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WEIERSTRASS



Karl Theodor Wilhelm Weierstrass (1815 – 1897) was a German mathematician who is often cited as the "father of modern analysis". 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 Munster (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 Munster. 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.
From the end of 1850s the international popularity of Weierstrass quickly grows. It is obliged by it to magnificent quality of the lectures. Here the list of subjects of its rates:
– Introduction in the theory of analytical functions, including the theory of real numbers.
– The theory of elliptic functions, appendices of elliptic functions to problems of geometry and mechanics.
– The theory of abelian integrals and functions.
– Calculus of variations.
Health of Weierstrass leaves much to be desired – constant overfatigue in young years affects. In 1861 during performance at it the strong attack of dizziness has begun and it was necessary to interrupt lecture. It is More Weierstrass never read lectures costing – it invariable sat, and one of the best students wrote for it on a board.
1861: he is selected by a member of the Bavarian academy of sciences.
1864: he is appointed by the ordinary professor.
1868: he is selected by member-correspondent of the Parisian academy of sciences.
1870: he gets acquainted with the twenty years' Sofia Kovalevskaja who has arrived to Berlin for preparation of the dissertation. Tender feeling to Sonja Weierstrass has carried by through all life (it and did not marry). Weierstrass helps Kovalevskaja to choose subject matter of the dissertation and a method of the approach to the decision, in the further on a regular basis advises it on complicated questions of the analysis, assists in reception of a scientific recognition. After protection of dissertation Kovalevskaja has left, letters of the teacher answered seldom and reluctantly, except for situations when to it consultation promptly was required.
1873: he is selected by the rector of the Berlin university.
1881: he is selected by a member of the London royal society.
1883: after suicide of Kovalevskaja husband, remained without means with the five years' daughter, comes to Berlin and stops at Weierstrass. By huge efforts, using all authority and communications, Weierstrass manages to obtain to her the professor seat at the Stockholm university.
1885: the 70 anniversary of the glorified mathematician solemnly is marked in the all-European scale.
1891: Sofia Kovalevskaja unexpectedly dies. Shaken Weierstrass sends flowers on its tomb and burns all letters from Sofia Kovalevskaja (letters from her were kept and in the beginning of XX century have been published). Condition Weierstrass worsens noticeably, it seldom rises, is engaged in editing of the collection of works.
1897: after long-term illness Weierstrass has died from flu complications.
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.
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 rigor. 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 ()-definition of limit, in the context of formally defining the derivative, in the 1820s, 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 (point wise) limit of (point wise) continuous functions was itself (point wise) 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' 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, was formulated by Weierstrass.
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.
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 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.
Weierstrass himself did not care of publications of the protruding lectures. However during lifetime of the beginning to leave the assembly of its works; in total there were 7 volumes (last — in 1927). Many protruding mathematicians have passed Weierstrass' school. To number of its disciples concern: Edmund Husserl, Philip Ludwig Zeydel, Georg Cantor, Sofja Kovalevskaja, Magnus Gesta Mittag-Leffler, Dmitry Selivanov, Ferdinand Georg Frobenius, Lazarus Immanuel Fuchs, Karl Hermann Amandus Schwartz.


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