Karl Wilhelm Theodor Weierstrass

Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (Weierstraß)
Born	October 31 1815(1815--) Ostenfelde, Westphalia
Died	February 19 1897 (aged 81) Berlin, Germany
Residence	Germany
Nationality	German
Field	Mathematician
Institutions	Gewerbeinstitut
Alma mater	University of Bonn Münster Academy
Academic advisor	Christoph Gudermann
Notable students	Georg Cantor Georg Frobenius Lazarus Fuchs Wilhelm Killing Leo Königsberger Mathias (Matyas) Lerch Hans von Mangoldt Richard Müller Carl Runge Arthur Schoenflies Friedrich Schottky Hermann Schwarz
Known for	Weierstrass function

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

Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany).

He 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.

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:

Failed to parse (unknown function\displaystyle): \displaystyle f(x)

is continuous at Failed to parse (unknown function\displaystyle): \displaystyle x = x_0
if for every Failed to parse (unknown function\displaystyle): \displaystyle \epsilon > 0\, \exists\,\delta > 0
such that

Failed to parse (unknown function\displaystyle): \displaystyle |x-x_0| < \delta \Rightarrow |f(x) - f(x_0)| < \epsilon.

Weierstrass also formulated similar 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.

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

Students of Karl Weierstrass

• Edmund Husserl
• Sofia Kovalevskaya
• Gösta Mittag-Leffler

See also

• Bolzano–Weierstrass theorem
  • Heine-Borel theorem
• Weierstrass extreme value theorem
• 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

External links

• O'Connor, John J; Edmund F. Robertson "Karl Weierstrass". MacTutor History of Mathematics archive.  
• Karl Weierstrass at the Mathematics Genealogy Project
• Digitalized versions of Weierstraß' original publications are freely available online from the library of the Berlin Brandenburgische Akademie der Wissenschaften.

pms:Karl Weierstrass

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