View source for Karl Weierstrass

{{short description|German mathematician (1815–1897)}}
{{Infobox scientist
| name              = Karl Weierstrass
| native_name       = Karl Weierstraß
| image             = Karl Weierstrass.jpg
| caption           = 
| birth_date        = {{birth date|1815|10|31|df=y}}
| birth_place       = [[Ennigerloh]], [[Province of Westphalia]], [[Kingdom of Prussia]]
| death_date        = {{death date and age|1897|2|19|1815|10|31|df=y}}
| death_place       = [[Berlin]], Kingdom of Prussia, [[German Empire]]
| nationality       = German
| field             = [[Mathematics]]
| work_institution  = [[Technical University of Berlin|Gewerbeinstitut]], [[Humboldt University of Berlin|Friedrich Wilhelm University]]
| alma_mater        = {{unbulleted list
|[[University of Bonn]]
|[[University of Münster|Münster Academy]]}}
| academic_advisors = <!--Weierstrass was awarded an honorary doctorate (no formal mentors)-->[[Christoph Gudermann]]
| doctoral_students = {{unbulleted list|item_style=margin:0
|[[Nikolai Bugaev]]
|[[Georg Cantor]]
|[[Georg Frobenius]]
|[[Lazarus Fuchs]]
|[[Wilhelm Killing]]
|[[Johannes Knoblauch]]
|[[Leo Königsberger]]
|[[Ernst Kötter]]
|[[Sofia Kovalevskaya]]
|[[Mathias Lerch]]
|[[Hans von Mangoldt]]
|[[Eugen Netto]]
|[[Adolf Piltz]]
|[[Carl Runge]]
|[[Arthur Schoenflies]]
|[[Friedrich Schottky]]
|[[Hermann Schwarz]]
|[[Ludwig Stickelberger]]}}
| known_for         = {{unbulleted list|item_style=margin:0
|[[Weierstrass function]]
|[[Weierstrass product inequality]]
|[[(ε, δ)-definition of limit]]
|[[Weierstrass–Erdmann condition]]
|[[Weierstrass theorem (disambiguation)|Weierstrass theorems]]
|[[Bolzano–Weierstrass theorem]]}}
| prizes            = {{unbulleted list
|[[PhD (Hon)]]: [[University of Königsberg]] (1854)
|[[Copley Medal]] (1895)}}
}}

'''Karl Theodor Wilhelm Weierstrass''' ({{IPAc-en|ˈ|v|aɪ|ər|ˌ|s|t|r|ɑː|s|,_|-|ˌ|ʃ|t|r|ɑː|s}};<ref>[http://dictionary.reference.com/browse/weierstrass "Weierstrass"]. ''[[Random House Webster's Unabridged Dictionary]]''.</ref> {{langx|de|link=no|Weierstraß}} {{IPA|de|ˈvaɪɐʃtʁaːs|}};<ref>''Duden. Das Aussprachewörterbuch.'' 7.&nbsp;Auflage. Bibliographisches Institut, Berlin 2015, {{ISBN|978-3-411-04067-4}}</ref> 31 October 1815 – 19 February 1897) was a German [[mathematician]] often cited as the "[[list of people considered father or mother of a field|father of]] modern [[mathematical analysis|analysis]]". Despite leaving university without a degree, he studied mathematics and trained as a school teacher, eventually teaching [[mathematics]], [[physics]], [[botany]] and [[gymnastics]].<ref>Weierstrass, Karl Theodor Wilhelm. (2018). In Helicon (Ed.), ''The Hutchinson unabridged encyclopedia with atlas and weather guide''. [Online]. Abington: Helicon. Available from: <nowiki>http://libezproxy.open.ac.uk/login?url=</nowiki> [https://search.credoreference.com/content/entry/heliconhe/weierstrass_karl_theodor_wilhelm/0?institutionId=292 Link] Accessed 8 July 2018.</ref> He later received an honorary doctorate and became professor of mathematics in Berlin.

Among many other contributions, Weierstrass  formalized the definition of the [[Continuous function|continuity of a function]] and [[complex analysis]], proved the [[intermediate value theorem]] and the [[Bolzano–Weierstrass theorem]], and used the latter to study the properties of continuous functions on closed bounded intervals.

== Biography ==
Weierstrass was born into a [[Roman Catholic]] family in Ostenfelde, a village near [[Ennigerloh]], in the [[Province of Westphalia]].<ref name=StAndrews>{{cite web | url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Weierstrass.html | title=Karl Theodor Wilhelm Weierstrass | publisher=School of Mathematics and Statistics, University of St Andrews, Scotland |author1=O'Connor, J. J. |author2=Robertson, E. F. | date=October 1998 | access-date=7 September 2014 }}</ref>

Karl Weierstrass was the son of Wilhelm Weierstrass and Theodora Vonderforst, the former of whom was a government official and both of whom were Catholic [[Rhineland|Rhinelanders]]. His interest in mathematics began while he was a [[Gymnasium (Germany)|gymnasium]] student at the [[Gymnasium Theodorianum|Theodorianum]] in [[Paderborn]].  He was sent to the [[University of Bonn]] upon graduation, to prepare for a government position; to this end, his studies were to be in the fields of law, economics, and finance—a situation immediately in conflict with his own hopes to study mathematics. He resolved the conflict by paying little heed to his planned course of study but continuing to study mathematics in private, which ultimately resulted in his leaving the university without a degree. 

Weierstrass continued to study mathematics at the [[University of Münster|Münster Academy]] (an institution even then famous for mathematics), and his father was able to obtain a place for him in a teacher-training school in [[Münster]]; his efforts there did, eventually, lead to his certification 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.

In 1843 he taught in [[Wałcz|Deutsch Krone]] in [[West Prussia]], and from 1848 he taught at the [[Collegium Hosianum|Lyceum Hosianum]] in [[Braunsberg]].<ref>{{Citation |last=Elstrodt |first=Jürgen |title=Die prägenden Jahre im Leben von Karl Weierstraß |date=2016 |url=http://link.springer.com/10.1007/978-3-658-10619-5_2 |work=Karl Weierstraß (1815–1897) |pages=11–51 |editor-last=König |editor-first=Wolfgang |access-date=2023-08-12 |place=Wiesbaden |publisher=Springer Fachmedien Wiesbaden |language=de |doi=10.1007/978-3-658-10619-5_2 |isbn=978-3-658-10618-8 |editor2-last=Sprekels |editor2-first=Jürgen}}</ref> Besides mathematics, he also taught physics, botany, and gymnastics.<ref name=StAndrews/>  At some point, Weierstrass may have had an illegitimate child ("Franz") with the widow of his friend [[Carl Wilhelm Borchardt]].<ref>{{cite book | chapter-url=https://www.ams.org/mathscinet-getitem?mr=1388786 |author1=Biermann, Kurt-R. |author2=Schubring, Gert | chapter=Einige Nachträge zur Biographie von Karl Weierstraß. (German) [Some postscripts to the biography of Karl Weierstrass] | title=History of mathematics | pages=65–91 | publisher=Academic Press | location=San Diego, CA | date=1996}}</ref>{{dubious|date=December 2024}}

After 1850, Weierstrass suffered from a long period of illness, but was yet able to publish mathematical articles of sufficient quality and originality to bring him fame and distinction. The [[University of Königsberg]] conferred an [[honorary doctorate]] on him on 31 March 1854. In 1856 he took a chair at the ''Gewerbeinstitut'' in Berlin (an institute to educate technical workers, which would later merge with the ''Bauakademie'' to form the ''[[Technische Hochschule]]'' in Charlottenburg; now [[Technische Universität Berlin]]). In 1864 he became professor at the Friedrich-Wilhelms-Universität Berlin, which later became the [[Humboldt University of Berlin|Humboldt Universität zu Berlin]].

In 1870, at the age of fifty-five, Weierstrass met [[Sofia Kovalevskaya]] whom he tutored privately after failing to secure her admission to the university.  They had a fruitful intellectual, and kindly personal relationship that "far transcended the usual teacher-student relationship". He mentored her for four years, and regarded her as his best student, helping to secure her a doctorate from Heidelberg University without the need for an oral thesis defense.

From 1870 until her death in 1891, Kovalevskaya corresponded with Weierstrass. Upon learning of her death, he burned her letters. About 150 of his letters to her have been preserved. Professor {{ill|Reinhard Bölling|de}} discovered the draft of the letter she wrote to Weierstrass when she arrived in Stockholm in 1883 upon her appointment as ''[[Privatdocent]]'' at [[Stockholm University]].<ref>{{cite book|chapter-url=https://books.google.com/books?id=fyGtticogaQC&pg=PA1|chapter=''The life of S. V. Kovalevskaya'' by Roger L. Cooke|title=The Kowalevski Property (Leeds, 2000) CRM Proceedings & Lecture Notes, vol. 32|pages=1–19|year=2002|isbn=978-0-8218-7330-4|editor=Kuznetsov, Vadim B.|publisher=American Mathematical Soc.|postscript=; See p. 7 in 2002 book.}} [https://web.archive.org/web/20060105083925/http://www.emba.uvm.edu/~cooke/svklife.pdf online text]</ref>

Weierstrass was immobile for the last three years of his life, and died in Berlin from [[pneumonia]] on the 19th of February, 1897.<ref>{{Cite book|title=Dictionary of scientific biography|others=Gillispie, Charles Coulston,, American Council of Learned Societies.|year=1970 |isbn=978-0-684-12926-6|location=New York|pages=223|oclc=89822}}</ref>

== Mathematical contributions ==

=== Soundness of calculus ===
Weierstrass was interested in the [[soundness]] of calculus, and at the time there were somewhat ambiguous definitions of the foundations of calculus so that important theorems could not be proven with sufficient rigour.  Although [[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,
and many mathematicians had only vague definitions of [[Limit of a function|limits]] and [[Continuous function|continuity]] of functions.

The basic idea behind [[(ε, δ)-definition of limit|Delta-epsilon]] proofs is, arguably, first found in the works of [[Augustin-Louis Cauchy|Cauchy]] in the 1820s.<ref>{{citation
|title=Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus
|first=Judith V.
|last=Grabiner
|journal=The American Mathematical Monthly
|date=March 1983
|volume=90
|pages=185–194
|url=http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Grabiner185-194.pdf |archive-url=https://web.archive.org/web/20141129124944/http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Grabiner185-194.pdf |archive-date=2014-11-29 |url-status=live
|doi=10.2307/2975545
|issue=3
|jstor=2975545
}}</ref><ref>{{citation
 |first        = A.-L.
 |last         = Cauchy
 |author-link  = Augustin-Louis Cauchy
 |title        = Résumé des leçons données à l'école royale polytechnique sur le calcul infinitésimal
 |place        = Paris
 |year         = 1823
 |url          = http://math-doc.ujf-grenoble.fr/cgi-bin/oeitem?id=OE_CAUCHY_2_4_9_0
 |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
 |chapter-url   = http://gallica.bnf.fr/ark:/12148/bpt6k90196z/f45n5.capture
 |page   = [http://gallica.bnf.fr/ark:/12148/bpt6k90196z.image.f47 44]
 |access-date  = 2009-05-01
 |archive-url  = https://www.webcitation.org/5gVUmywgY?url=http://math-doc.ujf-grenoble.fr/cgi-bin/oeitem?id=OE_CAUCHY_2_4_9_0
 |archive-date = 2009-05-04
 |url-status     = dead
}}</ref>
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 that is false in general. 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).
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:

<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>, &nbsp; <math> \displaystyle \ |x-x_0| < \delta \Rightarrow |f(x) - f(x_0)| < \varepsilon.</math> In simple English,  <math>\displaystyle f(x)</math> is continuous at a point <math>\displaystyle x = x_0</math> if for each <math>x</math> close enough to <math>x_0</math>, the function value <math>f(x)</math> is very close to <math>f(x_0)</math>, where the "close enough" restriction typically depends on the desired closeness of <math>f(x_0)</math>  to <math>f(x).</math>
Using this definition, he proved the [[intermediate value theorem|Intermediate Value Theorem.]] He also proved the [[Bolzano–Weierstrass theorem]] and used it to study the properties of continuous functions on closed and bounded intervals.

=== Calculus of variations ===

Weierstrass also made advances 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 that paved the way for the modern study of the calculus of variations. Among several 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 extremum and allows one to find a minimizing curve for a given integral.

=== Other analytical theorems ===
<!--The items listed here are important and ought to be described in the article-->
{{see also|List of things named after Karl Weierstrass}}
* [[Bolzano–Weierstrass theorem]]
* [[Stone–Weierstrass theorem]]
* [[Casorati–Weierstrass theorem]]
* [[Weierstrass elliptic function]]
* [[Weierstrass function]]
* [[Weierstrass M-test]]
* [[Weierstrass preparation theorem]]
* [[Lindemann–Weierstrass theorem]]
* [[Weierstrass factorization theorem]]
* [[Weierstrass–Enneper parameterization]]

== Students ==
* [[Edmund Husserl]]

== Honours and awards ==
The lunar [[Impact crater|crater]] [[Weierstrass (crater)|Weierstrass]] and the [[asteroid]] [[14100 Weierstrass]] are named after him. Also, there is the [[Weierstrass Institute for Applied Analysis and Stochastics]] in Berlin.

== Selected works ==
* ''Zur Theorie der Abelschen Funktionen'' (1854)
* ''Theorie der Abelschen Funktionen'' (1856)
* ''[http://name.umdl.umich.edu/AAN8481.0001.001 Abhandlungen-1]'', Math. Werke. Bd. 1. Berlin, 1894
* ''[http://name.umdl.umich.edu/AAN8481.0002.001 Abhandlungen-2]'', Math. Werke. Bd. 2. Berlin, 1895
* ''[http://name.umdl.umich.edu/AAN8481.0003.001 Abhandlungen-3]'', Math. Werke. Bd. 3. Berlin, 1903
* ''[http://name.umdl.umich.edu/AAN8481.0004.001 Vorl. ueber die Theorie der Abelschen Transcendenten]'', Math. Werke. Bd. 4. Berlin, 1902
* ''[http://name.umdl.umich.edu/AAN8481.0007.001 Vorl. ueber Variationsrechnung]'', Math. Werke. Bd. 7. Leipzig, 1927

== See also ==
* [[List of things named after Karl Weierstrass]]

== References ==
{{Reflist}}

== External links ==
{{commons|Karl Weierstrass}}
{{Wikiquote}}
*{{MacTutor Biography|id=Weierstrass}}
* [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]''.
* {{Gutenberg author |id=34161| name=Karl Weierstrass}}
* {{Internet Archive author |sname=Karl Weierstrass}}

{{Copley Medallists 1851-1900}}

{{Authority control}}

{{DEFAULTSORT:Weierstrass, Karl}}
[[Category:1815 births]]
[[Category:1897 deaths]]
[[Category:19th-century German mathematicians]]
[[Category:German mathematical analysts]]
[[Category:People from the Province of Westphalia]]
[[Category:People from Braniewo]]
[[Category:Recipients of the Copley Medal]]
[[Category:University of Bonn alumni]]
[[Category:University of Königsberg alumni]]
[[Category:University of Münster alumni]]
[[Category:Academic staff of the Humboldt University of Berlin]]
[[Category:Academic staff of Technische Universität Berlin]]
[[Category:Foreign members of the Royal Society]]
[[Category:Foreign associates of the National Academy of Sciences]]
[[Category:Corresponding members of the Saint Petersburg Academy of Sciences]]
[[Category:Honorary members of the Saint Petersburg Academy of Sciences]]
[[Category:Recipients of the Pour le Mérite (civil class)]]
[[Category:German Roman Catholics]]
[[Category:Deaths from pneumonia in Germany]]
[[Category:Recipients of the Cothenius Medal]]