# Karl Weierstrass

**Karl Theodor Wilhelm Weierstrass** (Template:Lang-de; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a teacher, eventually teaching mathematics, physics, botany and gymnastics.

Weierstrass formalized the definition of the continuity of a function, and used it and the concept of uniform convergence to prove the Bolzano–Weierstrass theorem and Heine–Borel theorem.

## Contents

## Biography

Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia.^{[1]}

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 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 in Münster. 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.^{[1]}

Weierstrass may have had an illegitimate child named Franz with the widow of his friend Carl Wilhelm Borchardt.^{[2]}

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, and 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.

Delta-epsilon proofs are first found in the works of Cauchy in the 1820s.^{[3]}^{[4]}
Cauchy did not clearly distinguish between continuity and uniform continuity on an interval. Notably, in his 1821 *Cours d'analyse,* Cauchy argued that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous, a statement interpreted as being incorrect by many scholars. The correct statement is rather that the *uniform* limit of continuous functions is continuous (also, 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:

is continuous at if such that for every in the domain of ,

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 gives 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

## Selected works

*Zur Theorie der Abelschen Funktionen*(1854)*Theorie der Abelschen Funktionen*(1856)*Abhandlungen-1*// Math. Werke. Bd. 1. Berlin, 1894*Abhandlungen-2*// Math. Werke. Bd. 2. Berlin, 1895*Abhandlungen-3*// Math. Werke. Bd. 3. Berlin, 1903*Vorl. ueber die Theorie der Abelschen Transcendenten*// Math. Werke. Bd. 4. Berlin, 1902*Vorl. ueber Variationsrechnung*// Math. Werke. Bd. 7. Leipzig, 1927

## Students of Karl Weierstrass

## Honours and awards

The lunar crater Weierstrass is named after him.

## See also

## References

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## External links

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- Template:MacTutor Biography
- Karl Weierstrass at the Mathematics Genealogy Project
- Digitalized versions of Weierstrass's original publications are freely available online from the library of the
*Berlin Brandenburgische Akademie der Wissenschaften*. - Works by Karl Weierstrass at Project Gutenberg

Template:Copley Medallists 1851-1900

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- Pages using authority control with parameters
- VIAF different on Wikidata
- 1815 births
- 1897 deaths
- German mathematicians
- 19th-century German mathematicians
- Mathematical analysts
- People from the Province of Westphalia
- People from Braniewo
- Recipients of the Copley Medal
- University of Bonn alumni
- University of Königsberg alumni
- University of Münster alumni
- Humboldt University of Berlin faculty
- Technical University of Berlin faculty
- Foreign Members of the Royal Society
- Corresponding Members of the St Petersburg Academy of Sciences
- Honorary Members of the St Petersburg Academy of Sciences
- German Roman Catholics