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Karl Weierstrass

[vahy-er-strahs, -shtrahs; Ger. vahy-uhr-shtrahs]

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, and 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 leaving the university without a degree. After that he studied mathematics at the University of Münster which was even to this time very famous for mathematics and his father was able to obtain a place for him in a teacher training school in Münster, and he later 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 ambiguous definitions regarding the fundamentals of calculus, hence theorems could not be properly proven. 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 other eminent mathematicians such as Cauchy had only vague definitions of limits and continuity of functions. Weierstrass defined continuity as follows:

$displaystyle f(x)$ is continuous at $displaystyle x = x_0$ if for every $displaystyle epsilon > 0, exists,delta > 0$ such that

displaystyle |x-x_0| < delta Rightarrow |f(x) - f(x_0)| < epsilon.

Weierstrass also formulated similar (ε, δ)-definitions of limit and derivative still taught today.

With these new definitions he was able to write proofs of several then-unproven theorems such as the intermediate value theorem, 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 corner conditions which give sufficient conditions for an extremal to have a corner.

Other analytical theorems

Selected works

Zur Theorie der Abelschen Functionen (1854)
Theorie der Abelschen Functionen (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