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Karl Weierstrass

Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the "father of modern analysis".

Biography

Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia.

Weierstrass was the son of Wilhelm Weierstrass, a government official, and Theodora Vonderforst. His interest in mathematics began while he was a Gymnasium student. 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.

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.

Mathematical contributions

Soundness of calculus

Weierstrass was interested in the soundness of calculus. 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.

Cauchy gave a form of the (ε, δ)-definition of limit, in the context of formally defining the derivative, in the 1820s,[1][2] but did not correctly distinguish between continuity at a point versus uniform continuity on an interval, due to insufficient rigor. Notably, in his 1821 Cours d'analyse, Cauchy gave a famously incorrect proof that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous. The correct statement is rather that 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 (ε, δ)-definition of limit, as formulated by Weierstrass, is as follows:

is continuous at if for every such that

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 given an insufficiently 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 gave way for the modern study of calculus of variations. Among several significant results, Weierstrass established a necessary condition for the existence of strong extrema of variational problems. He also helped devise the Weierstrass–Erdmann condition which give sufficient conditions for an extremal to have a corner.

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

Selected works

* 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, 1897
* Abhandlungen-3// Math. Werke. Bd. 3. Berlin, 1915
* Vorl. ueber die Theorie der Abelschen Transcendenten// Math. Werke. Bd. 4. Berlin, 1902
* Vorl. ueber Variationsrechnung// Math. Werke. Bd. 6. Berlin, 1927

Students of Karl Weierstrass

* Edmund Husserl
* Sofia Kovalevskaya
* Gösta Mittag-Leffler
* Hermann Schwarz

Honours and awards

The lunar crater Weierstrass is named after him.

References

1. ^ 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, http://www.maa.org/pubs/Calc_articles/ma002.pdf
2. ^ 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, http://gallica.bnf.fr/ark:/12148/bpt6k90196z/f45n5.capture, p. 44.