any assigned degree of accuracy) of an arbitrary function over a closed interval. A central place in his works is occupied by the theory of analytic functions, at the basis of which Weierstrass placed power series. Weierstrass was also involved in studying the behavior of analytic functions in the neighborhood of an isolated, singular point, the construction of a theory of analytic continuation, formulating a theorem concerning the analytic sum of a uniformly converging series of analytic functions, the expansion of entire functions into infinite products, the fundamentals of a theory of analytic functions of many variables, a new construction of the theory of elliptic functions, the theory of algebraic functions, and Abelian integrals. His investigations in the calculus of variations include the study of the sufficient conditions for the extremum of an integral (Weierstrass’ condition), construction of a calculus of variations for the case of the parametric representation of functions, and a study of “discontinuous” solutions in problems of the calculus of variations. In differential geometry Weierstrass studied geodetic lines (the shortest lines on a surface) and minimum surfaces (the surfaces of a minimum area, stretched on a given boundary). In linear algebra he is responsible for the construction of the theory of elementary divisors.

### WORKS

*Mathematische Werke*, vols. 1-7. Berlin-Leipzig, 1894-1927.

*Formeln und Lehrsätze zum Gebrauche der elliptischen Functionen*, 2nd ed, part 1. Edited and published by H. A. Schwarz. Berlin, 1893.

### REFERENCE

Klein, F.*Lektsii o razvitii matematiki v 19 stoletii*, part 1. Moscow-Leningrad, 1937. (Translated from German.)