Karl Theodor Wilhelm Weierstrass (Weierstraß)
31 October 1815|
Ostenfelde, Province of Westphalia, Kingdom of Prussia
|Died||19 February 1897
Berlin, Province of Brandenburg, Kingdom of Prussia
|Alma mater||University of Bonn
|Doctoral advisor||Christoph Gudermann|
|Doctoral students||Nikolai Bugaev
Hans von Mangoldt
|Known for||Weierstrass function|
|Notable awards||Copley Medal (1895)|
||This article's lead section may not adequately summarize key points of its contents. (June 2014)|
Weierstrass was the son of Wilhelm Weierstrass, a government official, and Theodora Vonderforst. His interest in mathematics began while he was a Gymnasium student at Theodorianum in Paderborn. He was sent to the University of Bonn upon graduation to prepare for a government position. Because his studies were to be in the fields of law, economics, and finance, he was immediately in conflict with his hopes to study mathematics. He resolved the conflict by paying little heed to his planned course of study, but continued private study in mathematics. The outcome was to leave the university without a degree. After that he studied mathematics at the University of Münster (which was even at this time very famous for mathematics) and his father was able to obtain a place for him in a teacher training school in Münster. Later he was certified as a teacher in that city. During this period of study, Weierstrass attended the lectures of Christoph Gudermann and became interested in elliptic functions. In 1843 he taught in Deutsch-Krone in Westprussia and since 1848 he taught at the Lyceum Hosianum in Braunsberg. Besides mathematics he also taught physics, botanics and gymnastics.
Weierstrass may have had an illegitimate child named Franz with the widow of his friend Borchardt.
After 1850 Weierstrass suffered from a long period of illness, but was able to publish papers that brought him fame and distinction. He took a chair at the Technical University of Berlin, then known as the Gewerbeinstitut. He was immobile for the last three years of his life, and died in Berlin from pneumonia.
Soundness of calculus
Weierstrass was interested in the soundness of calculus, and at the time, there were somewhat ambiguous definitions regarding the foundations of calculus, and hence important theorems could not be proven with sufficient rigour. While Bolzano had developed a reasonably rigorous definition of a limit as early as 1817 (and possibly even earlier) his work remained unknown to most of the mathematical community until years later, and many had only vague definitions of limits and continuity of functions.
Delta-epsilon proofs are first found in the works of Cauchy in the 1820s. Cauchy did not clearly distinguish between continuity and uniform continuity on an interval. Notably, in his 1821 Cours d'analyse, Cauchy argued that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous, a statement interpreted as being incorrect by many scholars. The correct statement is rather that the uniform limit of continuous functions is continuous (also, the uniform limit of uniformly continuous functions is uniformly continuous). This required the concept of uniform convergence, which was first observed by Weierstrass's advisor, Christoph Gudermann, in an 1838 paper, where Gudermann noted the phenomenon but did not define it or elaborate on it. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus.
The formal definition of continuity of a function, as formulated by Weierstrass, is as follows:
is continuous at if such that for every in the domain of ,
Using this definition and the concept of uniform convergence, Weierstrass was able to write proofs of several then-unproven theorems such as the intermediate value theorem (for which Bolzano had already given a rigorous proof), the Bolzano–Weierstrass theorem, and Heine–Borel theorem.
Calculus of variations
Weierstrass also made significant advancements in the field of calculus of variations. Using the apparatus of analysis that he helped to develop, Weierstrass was able to give a complete reformulation of the theory which paved the way for the modern study of the calculus of variations. Among the several significant axioms, Weierstrass established a necessary condition for the existence of strong extrema of variational problems. He also helped devise the Weierstrass–Erdmann condition, which gives sufficient conditions for an extremal to have a corner along a given extrema, and allows one to find a minimizing curve for a given integral.
Other analytical theorems
- Stone–Weierstrass theorem
- Weierstrass–Casorati theorem
- Weierstrass's elliptic functions
- Weierstrass function
- Weierstrass M-test
- Weierstrass preparation theorem
- Lindemann–Weierstrass theorem
- Weierstrass factorization theorem
- Enneper–Weierstrass parameterization
- Sokhatsky–Weierstrass theorem
- Zur Theorie der Abelschen Funktionen (1854)
- Theorie der Abelschen Funktionen (1856)
- Abhandlungen-1// Math. Werke. Bd. 1. Berlin, 1894
- Abhandlungen-2// Math. Werke. Bd. 2. Berlin, 1895
- Abhandlungen-3// Math. Werke. Bd. 3. Berlin, 1903
- Vorl. ueber die Theorie der Abelschen Transcendenten// Math. Werke. Bd. 4. Berlin, 1902
- Vorl. ueber Variationsrechnung// Math. Werke. Bd. 7. Leipzig, 1927
Students of Karl Weierstrass
Honours and awards
- See here
- Grabiner, Judith V. (March 1983), "Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus", The American Mathematical Monthly 90 (3): 185–194, doi:10.2307/2975545, JSTOR 2975545
- Cauchy, A.-L. (1823), "Septième Leçon – Valeurs de quelques expressions qui se présentent sous les formes indéterminées Relation qui existe entre le rapport aux différences finies et la fonction dérivée", Résumé des leçons données à l’école royale polytechnique sur le calcul infinitésimal, Paris, p. 44.
|Wikimedia Commons has media related to Karl Weierstrass.|
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- O'Connor, John J.; Robertson, Edmund F., "Karl Weierstrass", MacTutor History of Mathematics archive, University of St Andrews.
- Karl Weierstrass at the Mathematics Genealogy Project
- Digitalized versions of Weierstrass's original publications are freely available online from the library of the Berlin Brandenburgische Akademie der Wissenschaften.
- Works by Karl Weierstrass at Project Gutenberg