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Encyclopedia > Karl Weierstrass
Born Karl Theodor Wilhelm Weierstrass (Weierstraß) October 31, 1815 Ostenfelde, Westphalia February 19, 1897 Berlin, Germany Germany German Mathematician Gewerbeinstitut University of Bonn Münster Academy Christoph Gudermann Georg Cantor Georg Frobenius Lazarus Fuchs Wilhelm Killing Leo Königsberger Mathias (Matyas) Lerch Hans von Mangoldt Richard Müller Carl Runge Arthur Schoenflies Friedrich Schottky Image File history File links Download high-resolution version (1000x1431, 62 KB) Summary Source from de:Image:Karl_Weierstrass. ... October 31 is the 304th day of the year (305th in leap years) in the Gregorian calendar, with 61 days remaining. ... April 5-12: Mount Tambora explodes, changing climate. ... Westphalia (German: Westfalen) is a region in Germany, centred on the cities of Bielefeld, Dortmund, Gelsenkirchen, MÃ¼nster, and OsnabrÃ¼ck and included in the states of North Rhine-Westphalia and Lower Saxony. ... February 19 is the 50th day of the year in the Gregorian calendar. ... 1897 (MDCCCXCVII) was a common year starting on Friday (see link for calendar). ... Berlin is the capital city and one of the sixteen states of the Federal Republic of Germany. ... Image File history File links Flag_of_Germany. ... Image File history File links Flag_of_Germany. ... Leonhard Euler is considered by many to be one of the greatest mathematicians of all time A mathematician is the person whose primary area of study and research is the field of mathematics. ... South Side of the Main Building Areaway of the Main Building The Technical University of Berlin (TUB, TU Berlin, German: Technische UniversitÃ¤t Berlin) is located in Berlin in Germany. ... The main building, viewed from the Hofgarten. ... The University of MÃ¼nster (German WestfÃ¤lische Wilhelms-UniversitÃ¤t MÃ¼nster, WWU) is a public university located in the city of MÃ¼nster, North Rhine-Westphalia in Germany. ... Christoph Gudermann (March 25, 1798 - September 25, 1852) was born in Vienenburg, Germany. ... Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ... Picture of Frobenius Ferdinand Georg Frobenius (October 26, 1849 - August 3, 1917) was a German mathematician, best-known for his contributions to the theory of differential equations and to group theory. ... Immanuel Lazarus Fuchs (5 May 1833 - 26 April 1902) was a German mathematician. ... Wilhelm Karl Joseph Killing (1847 May 10 – 1923 February 11) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry. ... Photograph of Leo KÃ¶nigsberger, 1886 Leo KÃ¶nigsberger (October 15, 1837â€“December 15, 1921) was a German mathematician, and historian of science. ... Mathias Lerch (Matyas Lerch) (1860-1922) was an eminent Czech mathematician who published about 250 papers, some fifty of which were devoted to number theory. ... Hans Carl Friedrich von Mangoldt (1854-1925) was a German mathematician who contributed to the solution of the prime number theorem. ... Richard Muller For Richard MÃ¼ller, the German dentist who blew up his wife and car, see Richard MÃ¼ller (murderer). ... Carle David Tolmé Runge (August 30, 1856 – January 3, 1927) was a German mathematician, physicist, and spectroscopist. ... Arthur Moritz Schönflies (April 17, 1853 Landsberg an der Warthe(Gorzów) – May 27, 1928) was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology. ... Friedrich Hermann Schottky (July 24, 1851 - August 12, 1935) was a German mathematician who worked on elliptic, abelian, and theta functions and invented Schottky groups. ... Hermann Schwarz Weierstrass function

Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815February 19, 1897) was a German mathematician who is often cited as the "father of modern analysis". Karl Hermann Amandus Schwarz (25 January 1843 â€“ 30 November 1921) was a German mathematician, known for his work in complex analysis. ... In BIOLOGY, the SUMMER VACATION function was the first example found of a Chumba wumbafunction with the property that it is continuous everywhere but differentiable nowhere. ... October 31 is the 304th day of the year (305th in leap years) in the Gregorian calendar, with 61 days remaining. ... April 5-12: Mount Tambora explodes, changing climate. ... February 19 is the 50th day of the year in the Gregorian calendar. ... 1897 (MDCCCXCVII) was a common year starting on Friday (see link for calendar). ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ...

## Contents

Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). Westphalia (German: Westfalen) is a region in Germany, centred on the cities of Bielefeld, Dortmund, Gelsenkirchen, MÃ¼nster, and OsnabrÃ¼ck and included in the states of North Rhine-Westphalia and Lower Saxony. ...

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. A gymnasium (pronounced with or, in Swedish, as opposed to ) is a type of school providing secondary education in some parts of Europe, comparable to English Grammar Schools and U.S. High Schools. ... The main building, viewed from the Hofgarten. ... Lady Justice or Justitia is a personification of the moral force that underlies the legal system (particularly in Western art). ... The University of MÃ¼nster (German WestfÃ¤lische Wilhelms-UniversitÃ¤t MÃ¼nster, WWU) is a public university located in the city of MÃ¼nster, North Rhine-Westphalia in Germany. ... For other places with the same or similar names, and other uses of the word, see Munster (disambiguation) MÃ¼nster is a city in North Rhine-Westphalia, Germany. ... Christoph Gudermann (March 25, 1798 - September 25, 1852) was born in Vienenburg, Germany. ... In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. ...

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. South Side of the Main Building Areaway of the Main Building The Technical University of Berlin (TUB, TU Berlin, German: Technische UniversitÃ¤t Berlin) is located in Berlin in Germany. ... Pneumonia is an illness of the lungs and respiratory system in which the alveoli (microscopic air-filled sacs of the lung responsible for absorbing oxygen from the atmosphere) become inflamed and flooded with fluid. ...

## Soundness of calculus

Weierstrass was interested in the soundness of calculus. At the time, there were ambiguous definitions regarding the fundaments 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: (This article discusses the soundess notion of informal logic. ... Bernard Bolzano Bernard Placidus Johann Nepomuk Bolzano (October 5, 1781 â€“ December 18, 1848) was a Czech mathematician of German mother tongue, theologian, philosopher and logician. ... In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ... Augustin Louis Cauchy (August 21, 1789 â€“ May 23, 1857) was a French mathematician. ... In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

$displaystyle f(x)$ 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 definitions of limit and derivative still taught today. In some places this article assumes an acquaintance with algebra, analytic geometry, or the limit. ...

With these new definitions he was able to write proofs of several at the time unproven theorems such as the intermediate value theorem, Bolzano-Weierstrass theorem and Heine-Borel theorem. In analysis, the intermediate value theorem is either of two theorems of which an account is given below. ... The Bolzano-Weierstrass theorem in real analysis states that every bounded sequence of real numbers contains a convergent subsequence. ... In mathematical analysis, the Heine-Borel theorem, named after Eduard Heine and Ã‰mile Borel, states: A subset of the real numbers R is compact iff it is closed and bounded. ...

## 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 Gustav Albrecht Husserl (April 8, 1859, ProstÄ›jov â€“ April 26, 1938, Freiburg) was a German philosopher, known as the father of phenomenology. ... Sofia Vasilyevna Kovalevskaya (Ð¡Ð¾Ñ„ÑŒÑ Ð’Ð°ÑÐ¸Ð»ÑŒÐµÐ²Ð½Ð° ÐšÐ¾Ð²Ð°Ð»ÐµÐ²ÑÐºÐ°Ñ) (also known as Sonia Kovalevsky) (January 15, 1850 â€“ February 10, 1891) was the first major Russian female mathematician and a student of Karl Weierstrass in Berlin. ... Magnus Gustaf (GÃ¶sta) Mittag-Leffler (16 March 1846â€“7 July 1927) was a Swedish mathematician. ...

In real analysis, the Bolzanoâ€“Weierstrass theorem is an important theorem characterizing sequentially compact sets. ... In mathematical analysis, the Heine-Borel theorem, named after Eduard Heine and Ã‰mile Borel, states: A subset of the real numbers R is compact iff it is closed and bounded. ... In calculus, the extreme value theorem states that if a function f(x) is continuous in the closed interval [a,b] then f(x) must attain its maximum and minimum value, each at least once. ... In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. ... The Weierstrass-Casorati theorem in complex analysis describes the remarkable behavior of holomorphic functions near essential singularities. ... In mathematics, Weierstrasss elliptic functions are a standard type of elliptic functions (the other is the Jacobis elliptic functions). ... In BIOLOGY, the SUMMER VACATION function was the first example found of a Chumba wumbafunction with the property that it is continuous everywhere but differentiable nowhere. ... In mathematics, the Weierstrass M-test is an analogue of the comparison test for infinite series, and applies to a series whose terms are themselves functions of a real variable. ... In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point P. It states that such a function is, up to multiplication by a function not zero at P, a polynomial in one fixed variable z, which is monic... In mathematics, the Lindemann-Weierstrass theorem states that if &#945;1,...,&#945;n are algebraic numbers which are linearly independent over the rational numbers, then are algebraically independent over the algebraic numbers; in other words the set has transcendence degree n over . ... In mathematics, the Weierstrass factorization theorem in complex analysis, named after Karl Weierstrass, asserts that entire functions can be represented by a product involving their zeroes. ... In mathematics, the Enneper-Weierstrass parameterization of minimal surfaces is a classical piece of differential geometry. ...

Results from FactBites:

 Weierstrass biography (2615 words) Weierstrass had made a decision to become a mathematician but he was still supposed to be on a course studying public finance and administration. Weierstrass attended Gudermann's lectures on elliptic functions, some of the first lectures on this topic to be given, and Gudermann strongly encouraged Weierstrass in his mathematical studies. The concepts on which Weierstrass based his theory of functions of a complex variable in later years after 1857 are found explicitly in his unpublished works written in Münster from 1841 through 1842, while still under the influence of Gudermann.
 Karl Weierstrass - Wikipedia, the free encyclopedia (439 words) 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". Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). 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.
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