Karl Weierstrass

Karl Weierstrass (born Oct. 31, 1815, Ostenfelde, Bavaria [Germany]—died Feb. 19, 1897, Berlin) was a German mathematician, one of the founders of the modern theory of functions.

His domineering father sent him to the University of Bonn at age 19 to study law and finance in preparation for a position in the Prussian civil service. Weierstrass pursued four years of intensive fencing and drinking and returned home with no degree. He then entered the Academy of Münster in 1839 to prepare for a career as a secondary school teacher. At Münster he came under the influence of Cristof Gudermann, professor of mathematics, who was particularly interested in the theory of elliptic functions. Gudermann cultivated Weierstrass’s interest in the theory of functions with emphasis on the expansion of functions by power series.

In 1841 Weierstrass obtained his teacher’s certificate and began a 14-year career as a teacher of mathematics at the Pro-Gymnasium in Deutsche Krone (1842–48) and at the Collegium Hoseanum in Braunsberg (1848–56). During this time of isolation from other mathematicians—his salary was so small that he could not even correspond with his fellows—Weierstrass worked unceasingly on analysis. He conceived and in large part carried out a program known as the arithmetization of analysis, under which analysis is based on a rigorous development of the real number system. His preoccupation with rigour in mathematics is illustrated by his later development (1861) of a function that, though continuous, had no derivatives at any point. This idiosyncrasy of an apparently differentiable function caused consternation among the school of analysts who depended heavily upon intuition.

Weierstrass’s work on the theory of functions was guided by his desire to complete the work begun by Niels Abel of Norway and Carl Jacobi of Prussia, primarily Abel’s theorem that the number of independent integrals of algebraic functions is finite and Jacobi’s discovery of multiple periodic functions of many variables.

In 1854 Weierstrass burst from obscurity when his unexpected memoir on Abelian functions was published in Crelle’s Journal. The University of Königsberg conferred upon him an honorary doctor’s degree, and in 1856 a position was found for him at the Royal Polytechnic School in Berlin. Weierstrass contributed few papers to scholarly journals; his work was embodied in his lectures, which were collected in Gesammelte Abhandlungen, 8 vol. (1894–1927; “Collected Works”).

Known as the father of modern analysis, Weierstrass devised tests for the convergence of series and contributed to the theory of periodic functions, functions of real variables, elliptic functions, Abelian functions, converging infinite products, and the calculus of variations. He also advanced the theory of bilinear and quadratic forms. His greatest influence was felt through his students (among them Sofya Kovalevskaya), many of whom became creative mathematicians.