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| {{Infobox scientist
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| | name = Karl Weierstrass
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| | image = Karl Weierstrass.jpg|300px
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| | caption = Karl Theodor Wilhelm Weierstrass (Weierstraß)
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| | birth_date = {{birth date|1815|10|31|df=y}}
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| | birth_place = [[Ennigerloh|Ostenfelde]], [[Province of Westphalia]], [[Kingdom of Prussia]]
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| | death_date = {{death date and age|1897|2|19|1815|10|31|df=y}}
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| | death_place = [[Berlin]], [[Province of Brandenburg]], [[Kingdom of Prussia]]
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| | residence = [[Germany]]
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| | nationality = [[Germans|German]]
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| | field = [[Mathematics]]
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| | work_institution = [[Technical University of Berlin|Gewerbeinstitut]]
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| | alma_mater = [[University of Bonn]]<br>[[University of Münster|Münster Academy]]
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| | doctoral_advisor = [[Christoph Gudermann]]
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| | doctoral_students = [[Nikolai Bugaev]]<br>[[Georg Cantor]]<br>[[Georg Frobenius]]<br>[[Lazarus Fuchs]]<br>[[Wilhelm Killing]]<br>[[Leo Königsberger]]<br>[[Sofia Kovalevskaya]]<br>[[Mathias Lerch]]<br>[[Hans von Mangoldt]]<br>[[Eugen Netto]]<br>[[Adolf Piltz]]<br>[[Carl Runge]]<br>[[Arthur Schoenflies]]<br>[[Friedrich Schottky]]<br>[[Hermann Schwarz]]<br>[[Ludwig Stickelberger]]
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| | known_for = [[Weierstrass function]]
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| | prizes =
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| | religion =
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| | footnotes =
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| }}
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| '''Karl Theodor Wilhelm Weierstrass''' ({{lang-de|Weierstraß}}; 31 October 1815 – 19 February 1897) was a [[Germany|German]] [[mathematics|mathematician]] who is often cited as the "father of modern [[mathematical analysis|analysis]]".
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| == Biography ==
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| Weierstrass was born in Ostenfelde, part of [[Ennigerloh]], [[Province of Westphalia]].
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| Weierstrass was the son of Wilhelm Weierstrass, a government official, and Theodora Vonderforst. His interest in mathematics began while he was a ''[[Gymnasium (school)|Gymnasium]]'' student at [[:de:Gymnasium Theodorianum|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 function]]s.
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| 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.
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| Weierstrass may have had an illegitimate child named Franz with the widow of his friend Borchardt.<ref>See [http://www.ams.org/mathscinet-getitem?mr=1388786 here]</ref>
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| 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]].
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| == Mathematical contributions ==
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| === Soundness of calculus ===
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| Weierstrass was interested in the [[soundness]] of calculus. 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 [[Bernard Bolzano|Bolzano]] had developed a reasonably rigorous definition of a [[Limit of a function|limit]] as early as 1817 (and possibly even earlier) his work remained unknown to most of the mathematical community until years later,
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| and many had only vague definitions of [[Limit of a function|limits]] and [[Continuous function|continuity]] of functions.
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| [[(ε, δ)-definition of limit|Delta-epsilon]] proofs are first found in the works of [[Augustin Louis Cauchy|Cauchy]] in the 1820s.<ref>{{citation
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| |title=Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus
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| |first=Judith V.
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| |last=Grabiner
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| |journal=The American Mathematical Monthly
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| |date=March 1983
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| |volume=90
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| |pages=185–194
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| |url=http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Grabiner185-194.pdf
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| |doi=10.2307/2975545
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| |issue=3
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| |jstor=2975545
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| }}</ref><ref>{{citation
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| |first=A.-L.
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| |last=Cauchy
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| |author-link=Augustin Louis Cauchy
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| |title=Résumé des leçons données à l’école royale polytechnique sur le calcul infinitésimal
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| |place=Paris
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| |year=1823
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| |url=http://math-doc.ujf-grenoble.fr/cgi-bin/oeitem?id=OE_CAUCHY_2_4_9_0
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| |chapter=Septième Leçon – Valeurs de quelques expressions qui se présentent sous les formes indéterminées <math>\frac{\infty}\infty, \infty^0, \ldots</math> Relation qui existe entre le rapport aux différences finies et la fonction dérivée
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| |chapterurl=http://gallica.bnf.fr/ark:/12148/bpt6k90196z/f45n5.capture
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| |postscript=, [http://gallica.bnf.fr/ark:/12148/bpt6k90196z.image.f47 p. 44].
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| }}</ref>
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| 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|''uniform'' limit]] of continuous functions is continuous (also, the uniform limit of uniformly continuous functions is uniformly continuous).
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| 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.
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| The formal definition of continuity of a function, as formulated by Weierstrass, is as follows:
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| <math>\displaystyle f(x)</math> is continuous at <math>\displaystyle x = x_0</math> if <math> \displaystyle \forall \ \varepsilon > 0\ \exists\ \delta > 0</math> such that for every <math>x</math> in the domain of <math>f</math>, <math> \displaystyle \ |x-x_0| < \delta \Rightarrow |f(x) - f(x_0)| < \varepsilon.</math>
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| Using this definition and the concept of uniform convergence,
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| Weierstrass was able to write proofs of several then-unproven theorems such as the [[intermediate value theorem]] (for which [[Bernard Bolzano|Bolzano]] had already given a rigorous proof), the [[Bolzano–Weierstrass theorem]], and [[Heine–Borel theorem]].
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| === Calculus of variations ===
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| 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.
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| === Other analytical theorems ===
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| :: ''See also'' [[List of things named after Karl Weierstrass]].
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| * [[Stone–Weierstrass theorem]]
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| * [[Weierstrass–Casorati theorem]]
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| * [[Weierstrass's elliptic functions]]
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| * [[Weierstrass function]]
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| * [[Weierstrass M-test]]
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| * [[Weierstrass preparation theorem]]
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| * [[Lindemann–Weierstrass theorem]]
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| * [[Weierstrass factorization theorem]]
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| * [[Enneper–Weierstrass parameterization]]
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| * [[Sokhatsky–Weierstrass theorem]]
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| == Selected works ==
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| *''Zur Theorie der Abelschen Funktionen'' (1854)
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| *''Theorie der Abelschen Funktionen'' (1856)
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| * ''[http://name.umdl.umich.edu/AAN8481.0001.001 Abhandlungen-1]''// Math. Werke. Bd. 1. Berlin, 1894
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| * ''[http://name.umdl.umich.edu/AAN8481.0002.001 Abhandlungen-2]''// Math. Werke. Bd. 2. Berlin, 1895
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| * ''[http://name.umdl.umich.edu/AAN8481.0003.001 Abhandlungen-3]''// Math. Werke. Bd. 3. Berlin, 1903
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| * ''[http://name.umdl.umich.edu/AAN8481.0004.001 Vorl. ueber die Theorie der Abelschen Transcendenten]''// Math. Werke. Bd. 4. Berlin, 1902
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| * ''[http://name.umdl.umich.edu/AAN8481.0007.001 Vorl. ueber Variationsrechnung]''// Math. Werke. Bd. 7. Leipzig, 1927
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| ==Students of Karl Weierstrass==
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| * [[Edmund Husserl]]
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| * [[Sofia Kovalevskaya]]
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| * [[Gösta Mittag-Leffler]]
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| * [[Hermann Schwarz]]
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| * [[Carl Johannes Thomae]]
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| ==Honours and awards==
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| The lunar [[Impact crater|crater]] [[Weierstrass (crater)|Weierstrass]] is named after him.
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| == See also ==
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| * [[List of things named after Karl Weierstrass]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| {{commons|Karl Weierstrass}}
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| {{Wikiquote}}
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| *{{MacTutor Biography|id=Weierstrass}}
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| *{{MathGenealogy |id=7486}}
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| * [http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/autoren/weierstr/ Digitalized versions of Weierstrass's original publications] are freely available online from the library of the ''[http://bibliothek.bbaw.de/bibliothek-digital Berlin Brandenburgische Akademie der Wissenschaften]''.
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| *{{gutenberg author | id=Karl_Weierstrass | name=Karl Weierstrass }}
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| {{Copley Medallists 1851-1900}}
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| {{Authority control|PND=11876618X|LCCN=n/84/806249|VIAF=36999173|SELIBR=238824}}
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| {{Persondata<!-- Metadata: see [[Wikipedia:Persondata]] -->
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| |NAME= Weierstrass, Karl
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| |ALTERNATIVE NAMES=
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| |SHORT DESCRIPTION= [[Mathematician]]
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| |DATE OF BIRTH= 31 October 1815
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| |PLACE OF BIRTH= [[Ostenfelde]], [[Westphalia]]
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| |DATE OF DEATH= 19 February 1897
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| |PLACE OF DEATH= [[Berlin]], [[Germany]]
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| }}
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| {{DEFAULTSORT:Weierstrass, Karl}}
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| [[Category:1815 births]]
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| [[Category:1897 deaths]]
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| [[Category:19th-century German mathematicians]]
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| [[Category:Mathematical analysts]]
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| [[Category:People from the Province of Westphalia]]
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| [[Category:People from Braniewo]]
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| [[Category:Recipients of the Copley Medal]]
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| [[Category:University of Bonn alumni]]
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| [[Category:University of Königsberg alumni]]
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| [[Category:University of Münster alumni]]
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| [[Category:Humboldt University of Berlin faculty]]
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| [[Category:Berlin Institute of Technology faculty]]
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| [[Category:Foreign Members of the Royal Society]]
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| [[Category:Corresponding Members of the St Petersburg Academy of Sciences]]
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| [[Category:Honorary Members of the St Petersburg Academy of Sciences]]
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| [[Category:German Roman Catholics]]
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