# Karl Weierstrasse

Karl Weierstrasse
Karl Weierstraß is named on the honor roll of former students of the Theodorianum high school in Paderborn. (left side, second name from above)
Karl Weierstraß on the honor roll Lyceum Hosianum in Braniewo

Karl Theodor Wilhelm Weierstrass (born October 31, 1815 in Ostenfelde near Ennigerloh , Münsterland , † February 19, 1897 in Berlin ) was a German mathematician who has made a name for himself in the logically founded analysis of analysis .

## Life

At the time of Karl Weierstrasse's birth, his father Wilhelm was secretary to the mayor of Ostenfelde. When Karl was eight years old, his father became a tax inspector, which meant that the family had to move around a lot in Prussia. In 1827, the year his mother died, his father was given a permanent post in Paderborn , so that Karl could attend the “Akademische Gymnasium” there (today Theodorianum ). On the side he had to work in the bookkeeping to improve the family finances, but still had good grades and read the leading German mathematics journal Crelles Journal . At his father's request, Weierstrass studied law and finance at the Rheinische Friedrich-Wilhelms-Universität Bonn from 1834 to 1838 in order to prepare for a career as a Prussian administrative officer. Since 1836 he was a member of the Corps Saxonia Bonn , in which, according to Felix Klein's description, he became a little too much. On the side, however, he read works by Pierre-Simon Laplace , Niels Henrik Abel and Carl Gustav Jacob Jacobi , which encouraged him to turn to mathematics . After he left the University of Bonn in 1838 without a degree, his angry father was persuaded to let him study mathematics and physics , which were more in keeping with his inclinations , from 1838 to 1840 at the Münster Academy . He heard the theory of elliptical functions from Christoph Gudermann , who was very impressed by Weierstrass. He prepared for his exams by self-study in Westernkotten near Lippstadt , where his father was the director of a salt works .

After passing his exams, he taught at grammar schools in Münster in 1841/42. Here he also developed the foundations of his later theory of complex functions, but did not publish anything. From Easter 1843 he worked in Deutsch Krone in West Prussia and from 1848 in Braunsberg at the Lyceum Hosianum . In addition to mathematics, he also taught various other subjects such as physics, botany and gymnastics . There was a special reason for the latter subject: when physical education was to be introduced in Deutsch-Krone in 1844, only Weierstrass came into consideration as a suitable physical education teacher. He had done gymnastics himself at a young age and was familiar with Carl Euler's book Die deutsche Turnkunst . At the end of July 1844 he traveled to Berlin and trained there to become a gym teacher.

In complete isolation from the mathematical world, he worked intensively on his theory of Abelian functions (the direct generalizations of elliptical functions) and published in the journal of his school. However, it was not until an article in Crelle's Journal in 1854 on the theory of Abelian functions that attracted attention , which was followed by a more detailed work in 1856.

As a result, he received an honorary doctorate from the Albertus University in Königsberg in the same year , and the leading Berlin mathematicians Peter Gustav Lejeune Dirichlet and Ernst Eduard Kummer tried to pull him to Berlin. From 1856 he taught mathematics at the Royal Trade Institute (integrated into the Technical University of Berlin in 1879 ), but in the same year became a professor at the Friedrich-Wilhelms-Universität Berlin , while at the same time intensive efforts were made to bring him to Austria. In Berlin a large school soon formed around him, the hallmark of which was the introduction of " Weierstrasse strictness " into analysis. He was even more effective than his publications through the numerous, widely circulated transcripts of his lectures by his students, such as Wilhelm Killing or Adolf Hurwitz . At first he got along well with his Berlin colleague Leopold Kronecker , but in 1877 a rift broke out because of his rejection of set theory by Weierstrass's student Georg Cantor .

Weierstraß, who never married, had a special relationship with his Russian student Sofia Kowalewskaja , whom he taught privately from 1870 on, as she was not admitted to the university as a woman. He exercised his influence so that she was able to do her doctorate in Göttingen in 1874 and take up a position as a private lecturer in Stockholm in 1884. He remained in constant correspondence with her until her death in 1891.

Heliogravure
after R. von Voigtländer
Pastel picture
by Conrad Fehr
Etching
by Conrad Fehr

Already in his time in Braunsberg he suffered from health problems, and in late 1861 he suffered a complete breakdown.

On his 70th birthday, as a token of admiration and gratitude, he was given a photo album with portraits of many of his students, friends and colleagues. On the occasion of his 80th birthday, two paintings were made: one by Rudolf von Voigtländer (after whom the well-known heliogravure was made) and the other by the painter, graphic artist and sculptor Conrad Fehr (1854–1933), and an etching after Fehr made. For his anniversary, Weierstrass had been dependent on a wheelchair for a year and, on medical advice, was only able to receive congratulations from students, friends and colleagues in his apartment for two hours while sitting in an armchair. Although he was marked by physical suffering, he responded quickly and appropriately to the speeches he had given.

In 1856 he was accepted as a full member of the Prussian Academy of Sciences . In 1864 he became a corresponding and in 1895 honorary member of the Russian Academy of Sciences in St. Petersburg . Since 1868 he was a corresponding and since 1879 an external member of the Académie des sciences . In 1881 he became a foreign member of the Royal Society , whose Copley Medal he received in 1895. In 1883 he was elected a member of the Leopoldina . In 1892 he was elected to the National Academy of Sciences , in 1896 to the American Academy of Arts and Sciences . In 1887 he received the Cothenius Medal of the Leopoldina.

He died of pneumonia in Berlin on February 19, 1897 and was buried in the St. Hedwigs Cemetery in Berlin. His tombstone was moved to the old cemetery wall in 1961 when the Berlin Wall was built; the grave is located in the former death strip. Until 2014, the new location of the tombstone was designated as the honor grave of the city of Berlin .

The lunar crater Weierstrass and the asteroid (14100) Weierstrass were named after him. There is also the Weierstrass Institute for Applied Analysis and Stochastics in Berlin , a Leibniz Institute in the Forschungsverbund Berlin e. V.

## plant

His main work was the logically correct foundation of analysis (first in lectures in 1859/60) and the development of function theory on the basis of power series expansions . He made important contributions to the theory of elliptic functions , differential geometry and calculus of variations .

Many important concepts of the analysis taught today come from him, e.g. B. Convergence criteria for series, the treatment of infinite products and the concept of uniform convergence (Weierstrass criterion). He wrote the Bolzano-Weierstrass theorem , which says that every restricted sequence has at least one accumulation point . ${\ displaystyle \ mathbb {R} ^ {n}}$

Weierstraß also gave one of the first axiomatizations of real numbers in the context of his strict justification of analysis (although the strict axiomatization of the natural numbers on which these are based only began later with Giuseppe Peano ). Its access was via convergent infinite series in the reciprocal values ​​of natural numbers with bounded partial sums. He knew it already at the beginning of the 1860s, but it was only published in 1872 by Ernst Kossak after Weierstrass's lecture on the introduction to function theory from 1865/66, which he included every two years (with modifications) in his lecture cycle from 1857 to 1887 Function theory held. Weierstrass, however, saw Kossak's publication as inadequate and distorting. Weierstrass's method was published several times by others, but mostly the methods of Dedekind or Cantor were preferred. Before Weierstrass, Richard Dedekind (1872), Georg Cantor and Charles Méray (1869) published a strict theory of real numbers. In an elaboration of the same lecture by Weierstraß by Moritz Pasch , the term accumulation point is also found for the first time (taken from Georg Cantor in 1872).

In 1868 he found the Jordan normal form for matrices over the complex numbers ( Camille Jordan introduced it for matrices over finite fields in 1870), using the language of bilinear forms and the Weierstraß normal form (see Frobenius normal form ) and elementary dividers . Henry John Stephen Smith found the elementary divisors independently (see Smith normal form ). Weierstraß also proved in 1863 that the field of complex numbers is the only finite-dimensional commutative upper body of real numbers (published in Hermann Hankel Theory of Complex Number Systems ).

In the calculus of variations, which Weierstrasse regularly read about, he gave the necessary conditions for extremes. His criticism of the Dirichlet principle , with which Bernhard Riemann founded his theory of functions, is also known.

In 1872, Weierstrass found a function that was constant everywhere , but nowhere differentiable. Bernard Bolzano had already given such an example in 1834, but the mathematical experts had not taken note of it. Subsequently, other mathematicians discovered such monster curves , so named because their existence contradicted intuition.

Weierstraß, who also edited the works of Jakob Steiner and Carl Gustav Jacobi , also supervised the publication of the first volumes of his own works, in which his lectures, which contained a lot of unpublished material, were to be published.

In didactics he made a contribution to maeutics in 1845 . In About the Socratic Teaching Method and its Applicability in School Teaching , he praised the method, but was skeptical about its use in school.

### sentences

Furthermore come from him

## Fonts

• Contribution to the theory of Abel integrals. In: Annual report on the Königl. Katholische Gymnasium zu Braunsberg 1848/49, pp. 1–23. ( Digitized and full text in the German text archive )
• Collected works , 7 vols., Berlin: Mayer and Müller, 1894–1927 (reprint Hildesheim: Olms 1967), especially:
• Selected chapters from the theory of functions (lecture, held in Berlin 1886, with the academic inaugural speech, Berlin 1857 and three other original papers), Teubner Archive Mathematics, Leipzig 1988 (Ed. R. Siegmund-Schultze)
• Treatises from the theory of functions, Springer, Berlin 1886
• Introduction to the theory of analytical functions, Lecture Berlin 1878, Documents on the History of Mathematics 4, Vieweg 1988 (transcript by Adolf Hurwitz)
• Formulas and theorems for the use of elliptical functions, based on lectures and notes by K. Weierstrass, Berlin 1893 (Ed. Hermann Amandus Schwarz, only the 1st section published), reprint Würzburg: Physica Verlag 1962
• Theory of Abelian Functions, Crelle J. 1856, Project Gutenberg

## Honors

At the University of Paderborn in the Faculty of Electrical Engineering, Computer Science and Mathematics, the Weierstraß Prize for outstanding teaching is awarded annually to one lecturer and one employee. In 1875 he received the Pour le Mérite .

The Weierstrasse hall in the main building of the Humboldt University in Berlin is named after him.

## Web links

Wikisource: Karl Weierstraß  - Sources and full texts
Commons : Karl Weierstraß  - Collection of images, videos and audio files

## swell

1. Kösener Corps lists 1910, 27/22
2. ^ Felix Klein: Lectures on the Development of Mathematics in the 19th Century Springer, Berlin 1979, ISBN 3-540-09235-8 (reprint of the Berlin 1926/27 edition)
3. Renate Tobies (Ed.): "All male culture in spite of". Women in math and science . With a foreword by Knut Radbruch . Campus, Frankfurt a. M./New York 1997, ISBN 3-593-35749-6 , pp. 132 ( limited preview in Google Book search).
4. ^ List of former members since 1666: Letter W. Académie des sciences, accessed on March 14, 2020 (French).
5. ^ Entry on Weierstrass, Carl Wilhelm (1815 - 1897) in the archive of the Royal Society , London
6. ^ Member entry by Karl Weierstrass at the German Academy of Natural Scientists Leopoldina , accessed on June 10, 2016.
7. Klaus Kopfermann, Weierstrass lecture on functional theory, in Behnke, Kopfermann, Weierstrass-Festschrift, Westdeutscher Verlag 1966, p. 80
8. ^ Ernst Kossak: The elements of arithmetic , program Friedrich Werdersches Gymnasium, Berlin 1872
9. ^ As in 1920 by Gösta Mittag-Leffler in his book The Number
10. Dugac, foundations of analysis, in Dieudonné, history of mathematics, Vieweg 1990, p 387
11. Heinz-Wilhelm Alten u. a., 4000 years of algebra, Springer, 2008, p. 409
12. Eric W. Weisstein : Weierstrass Sigma Function . In: MathWorld (English).
13. Eric W. Weisstein : Weierstrass Constant . In: MathWorld (English).
14. Eric W. Weisstein : Weierstrass Zeta Function . In: MathWorld (English).
15. ^ Weierstraß in the order Pour le Mérite