VI.44 Karl Weierstrass
b. Ostenfelde, Germany, 1815; d. Berlin, 1897
Analysis
Weierstrass began his career studying finance and administration at the University of Bonn but his real interest was mathematics and he did not complete his course. He qualified as a teacher and taught in gymnasia for fourteen years. The turning point in his life occurred when, at the age of almost forty, he published a ground-breaking paper on Abelian functions, in which he solved the problem of inversion of hyperelliptic integrals. Shortly afterward he was offered a position at the University of Berlin. He demanded of himself the very strictest standards, with the result that he published little. His ideas, and his reputation, spread through his excellent lectures, which drew students and established mathematicians from around the world.
Weierstrass has been described as the “father of modern analysis.” He contributed to all branches of the subject: calculus, differential and integral equations, the CALCULUS OF VARIATIONS [III.94], infinite series, elliptic and Abelian functions, and real and complex analysis. His work is characterized by attention to foundations and by scrupulous logical reasoning. “Weierstrassian rigor” has come to denote rigor of the strictest standard.
Calculus in the seventeenth and eighteenth centuries was heuristic, lacking logical foundations. The nineteenth century ushered in a rigorous spirit in mathematics which included an examination of the foundation of various fields of mathematics. CAUCHY [VI.29] initiated this process in calculus in the 1820s. But there were several major foundational problems with his approach: verbal definitions of limit and continuity; frequent use of infinitesimals; and intuitive appeal to geometry in proving the existence of various limits.
Weierstrass and DEDEKIND [VI.50] (among others) determined to remedy this unsatisfactory situation, and set themselves the goal of establishing theorems in a “purely arithmetic” manner, as Dedekind put it. To that end, Weierstrass gave precise ε-δ definitions of LIMIT [I.3 §5.1] and CONTINUITY [I.3 §5.2] (those we still use today), thus banishing infinitesimals from analysis (until ROBINSON [VI.95] some hundred years later). He also defined the real numbers based on the rationals (although Dedekind’s and CANTOR’S [VI.54] approaches proved more accessible). He was thereby largely responsible for the “arithmetization of analysis” (a term coined by KLEIN [VI.5 7]). Among his remarkable contributions to real analysis are his introduction of uniform convergence (introduced independently by P. L. Seidel) and his example of an everywhere-continuous and nowhere-differentiable function (Cauchy and his contemporaries believed that a continuous function was differentiable except possibly at isolated points).
Both RIEMANN [VI.49] and Weierstrass (succeeding Cauchy) founded complex function theory, but they had fundamentally different approaches to the subject. Riemann’s global, geometric conception was based on the notion of a RIEMANN SURFACE [III.79] and on the DIRICHLET PRINCIPLE [IV.12 §3.5], while Weierstrass’s local algebraic theory was grounded in power series and ANALYTIC CONTINUATION [I.3 §5.6]. “The more I ponder the principles of function theory—and I do so incessantly—the more I am convinced that it must be founded on simple algebraic truths. . . ,” he asserted in a letter to H. A. Schwartz. He severely criticized the Dirichlet principle for being mathematically not well-grounded, and produced a counterexample, after which his approach to complex analysis became dominant until the early twentieth century. Klein commented on Weierstrass’s general approach to mathematics: “[He] is first of all a logician; he proceeds slowly, systematically, step-by-step. When he works, he strives for the definitive form.”
Weierstrass’s name is attached to various concepts and results, among them the Weierstrass approximation theorem, which says that a continuous function can be uniformly approximated by polynomials; the Bolzano-Weierstrass theorem, which states that every infinite, bounded set of real numbers has a limit point; the Weierstrass factorization theorem, which gives the representation of an entire function in terms of an infinite product of “prime functions”; the Casorati-Weierstrass theorem, which says that in every neighborhood of an isolated essential singularity an analytic function takes values arbitrarily close to any assigned complex number; the Weierstrass M-test, which deals with the comparison of series for convergence; and the Weierstrass p-function, an example of an ELLIPTIC FUNCTION [V.31] of order 2.
Weierstrass was most proud of his work on Abelian functions, and much of his fame in the nineteenth century rested on it. His results in this field are, however, less significant today. For us, his main legacy is his unrelenting insistence on maintaining high standards of rigor and seeking the fundamental ideas underlying mathematical concepts and theories.
Further Reading
Bottazzini, U. 1986. The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass. New York: Springer.