Karl Weierstrass
Karl Weierstrass  

Karl Theodor Wilhelm Weierstrass (Weierstraß)


Born  Ostenfelde, Province of Westphalia, Kingdom of Prussia 
31 October 1815
Died  19 February 1897 Berlin, Province of Brandenburg, Kingdom of Prussia 
(aged 81)
Residence  Germany 
Nationality  German 
Fields  Mathematics 
Institutions  Gewerbeinstitut 
Alma mater  University of Bonn Münster Academy 
Doctoral advisor  Christoph Gudermann 
Doctoral students  Nikolai Bugaev Georg Cantor Georg Frobenius Lazarus Fuchs Wilhelm Killing Leo Königsberger Sofia Kovalevskaya Mathias Lerch Hans von Mangoldt Eugen Netto Adolf Piltz Carl Runge Arthur Schoenflies Friedrich Schottky Hermann Schwarz Ludwig Stickelberger 
Known for  Weierstrass function 
Karl Theodor Wilhelm Weierstrass (German: Weierstraß; 31 October 1815 – 19 February 1897) was a German mathematician who is often cited as the "father of modern analysis".
Contents
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 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 DeutschKrone 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.^{1}
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,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.
Deltaepsilon proofs are first found in the works of Cauchy in the 1820s.^{2}^{3} 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 thenunproven 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 Mtest
 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)
 Abhandlungen1// Math. Werke. Bd. 1. Berlin, 1894
 Abhandlungen2// Math. Werke. Bd. 2. Berlin, 1895
 Abhandlungen3// 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
The lunar crater Weierstrass is named after him.
See also
References
 ^ 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.
External links
Wikimedia Commons has media related to Karl Weierstrass. 
Wikiquote has a collection of quotations related to: Karl Weierstrass 
 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

HPTS  Area Progetti  EduSoft  JavaEdu  N.Saperi  Ass.Scuola..  TS BCTV  TS VideoRes  TSODP  TRTWE  