KARL WILHELM THEODOR WEIERSTRASS

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SS-stokesKarl 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”.

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 Gymnasiumstudent 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 theUniversity 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 inMünster. 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.

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, 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 formal definition of continuity of a function, as formulated by Weierstrass, is as follows:

\displaystyle f(x) is continuous at \displaystyle x=x_0 if  \displaystyle \forall \ \varepsilon > 0\ \exists\ \delta > 0 such that for every x in the domain of f,    \displaystyle \ |x-x_0| < \delta \Rightarrow |f(x) - f(x_0)| < \varepsilon.

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 give 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

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