Karl Theodor Wilhelm Weierstrass (/ˈvaɪərˌstrɑːs, -ˌʃtrɑːs/;[1] German: Weierstraß [ˈvaɪɐʃtʁaːs];[2] 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis".
Among many other contributions, Weierstrass formalized the definition of the continuity of a function and complex analysis, proved the intermediate value theorem and the Bolzano–Weierstrass theorem, and used the latter to study the properties of continuous functions on closed bounded intervals.
Weierstrass was born into a Roman Catholic family in Ostenfelde, a village near Ennigerloh, in the Province of Westphalia.
His interest in mathematics began while he was a gymnasium student at the Theodorianum in Paderborn.
He was sent to the University of Bonn upon graduation, to prepare for a government position; to this end, his studies were to be in the fields of law, economics, and finance—a situation immediately in conflict with his own hopes to study mathematics.
He resolved the conflict by paying little heed to his planned course of study but continuing to study mathematics in private, which ultimately resulted in his leaving the university without a degree.
Weierstrass continued to study mathematics at the Münster Academy (an institution even then famous for mathematics), and his father was able to obtain a place for him in a teacher-training school in Münster; his efforts there did, eventually, lead to his certification as a teacher in that city.
During this period of study, Weierstrass attended the lectures of Christoph Gudermann and became interested in elliptic functions.
[4] At some point, Weierstrass may have had an illegitimate child ("Franz") with the widow of his friend Carl Wilhelm Borchardt.
[6][dubious – discuss] After 1850, Weierstrass suffered from a long period of illness, but was yet able to publish mathematical articles of sufficient quality and originality to bring him fame and distinction.
In 1856 he took a chair at the Gewerbeinstitut in Berlin (an institute to educate technical workers, which would later merge with the Bauakademie to form the Technische Hochschule in Charlottenburg; now Technische Universität Berlin).
In 1870, at the age of fifty-five, Weierstrass met Sofia Kovalevskaya whom he tutored privately after failing to secure her admission to the university.
He mentored her for four years, and regarded her as his best student, helping to secure her a doctorate from Heidelberg University without the need for an oral thesis defense.
Professor Reinhard Bölling [de] discovered the draft of the letter she wrote to Weierstrass when she arrived in Stockholm in 1883 upon her appointment as Privatdocent at Stockholm University.
[7] Weierstrass was immobile for the last three years of his life, and died in Berlin from pneumonia on the 19th of February, 1897.
Although 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 mathematicians had only vague definitions of limits and continuity of functions.
The basic idea behind Delta-epsilon proofs is, arguably, first found in the works of Cauchy in the 1820s.
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.
He also proved the Bolzano–Weierstrass theorem and used it to study the properties of continuous functions on closed and bounded intervals.
Using the apparatus of analysis that he helped to develop, Weierstrass was able to give a complete reformulation of the theory that paved the way for the modern study of the calculus of variations.
Among several axioms, Weierstrass established a necessary condition for the existence of strong extrema of variational problems.