Karl Weierstrass

Karl Weierstrass (31 October 1815–19 February 1897) was a German mathematician and a major figure in the development of modern mathematical analysis. His work combined rigorous foundations with research on complex analysis, elliptic and Abelian functions, and approximation. Particularly influential were his treatment of functions through convergent power series, his emphasis on precise definitions and proofs, and his example of a continuous function that is nowhere differentiable. Much of his influence spread through his teaching in Berlin rather than through promptly published papers. (mathshistory.st-andrews.ac.uk)

Life and career

Weierstrass was born in Ostenfelde, Westphalia, now part of Germany. In 1834 he entered the University of Bonn, where his father intended him to prepare for an administrative career by studying law and finance. He left without completing a degree. In 1839 he enrolled at the academy in Münster, studying with Christoph Gudermann, whose teaching on elliptic functions strongly influenced his mathematical development. After qualifying as a teacher, he worked in secondary schools in Deutsch Krone from 1842 and Braunsberg from 1848, pursuing research alongside extensive teaching duties. (mathshistory.st-andrews.ac.uk)

His 1854 paper Zur Theorie der Abelschen Functionen attracted international attention. It addressed the inversion of hyperelliptic integrals and brought him an honorary doctorate from the University of Königsberg. In 1856 he moved to Berlin, accepting appointments at the Industry Institute and subsequently the university. He became a full university professor in 1864. (mathshistory.st-andrews.ac.uk)

Rigorous foundations of analysis

Weierstrass helped establish an approach often called the arithmetization of analysis: grounding arguments about continuity and convergence in precise numerical conditions rather than relying on geometric intuition. His lectures developed a construction of the real numbers and emphasized explicit inequalities and the order of quantifiers in definitions involving limits. He was especially influential in establishing the epsilon–delta style of reasoning used in modern analysis. (math.nist.gov)

For example, the modern definition of continuity at a point $a$ states that, for every $\varepsilon>0$, there exists $\delta>0$ such that

The definition makes the relationship between a permitted output error and a sufficiently small input change explicit. This precision also clarifies distinctions between types of convergence. In pointwise convergence, the required stage of approximation may depend on the point; in uniform convergence, one stage works throughout the domain. Such distinctions determine when limiting operations preserve continuity or may be interchanged with other operations. (math.nist.gov)

Continuous but nowhere differentiable functions

In 1872 Weierstrass presented an example of a continuous function with no finite derivative at any point. It demonstrated that continuity does not imply differentiability, even on a small subinterval, and challenged assumptions encouraged by familiar smooth curves. (mathshistory.st-andrews.ac.uk)

The importance of the example lies in separating properties that visual intuition can easily conflate. A function may have no breaks while still failing to possess a well-defined tangent slope anywhere. Weierstrass’s construction thus made a foundational distinction concrete: conclusions about differentiation require hypotheses beyond continuity alone. (mathshistory.st-andrews.ac.uk)

Uniform convergence and the M-test

The Weierstrass M-test supplies a sufficient condition for uniform and absolute convergence of an infinite series of functions. If functions $f_n$ on a set $E$ satisfy

where the nonnegative numerical series $\sum_n M_n$ converges, then $\sum_n f_n(x)$ converges uniformly and absolutely on $E$. The essential feature is that the same bounds apply at every point of the domain. (maths.ox.ac.uk)

This criterion makes complicated functional convergence questions accessible through simpler numerical estimates. In particular, if all the summands are continuous, uniform convergence ensures that their sum is continuous. For complex analytic functions, corresponding convergence arguments also support the construction of analytic functions from series. (maths.ox.ac.uk)

Polynomial approximation

The Weierstrass approximation theorem, published in 1885, states that every continuous real-valued function on a finite closed interval can be approximated uniformly by polynomials. Equivalently, for every continuous $f:[a,b]\to\mathbb R$ and every $\varepsilon>0$, there is a polynomial $p$ such that

The theorem concerns simultaneous control of the approximation error over the whole interval, not merely agreement at selected points. (history-of-approximation-theory.com)

It does not assert that the function itself is a polynomial or that it equals its Taylor series. Its hypotheses require only continuity, not differentiability. Consequently, polynomial approximation reaches functions much less regular than those described by convergent Taylor expansions. (ocw.mit.edu)

Complex and elliptic functions

Weierstrass’s approach to analytic functions placed convergent power-series representations at the center of the theory. His research on Abelian functions, including the inversion of hyperelliptic integrals, was central to his early recognition and remained a major subject of his Berlin teaching. His lecture program also covered applications of elliptic functions to geometry and mechanics, as well as the calculus of variations. (mathshistory.st-andrews.ac.uk)

A prominent construction is the Weierstrass elliptic function, written $\wp$. It is a meromorphic function associated with a lattice in the complex plane and has two independent periods. Its defining series includes compensating terms that secure convergence. The function provides a fundamental analytic description of complex tori and elliptic curves, connecting function theory with geometry. (maths.ox.ac.uk)

Teaching and publication

Weierstrass’s lectures attracted an international audience and influenced mathematicians including Felix Klein, Hermann Amandus Schwarz, Gösta Mittag-Leffler, and Hermann Minkowski. With Ernst Eduard Kummer, he organized a mathematical seminar in Berlin in 1861. Lecture transcripts and the subsequent work of his students helped disseminate his methods. (mathshistory.st-andrews.ac.uk)

He privately taught Sofia Kovalevskaya because she was denied admission to university courses in Berlin. His reports and support contributed to her receiving a doctorate from Göttingen in 1874. (mathshistory.st-andrews.ac.uk)

Weierstrass repeatedly revised his work before publication. The first two volumes of his collected Mathematische Werke appeared in 1894 and 1895; further volumes, incorporating substantial material from his lectures, were published after his death. (mathshistory.st-andrews.ac.uk)

References

1. Karl Weierstrass (1815–1897) — Biography — MacTutor History of Mathematicsmathshistory.st-andrews.ac.uk
2. Karl Weierstrass — Dictionary of Scientific Biographymathshistory.st-andrews.ac.uk
3. Karl Weierstrass — NIST historical essaymath.nist.gov
4. Lecture Notes and Readings — Real Analysis — MIT OpenCourseWareocw.mit.edu
5. History of Approximation Theory: Historical Papershistory-of-approximation-theory.com
6. B3.2 Geometry of Surfaces — Alexander F. Rittermaths.ox.ac.uk
7. Sofia Kovalevskaya (1850–1891) — Biography — MacTutor History of Mathematicsmathshistory.st-andrews.ac.uk