Karl Weierstrass
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Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
often cited as the "father of modern
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
". Despite leaving university without a degree, he studied mathematics and trained as a school teacher, eventually teaching mathematics, physics,
botany Botany, also called , plant biology or phytology, is the science of plant life and a branch of biology. A botanist, plant scientist or phytologist is a scientist who specialises in this field. The term "botany" comes from the Ancient Greek w ...
and gymnastics. He later received an honorary doctorate and became professor of mathematics in Berlin. Among many other contributions, Weierstrass formalized the definition of the continuity of a function, proved the intermediate value theorem and the
Bolzano–Weierstrass theorem In mathematics, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space \R^n. The theorem states that each ...
, and used the latter to study the properties of continuous functions on closed bounded intervals.


Biography

Weierstrass was born into a
Roman Catholic Roman or Romans most often refers to: *Rome, the capital city of Italy * Ancient Rome, Roman civilization from 8th century BC to 5th century AD * Roman people, the people of ancient Rome *'' Epistle to the Romans'', shortened to ''Romans'', a let ...
family in Ostenfelde, a village near
Ennigerloh Ennigerloh () is a town in the district of Warendorf, in North Rhine-Westphalia, Germany. It is situated approximately 25 km northeast of Hamm and 30 km southeast of Münster. The town, located in an agricultural area and with a well-p ...
, in the Province of Westphalia. Weierstrass was the son of Wilhelm Weierstrass, a government official, and Theodora Vonderforst both of whom were catholic Rhinelanders. His interest in mathematics began while he was a gymnasium student at the Theodorianum in
Paderborn Paderborn (; Westphalian: ''Patterbuorn'', also ''Paterboärn'') is a city in eastern North Rhine-Westphalia, Germany, capital of the Paderborn district. The name of the city derives from the river Pader and ''Born'', an old German term for t ...
. He was sent to the
University of Bonn The Rhenish Friedrich Wilhelm University of Bonn (german: Rheinische Friedrich-Wilhelms-Universität Bonn) is a public research university located in Bonn, North Rhine-Westphalia, Germany. It was founded in its present form as the ( en, Rhine ...
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 study in mathematics. The outcome was that he left the university without a degree. He then studied mathematics at the Münster Academy (which was even then famous for mathematics) and his father was able to obtain a place for him in a teacher training school in
Münster Münster (; nds, Mönster) is an independent city (''Kreisfreie Stadt'') in North Rhine-Westphalia, Germany. It is in the northern part of the state and is considered to be the cultural centre of the Westphalia region. It is also a state di ...
. Later he was certified as a teacher in that city. During this period of study, Weierstrass attended the lectures of
Christoph Gudermann Christoph Gudermann (25 March 1798 – 25 September 1852) was a German mathematician noted for introducing the Gudermannian function and the concept of uniform convergence, and for being the teacher of Karl Weierstrass, who was greatly influen ...
and became interested in elliptic functions. In 1843 he taught in
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in
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and since 1848 he taught at the Lyceum Hosianum in
Braunsberg Braniewo () (german: Braunsberg in Ostpreußen, la, Brunsberga, Old Prussian: ''Brus'', lt, Prūsa), is a town in northern Poland, in Warmia, in the Warmian-Masurian Voivodeship, with a population of 16,907 as of June 2021. It is the capita ...
. Besides mathematics he also taught physics, botany, and gymnastics. Weierstrass may have had an illegitimate child named Franz with the widow of his friend
Carl Wilhelm Borchardt Carl Wilhelm Borchardt (22 February 1817 – 27 June 1880) was a German mathematician. Borchardt was born to a Jewish family in Berlin. His father, Moritz, was a respected merchant, and his mother was Emma Heilborn. Borchardt studied under ...
. After 1850 Weierstrass suffered from a long period of illness, but was able to publish mathematical articles that brought him fame and distinction. The
University of Königsberg The University of Königsberg (german: Albertus-Universität Königsberg) was the university of Königsberg in East Prussia. It was founded in 1544 as the world's second Protestant academy (after the University of Marburg) by Duke Albert of Pruss ...
conferred an honorary doctor's degree 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 Technical University of Berlin). In 1864 he became professor at the Friedrich-Wilhelms-Universität Berlin, which later became the
Humboldt Universität zu Berlin Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiative of ...
. In 1870, at the age of fifty-five, Weierstrass met Sofia Kovalevsky whom he tutored privately after failing to secure her admission to the University. They had a fruitful intellectual, but troubled personal, relationship that "far transcended the usual teacher-student relationship". The misinterpretation of this relationship and Kovalevsky's early death in 1891 was said to have contributed to Weierstrass' later ill-health. He was immobile for the last three years of his life, and died in Berlin from
pneumonia Pneumonia is an inflammatory condition of the lung primarily affecting the small air sacs known as alveoli. Symptoms typically include some combination of productive or dry cough, chest pain, fever, and difficulty breathing. The severi ...
.


Mathematical contributions


Soundness of calculus

Weierstrass was interested in the
soundness In logic or, more precisely, deductive reasoning, an argument is sound if it is both valid in form and its premises are true. Soundness also has a related meaning in mathematical logic, wherein logical systems are sound if and only if every formu ...
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
Bolzano Bolzano ( or ; german: Bozen, (formerly ); bar, Bozn; lld, Balsan or ) is the capital city of the province of South Tyrol in northern Italy. With a population of 108,245, Bolzano is also by far the largest city in South Tyrol and the third la ...
had developed a reasonably rigorous definition of a
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
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
limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
and continuity of functions. The basic idea behind Delta-epsilon proofs is, arguably, first found in the works of
Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
in the 1820s. 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 of continuous functions is continuous (also, the uniform limit of uniformly continuous functions is uniformly continuous). This required the concept of
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
, which was first observed by Weierstrass's advisor,
Christoph Gudermann Christoph Gudermann (25 March 1798 – 25 September 1852) was a German mathematician noted for introducing the Gudermannian function and the concept of uniform convergence, and for being the teacher of Karl Weierstrass, who was greatly influen ...
, 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: \displaystyle f(x) is continuous at \displaystyle x = x_0 if \displaystyle \forall \ \varepsilon > 0\ \exists\ \delta > 0 such that for every x in the domain of f,   \displaystyle \ , x-x_0, < \delta \Rightarrow , f(x) - f(x_0), < \varepsilon. In simple English, \displaystyle f(x) is continuous at a point \displaystyle x = x_0 if for each x close enough to x_0, the function value f(x) is very close to f(x_0), where the "close enough" restriction typically depends on the desired closeness of f(x_0) to f(x). Using this definition, he proved the Intermediate Value Theorem. He also proved the
Bolzano–Weierstrass theorem In mathematics, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space \R^n. The theorem states that each ...
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 The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
of variational problems. He also helped devise the
Weierstrass–Erdmann condition The Weierstrass–Erdmann condition is a mathematical result from the calculus of variations, which specifies sufficient conditions for broken extremals (that is, an extremal which is constrained to be smooth except at a finite number of "corners") ...
, 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

* Stone–Weierstrass theorem *
Casorati–Weierstrass theorem In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati. In Russian ...
*
Weierstrass elliptic function In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by t ...
*
Weierstrass function In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass. The Weierstr ...
*
Weierstrass M-test In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely. It applies to series whose terms are bounded functions with real or complex values, and is analogous t ...
*
Weierstrass preparation theorem 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 p ...
*
Lindemann–Weierstrass theorem In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following: In other words, the extension field \mathbb(e^, \dots, e^) has transce ...
* Weierstrass factorization theorem *
Weierstrass–Enneper parameterization In mathematics, the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of differential geometry. Alfred Enneper and Karl Weierstrass studied minimal surfaces as far back as 1863. Let f and g be functions on either ...


Students

*
Edmund Husserl , thesis1_title = Beiträge zur Variationsrechnung (Contributions to the Calculus of Variations) , thesis1_url = https://fedora.phaidra.univie.ac.at/fedora/get/o:58535/bdef:Book/view , thesis1_year = 1883 , thesis2_title ...


Honours and awards

The lunar crater
Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
and the asteroid
14100 Weierstrass 141 may refer to: * 141 (number), an integer * AD 141, a year of the Julian calendar * 141 BC __NOTOC__ Year 141 BC was a year of the pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Caepio and Pompeius (or, ...
are named after him. Also, there is the
Weierstrass Institute for Applied Analysis and Stochastics The Weierstrass Institute for Applied Analysis and Stochastics (WIAS), is a part of the Forschungsverbund Berlin e.V. and a member of the Leibniz Association. Based in Berlin’s district Mitte, the institute's research activities involve a ...
in Berlin.


Selected works

* ''Zur Theorie der Abelschen Funktionen'' (1854) * ''Theorie der Abelschen Funktionen'' (1856) *
Abhandlungen-1
', Math. Werke. Bd. 1. Berlin, 1894 *
Abhandlungen-2
', Math. Werke. Bd. 2. Berlin, 1895 *
Abhandlungen-3
', Math. Werke. Bd. 3. Berlin, 1903 *
Vorl. ueber die Theorie der Abelschen Transcendenten
', Math. Werke. Bd. 4. Berlin, 1902 *
Vorl. ueber Variationsrechnung
', Math. Werke. Bd. 7. Leipzig, 1927


See also

*
List of things named after Karl Weierstrass This is a list of things named after the German mathematician Karl Weierstrass. Mathematical concepts, theorems, and the like Named after Weierstrass and other persons Named after Weierstrass alone {{columns-list, colwidth=20em, * Weierst ...


References


External links

*
Digitalized versions of Weierstrass's original publications
are freely available online from the library of the
Berlin Brandenburgische Akademie der Wissenschaften
'. * * {{DEFAULTSORT:Weierstrass, Karl 1815 births 1897 deaths 19th-century German mathematicians Mathematical analysts People from the Province of Westphalia People from Braniewo Recipients of the Copley Medal University of Bonn alumni University of Königsberg alumni University of Münster alumni Humboldt University of Berlin faculty Technical University of Berlin faculty Foreign Members of the Royal Society Foreign associates of the National Academy of Sciences Corresponding members of the Saint Petersburg Academy of Sciences Honorary members of the Saint Petersburg Academy of Sciences Recipients of the Pour le Mérite (civil class) German Roman Catholics Deaths from pneumonia in Germany