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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
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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 continuing private
1220:. 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
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681:, 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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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
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From 1870 until her death in 1891, Kovalevsky corresponded with
Weierstrass. Upon learning of her death, he burned her letters. About 150 of his letters to her have been preserved. Professor
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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
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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
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a doctorate for her from
Heidelberg University without the need for an oral thesis defense. He was immobile for the last three years of his life, and died in Berlin from
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of continuous functions is continuous (also, the uniform limit of uniformly continuous functions is uniformly continuous). This required the concept of
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After 1850 Weierstrass suffered from a long period of illness, but was able to publish mathematical articles that brought him fame and distinction. The
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in the 1820s. Cauchy did not clearly distinguish between continuity and uniform continuity on an interval. Notably, in his 1821
1541:"Einige Nachträge zur Biographie von Karl Weierstraß. (German) [Some postscripts to the biography of Karl Weierstrass]"
513:(which was even then famous for mathematics) and his father was able to obtain a place for him in a teacher training school in
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430:". Despite leaving university without a degree, he studied mathematics and trained as a school teacher, eventually teaching
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Weierstrass was the son of
Wilhelm Weierstrass, a government official, and Theodora Vonderforst both of whom were Catholic
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discovered the draft of the letter she wrote to
Weierstrass when she arrived in Stockholm in 1883 upon her appointment as
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517:. Later he was certified as a teacher in that city. During this period of study, Weierstrass attended the lectures of
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study in mathematics. The outcome was that he left the university without a degree. He then studied mathematics at the
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1561:. Gillispie, Charles Coulston,, American Council of Learned Societies. New York. 1970. p. 223.
578:). In 1864 he became professor at the Friedrich-Wilhelms-Universität Berlin, which later became the
1810:
465:, and used the latter to study the properties of continuous functions on closed bounded intervals.
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1914:
1712:"Septième Leçon – Valeurs de quelques expressions qui se présentent sous les formes indéterminées
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1978:
947:{\displaystyle \displaystyle \ |x-x_{0}|<\delta \Rightarrow |f(x)-f(x_{0})|<\varepsilon .}
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The formal definition of continuity of a function, as formulated by
Weierstrass, is as follows:
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and used it to study the properties of continuous functions on closed and bounded intervals.
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Weierstrass may have had an illegitimate child named Franz with the widow of his friend
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in Berlin (an institute to educate technical workers which would later merge with the
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Résumé des leçons données à l'école royale polytechnique sur le calcul infinitésimal
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Relation qui existe entre le rapport aux différences finies et la fonction dérivée"
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The
Kowalevski Property (Leeds, 2000) CRM Proceedings & Lecture Notes, vol. 32
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797:{\displaystyle \displaystyle \forall \ \varepsilon >0\ \exists \ \delta >0}
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1457:. . Abington: Helicon. Available from: http://libezproxy.open.ac.uk/login?url=
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Among many other contributions, Weierstrass formalized the definition of the
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1478:. School of Mathematics and Statistics, University of St Andrews, Scotland
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1660:"Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus"
1505:(in German), Wiesbaden: Springer Fachmedien Wiesbaden, pp. 11–51,
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544:. Besides mathematics he also taught physics, botany, and gymnastics.
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Elstrodt, Jürgen (2016), König, Wolfgang; Sprekels, Jürgen (eds.),
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The
Hutchinson unabridged encyclopedia with atlas and weather guide
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35:
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Corresponding members of the Saint
Petersburg Academy of Sciences
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1751:{\displaystyle {\frac {\infty }{\infty }},\infty ^{0},\ldots }
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1453:
Weierstrass, Karl
Theodor Wilhelm. (2018). In Helicon (Ed.),
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Honorary members of the Saint
Petersburg Academy of Sciences
1817:
Digitalized versions of Weierstrass's original publications
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7. Auflage. Bibliographisches Institut, Berlin 2015,
1327:
Weierstrass Institute for Applied Analysis and Stochastics
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on him on 31 March 1854. In 1856 he took a chair at the
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Foreign associates of the National Academy of Sciences
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are freely available online from the library of the
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Vorl. ueber die Theorie der Abelschen Transcendenten
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had developed a reasonably rigorous definition of a
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Academic staff of the Humboldt University of Berlin
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Berlin Brandenburgische Akademie der Wissenschaften
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1224:of variational problems. He also helped devise the
585:In 1870, at the age of fifty-five, Weierstrass met
492:. His interest in mathematics began while he was a
418:; 31 October 1815 – 19 February 1897) was a German
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1499:"Die prägenden Jahre im Leben von Karl Weierstraß"
1474:O'Connor, J. J.; Robertson, E. F. (October 1998).
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658:proofs is, arguably, first found in the works of
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1547:. San Diego, CA: Academic Press. pp. 65–91.
2361:Academic staff of Technische Universität Berlin
1216:Weierstrass also made advances in the field of
2386:Recipients of the Pour le Mérite (civil class)
1862:
621:
1620:. American Mathematical Soc. pp. 1–19.
1422:Random House Webster's Unabridged Dictionary
1231:
1539:Biermann, Kurt-R.; Schubring, Gert (1996).
1469:
1467:
1398:List of things named after Karl Weierstrass
1238:List of things named after Karl Weierstrass
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1855:
1589:: CS1 maint: location missing publisher (
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1325:are named after him. Also, there is the
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1806:MacTutor History of Mathematics Archive
2326:People from the Province of Westphalia
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2163:Friedrich August Kekulé von Stradonitz
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1306:
1200:Using this definition, he proved the
1013:{\displaystyle \displaystyle x=x_{0}}
745:{\displaystyle \displaystyle x=x_{0}}
477:family in Ostenfelde, a village near
413:
2366:Foreign members of the Royal Society
1339:Zur Theorie der Abelschen Funktionen
1289:Weierstrass–Enneper parameterization
1387:, Math. Werke. Bd. 7. Leipzig, 1927
13:
2316:19th-century German mathematicians
1838:Works by or about Karl Weierstrass
1733:
1724:
1721:
1658:Grabiner, Judith V. (March 1983),
1559:Dictionary of scientific biography
1476:"Karl Theodor Wilhelm Weierstrass"
1379:, Math. Werke. Bd. 4. Berlin, 1902
1371:, Math. Werke. Bd. 3. Berlin, 1903
1363:, Math. Werke. Bd. 2. Berlin, 1895
1355:, Math. Werke. Bd. 1. Berlin, 1894
978:{\displaystyle \displaystyle f(x)}
778:
760:
710:{\displaystyle \displaystyle f(x)}
631:Weierstrass was interested in the
14:
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2401:Recipients of the Cothenius Medal
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1667:The American Mathematical Monthly
1608:Kuznetsov, Vadim B., ed. (2002).
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1284:Weierstrass factorization theorem
2396:Deaths from pneumonia in Germany
1435:Duden. Das Aussprachewörterbuch.
1345:Theorie der Abelschen Funktionen
343:
339:Karl Theodor Wilhelm Weierstrass
16:German mathematician (1815–1897)
2346:University of Königsberg alumni
1700:
1696:from the original on 2014-11-29
1274:Weierstrass preparation theorem
536:and from 1848 he taught at the
2336:Recipients of the Copley Medal
1651:
1612:The life of S. V. Kovalevskaya
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1384:Vorl. ueber Variationsrechnung
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580:Humboldt Universität zu Berlin
135:Weierstrass product inequality
1:
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1279:Lindemann–Weierstrass theorem
1259:Weierstrass elliptic function
1226:Weierstrass–Erdmann condition
576:Technische Universität Berlin
145:Weierstrass–Erdmann condition
2351:University of Münster alumni
2321:German mathematical analysts
1254:Casorati–Weierstrass theorem
473:Weierstrass was born into a
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216:Friedrich Wilhelm University
7:
1511:10.1007/978-3-658-10619-5_2
1503:Karl Weierstraß (1815–1897)
1391:
1294:
1244:Bolzano–Weierstrass theorem
1206:Bolzano–Weierstrass theorem
1202:Intermediate Value Theorem.
463:Bolzano–Weierstrass theorem
155:Bolzano–Weierstrass theorem
10:
2417:
2107:Jean-Baptiste Boussingault
2083:August Wilhelm von Hofmann
1640:: CS1 maint: postscript (
1235:
622:Mathematical contributions
459:intermediate value theorem
140:(ε, δ)-definition of limit
2341:University of Bonn alumni
1885:
1829:Works by Karl Weierstrass
1249:Stone–Weierstrass theorem
1232:Other analytical theorems
985:is continuous at a point
521:and became interested in
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1811:University of St Andrews
1630:; See p. 7 in 2002 book.
1161:{\displaystyle f(x_{0})}
1125:{\displaystyle f(x_{0})}
560:honorary doctor's degree
556:University of Königsberg
451:continuity of a function
175:University of Königsberg
2051:Julius Robert von Mayer
574:in Charlottenburg; now
2391:German Roman Catholics
2123:James Joseph Sylvester
1939:Michel Eugène Chevreul
1899:Alexander von Humboldt
1752:
1545:History of mathematics
1218:calculus of variations
1212:Calculus of variations
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549:Carl Wilhelm Borchardt
504:. He was sent to the
483:Province of Westphalia
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85:, Kingdom of Prussia,
61:Province of Westphalia
2227:George Gabriel Stokes
2131:Charles Adolphe Wurtz
2067:Hermann von Helmholtz
2035:Henri Victor Regnault
1915:Johannes Peter Müller
1907:Heinrich Wilhelm Dove
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1593:) CS1 maint: others (
1461:Accessed 8 July 2018.
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1193:{\displaystyle f(x).}
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627:Soundness of calculus
572:Technische Hochschule
528:In 1843 he taught in
415:[ˈvaɪɐʃtʁaːs]
2331:People from Braniewo
2211:Stanislao Cannizzaro
2179:Joseph Dalton Hooker
2043:James Prescott Joule
1955:Wilhelm Eduard Weber
1797:Robertson, Edmund F.
1716:
1264:Weierstrass function
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1089:{\displaystyle f(x)}
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954:In simple English,
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804:such that for every
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616:Stockholm University
422:often cited as the "
324:Ludwig Stickelberger
150:Weierstrass theorems
130:Weierstrass function
2283:Marcellin Berthelot
2275:John William Strutt
2259:Albert von Kölliker
2187:Thomas Henry Huxley
2171:Franz Ernst Neumann
2019:Karl Ernst von Baer
1931:Henri Milne-Edwards
1795:O'Connor, John J.;
1204:He also proved the
679:Christoph Gudermann
675:uniform convergence
519:Christoph Gudermann
226:Christoph Gudermann
2027:Charles Wheatstone
1801:"Karl Weierstrass"
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1614:by Roger L. Cooke"
1307:Honours and awards
1269:Weierstrass M-test
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506:University of Bonn
314:Friedrich Schottky
309:Arthur Schoenflies
279:Sofia Kovalevskaya
264:Johannes Knoblauch
109:University of Bonn
65:Kingdom of Prussia
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2099:James Dwight Dana
1878:Copley Medallists
1833:Project Gutenberg
1766:, Paris, p.
1727:
1627:978-0-8218-7330-4
1568:978-0-684-12926-6
1520:978-3-658-10618-8
1443:978-3-411-04067-4
1323:14100 Weierstrass
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1033:{\displaystyle x}
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837:{\displaystyle f}
824:in the domain of
817:{\displaystyle x}
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232:Doctoral students
222:Academic advisors
192:Scientific career
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346:
342:
328:
319:Hermann Schwarz
259:Wilhelm Killing
249:Georg Frobenius
212:Gewerbeinstitut
186:
159:
118:
114:Münster Academy
102:Alma mater
89:
80:
76:
67:
54:
53:31 October 1815
48:
46:
29:Karl Weierstraß
24:
17:
12:
11:
5:
2414:
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2255:
2251:Carl Gegenbaur
2247:
2239:
2231:
2223:
2219:Rudolf Virchow
2215:
2207:
2199:
2191:
2183:
2175:
2167:
2159:
2151:
2143:
2135:
2127:
2119:
2111:
2103:
2095:
2091:Claude Bernard
2087:
2079:
2071:
2063:
2055:
2047:
2039:
2031:
2023:
2015:
2011:Julius Plücker
2007:
2003:Michel Chasles
1999:
1995:Charles Darwin
1991:
1983:
1975:
1967:
1959:
1951:
1943:
1935:
1927:
1919:
1911:
1903:
1895:
1886:
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1866:
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1845:
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1835:
1826:
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1790:
1789:External links
1787:
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1673:(3): 185–194,
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1368:Abhandlungen-3
1364:
1360:Abhandlungen-2
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1352:Abhandlungen-1
1348:
1342:
1334:
1333:Selected works
1331:
1308:
1305:
1304:
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1301:Edmund Husserl
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1286:
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1276:
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1222:strong extrema
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738:
734:
730:
727:
705:
702:
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651:of functions.
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625:
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620:
475:Roman Catholic
470:
467:
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132:
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123:Known for
120:
119:
117:
116:
111:
105:
103:
99:
98:
95:
91:
90:
81:
79:(aged 81)
73:
69:
68:
55:
44:
40:
39:
31:
30:
26:
25:
22:
15:
9:
6:
4:
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2:
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2200:
2196:
2195:George Salmon
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2180:
2176:
2172:
2168:
2164:
2160:
2156:
2152:
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2144:
2140:
2139:Arthur Cayley
2136:
2132:
2128:
2124:
2120:
2116:
2112:
2108:
2104:
2100:
2096:
2092:
2088:
2084:
2080:
2076:
2075:Louis Pasteur
2072:
2068:
2064:
2060:
2056:
2052:
2048:
2044:
2040:
2036:
2032:
2028:
2024:
2020:
2016:
2012:
2008:
2004:
2000:
1996:
1992:
1988:
1987:Adam Sedgwick
1984:
1980:
1979:Thomas Graham
1976:
1972:
1971:Louis Agassiz
1968:
1964:
1963:Robert Bunsen
1960:
1956:
1952:
1948:
1947:Charles Lyell
1944:
1940:
1936:
1932:
1928:
1924:
1923:Léon Foucault
1920:
1916:
1912:
1908:
1904:
1900:
1896:
1892:
1888:
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1884:
1879:
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1824:
1823:
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1815:
1812:
1808:
1807:
1802:
1798:
1793:
1792:
1774:on 2009-05-04
1773:
1769:
1765:
1764:
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1742:
1737:
1729:
1709:
1708:Cauchy, A.-L.
1703:
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1680:
1676:
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1668:
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1468:
1460:
1456:
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1444:
1440:
1436:
1431:
1424:
1423:
1418:
1417:"Weierstrass"
1413:
1409:
1399:
1396:
1395:
1386:
1385:
1381:
1378:
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1373:
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1321:
1317:
1314:
1302:
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1290:
1287:
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1270:
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1257:
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1252:
1250:
1247:
1245:
1242:
1241:
1239:
1229:
1227:
1223:
1219:
1209:
1207:
1203:
1187:
1181:
1175:
1150:
1146:
1139:
1114:
1110:
1103:
1080:
1074:
1052:
1048:
1027:
1004:
1000:
996:
993:
968:
962:
940:
937:
934:
921:
917:
910:
907:
901:
895:
884:
881:
871:
867:
863:
860:
831:
811:
790:
787:
784:
772:
769:
766:
736:
732:
728:
725:
700:
694:
685:
682:
680:
676:
672:
670:
665:
661:
657:
656:Delta-epsilon
652:
650:
646:
642:
638:
634:
619:
617:
613:
612:
605:
600:
595:
593:
588:
583:
581:
577:
573:
569:
565:
561:
558:conferred an
557:
552:
550:
545:
543:
539:
535:
531:
530:Deutsch Krone
526:
524:
520:
516:
512:
507:
503:
499:
495:
491:
486:
484:
480:
476:
466:
464:
460:
457:, proved the
456:
452:
447:
445:
441:
437:
433:
429:
425:
421:
420:mathematician
416:
412:
407:
400:
340:
331:
325:
322:
320:
317:
315:
312:
310:
307:
305:
302:
300:
297:
295:
292:
290:
287:
285:
284:Mathias Lerch
282:
280:
277:
275:
272:
270:
267:
265:
262:
260:
257:
255:
254:Lazarus Fuchs
252:
250:
247:
245:
242:
240:
237:
236:
234:
230:
227:
224:
220:
217:
213:
210:
206:
203:
200:
196:
193:
189:
182:
179:
176:
172:
169:
168:
166:
162:
156:
153:
151:
148:
146:
143:
141:
138:
136:
133:
131:
128:
127:
125:
121:
115:
112:
110:
107:
106:
104:
100:
96:
92:
88:
87:German Empire
84:
74:
70:
66:
62:
58:
45:
41:
37:
32:
27:
20:
2242:
1891:Richard Owen
1820:
1804:
1776:, retrieved
1772:the original
1762:
1702:
1670:
1666:
1653:
1617:
1611:
1603:
1558:
1553:
1544:
1534:
1524:, retrieved
1502:
1492:
1480:. Retrieved
1454:
1449:
1434:
1430:
1420:
1412:
1382:
1374:
1366:
1358:
1350:
1344:
1338:
1310:
1215:
1020:if for each
686:
683:
668:
663:
653:
630:
611:Privatdocent
609:
596:
584:
570:to form the
567:
563:
553:
546:
534:West Prussia
527:
498:Theodorianum
490:Rhinelanders
487:
472:
448:
338:
337:
274:Ernst Kötter
244:Georg Cantor
208:Institutions
191:
181:Copley Medal
77:(1897-02-19)
2311:1897 deaths
2306:1815 births
2155:Carl Ludwig
1880:(1851–1900)
1647:online text
1482:7 September
1329:in Berlin.
1316:Weierstrass
602: [
568:Bauakademie
432:mathematics
299:Adolf Piltz
294:Eugen Netto
202:Mathematics
94:Nationality
2300:Categories
1778:2009-05-01
1526:2023-08-12
1404:References
1311:The lunar
1236:See also:
649:continuity
542:Braunsberg
479:Ennigerloh
444:gymnastics
411:Weierstraß
304:Carl Runge
57:Ennigerloh
49:1815-10-31
1746:…
1734:∞
1725:∞
1722:∞
1636:cite book
1585:cite book
938:ε
908:−
888:⇒
885:δ
864:−
844:,
785:δ
779:∃
767:ε
761:∀
633:soundness
592:pneumonia
502:Paderborn
494:gymnasium
481:, in the
469:Biography
424:father of
406:‹See Tfd›
171:PhD (Hon)
1710:(1823),
1691:archived
1392:See also
1320:asteroid
1318:and the
1295:Students
461:and the
428:analysis
409:German:
1840:at the
1687:2975545
669:uniform
637:Bolzano
515:Münster
436:physics
426:modern
2285:(1900)
2277:(1899)
2269:(1898)
2261:(1897)
2253:(1896)
2245:(1895)
2237:(1894)
2229:(1893)
2221:(1892)
2213:(1891)
2205:(1890)
2197:(1889)
2189:(1888)
2181:(1887)
2173:(1886)
2165:(1885)
2157:(1884)
2149:(1883)
2141:(1882)
2133:(1881)
2125:(1880)
2117:(1879)
2109:(1878)
2101:(1877)
2093:(1876)
2085:(1875)
2077:(1874)
2069:(1873)
2061:(1872)
2053:(1871)
2045:(1870)
2037:(1869)
2029:(1868)
2021:(1867)
2013:(1866)
2005:(1865)
1997:(1864)
1989:(1863)
1981:(1862)
1973:(1861)
1965:(1860)
1957:(1859)
1949:(1858)
1941:(1857)
1933:(1856)
1925:(1855)
1917:(1854)
1909:(1853)
1901:(1852)
1893:(1851)
1685:
1624:
1575:
1565:
1517:
1441:
1347:(1856)
1341:(1854)
1313:crater
853:
782:
776:
764:
660:Cauchy
645:limits
440:botany
198:Fields
183:(1895)
177:(1854)
164:Awards
97:German
83:Berlin
1694:(PDF)
1683:JSTOR
1663:(PDF)
1577:89822
671:limit
641:limit
606:]
1642:link
1622:ISBN
1595:link
1591:link
1573:OCLC
1563:ISBN
1515:ISBN
1484:2014
1459:Link
1439:ISBN
935:<
882:<
788:>
770:>
647:and
453:and
442:and
72:Died
43:Born
1831:at
1675:doi
1507:doi
1168:to
752:if
614:at
540:in
532:in
500:in
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345:/
341:(
51:)
47:(
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