Karl Weierstrass

Overview

**Karl Theodor Wilhelm Weierstrass**was a German

Germany

Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

mathematician

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

who is often cited as the "father of modern analysis

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

".

Weierstrass was born in Ostenfelde, part of Ennigerloh

Ennigerloh

Ennigerloh is a town in the district of Warendorf, in North Rhine-Westphalia, Germany. It is located approximately 25 km northeast of Hamm and 30 km southeast of Münster....

, Province of Westphalia

Province of Westphalia

The Province of Westphalia was a province of the Kingdom of Prussia and the Free State of Prussia from 1815 to 1946.-History:Napoleon Bonaparte founded the Kingdom of Westphalia, which was a client state of the First French Empire from 1807 to 1813...

.

Weierstrass was the eldest son of Wilhelm Weierstrass, a government official, and Theodora Vonderforst. The family soon moved to Westernkotten in Westphalia, where Weierstrass' three siblings, Peter, Klara, and Elise, were born; none of whom married.

Unanswered Questions

Encyclopedia

**Karl Theodor Wilhelm Weierstrass**was a German

Germany

Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

mathematician

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

who is often cited as the "father of modern analysis

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

".

## Biography

Weierstrass was born in Ostenfelde, part of EnnigerlohEnnigerloh

Ennigerloh is a town in the district of Warendorf, in North Rhine-Westphalia, Germany. It is located approximately 25 km northeast of Hamm and 30 km southeast of Münster....

, Province of Westphalia

Province of Westphalia

The Province of Westphalia was a province of the Kingdom of Prussia and the Free State of Prussia from 1815 to 1946.-History:Napoleon Bonaparte founded the Kingdom of Westphalia, which was a client state of the First French Empire from 1807 to 1813...

.

Weierstrass was the eldest son of Wilhelm Weierstrass, a government official, and Theodora Vonderforst. The family soon moved to Westernkotten in Westphalia, where Weierstrass' three siblings, Peter, Klara, and Elise, were born; none of whom married. Shortly after Elise's birth in 1826, Weierstrass' mother died, and a year later his father remarried. His interest in mathematics began while he was a

*Gymnasium*

student at Theodorianum in Paderborn

Gymnasium (school)

A gymnasium is a type of school providing secondary education in some parts of Europe, comparable to English grammar schools or sixth form colleges and U.S. college preparatory high schools. The word γυμνάσιον was used in Ancient Greece, meaning a locality for both physical and intellectual...

Paderborn

Paderborn is a city in North Rhine-Westphalia, Germany, capital of the Paderborn district. The name of the city derives from the river Pader, which originates in more than 200 springs near Paderborn Cathedral, where St. Liborius is buried.-History:...

. He was sent to the University of Bonn

University of Bonn

The University of Bonn is a public research university located in Bonn, Germany. Founded in its present form in 1818, as the linear successor of earlier academic institutions, the University of Bonn is today one of the leading universities in Germany. The University of Bonn offers a large number...

upon graduation to prepare for a government position. Because his studies were to be in the fields of law

Law

Law is a system of rules and guidelines which are enforced through social institutions to govern behavior, wherever possible. It shapes politics, economics and society in numerous ways and serves as a social mediator of relations between people. Contract law regulates everything from buying a bus...

, 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

University of Münster

The University of Münster is a public university located in the city of Münster, North Rhine-Westphalia in Germany. The WWU is part of the Deutsche Forschungsgemeinschaft, a society of Germany's leading research universities...

(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

Münster

Münster is an independent city in North Rhine-Westphalia, Germany. It is located in the northern part of the state and is considered to be the cultural centre of the Westphalia region. It is also capital of the local government region Münsterland...

. Later he was certified as a teacher in that city. During this period of study, Weierstrass attended the lectures of Christoph Gudermann

Christoph Gudermann

Christoph Gudermann was born in Vienenburg. He was the son of a school teacher and became a teacher himself after studying at the University of Göttingen, where his advisor was Karl Friedrich Gauss...

and became interested in elliptic function

Elliptic function

In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...

s.

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.

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

Technical University of Berlin

The Technische Universität Berlin is a research university located in Berlin, Germany. Translating the name into English is discouraged by the university, however paraphrasing as Berlin Institute of Technology is recommended by the university if necessary .The TU Berlin was founded...

, then known as the Gewerbeinstitut. He was immobile for the last three years of his life, and died in Berlin from pneumonia

Pneumonia

Pneumonia is an inflammatory condition of the lung—especially affecting the microscopic air sacs —associated with fever, chest symptoms, and a lack of air space on a chest X-ray. Pneumonia is typically caused by an infection but there are a number of other causes...

.

### Soundness of calculus

Weierstrass was interested in the soundnessSoundness

In mathematical logic, a logical system has the soundness property if and only if its inference rules prove only formulas that are valid with respect to its semantics. In most cases, this comes down to its rules having the property of preserving truth, but this is not the case in general. The word...

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 Bolzano

Bernard Bolzano

Bernhard Placidus Johann Nepomuk Bolzano , Bernard Bolzano in English, was a Bohemian mathematician, logician, philosopher, theologian, Catholic priest and antimilitarist of German mother tongue.-Family:Bolzano was the son of two pious Catholics...

had developed a reasonably rigorous definition of a limit

Limit of a function

In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

as early as 1817 (and possibly even earlier) his work remained unknown to most of the mathematical community until years later,

and many had only vague definitions of limits

Limit of a function

In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

and continuity

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

of functions.

Cauchy

Augustin Louis Cauchy

Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...

gave a form of the (ε, δ)-definition of limit, in the context of formally defining the derivative, in the 1820s,

but did not correctly distinguish between continuity at a point versus uniform continuity on an interval, due to insufficient rigor. Notably, in his 1821

*Cours d'analyse,*Cauchy gave a famously incorrect proof that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous. The correct statement is rather that 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

Christoph Gudermann

Christoph Gudermann was born in Vienenburg. He was the son of a school teacher and became a teacher himself after studying at the University of Göttingen, where his advisor was Karl Friedrich Gauss...

, 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 (ε, δ)-definition of limit, as formulated by Weierstrass, is as follows:

is continuous at if such that

Using this definition and the concept of uniform convergence,

Weierstrass was able to write proofs of several then-unproven theorems such as the intermediate value theorem

Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value....

(for which Bolzano

Bernard Bolzano

Bernhard Placidus Johann Nepomuk Bolzano , Bernard Bolzano in English, was a Bohemian mathematician, logician, philosopher, theologian, Catholic priest and antimilitarist of German mother tongue.-Family:Bolzano was the son of two pious Catholics...

had already given a rigorous proof), the Bolzano–Weierstrass theorem

Bolzano–Weierstrass theorem

In real analysis, the Bolzano–Weierstrass theorem is a fundamental result about convergence in a finite-dimensional Euclidean space Rn. The theorem states thateach bounded sequence in Rn has a convergent subsequence...

, and Heine–Borel theorem

Heine–Borel theorem

In the topology of metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:For a subset S of Euclidean space Rn, the following two statements are equivalent:*S is closed and bounded...

.

### Calculus of variations

Weierstrass also made significant advancements in the field of calculus of variationsCalculus of variations

Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...

. 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 several significant results, Weierstrass established a necessary condition for the existence of strong extrema of variational problems. He also helped devise the Weierstrass–Erdmann condition

Weierstrass–Erdmann condition

The Weierstrass–Erdmann condition is a technical tool from the calculus of variations. This condition gives the sufficient conditions for an extremal to have a corner.- Conditions :...

which give sufficient conditions for an extremal to have a corner.

### Other analytical theorems

- Stone–Weierstrass theorem
- Weierstrass–Casorati theoremWeierstrass–Casorati theoremIn complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of meromorphic functions near essential singularities...
- Weierstrass's elliptic functionsWeierstrass's elliptic functionsIn mathematics, Weierstrass's elliptic functions are elliptic functions that take a particularly simple form; they are named for Karl Weierstrass...
- Weierstrass functionWeierstrass functionIn mathematics, the Weierstrass function is a pathological example of a real-valued function on the real line. The function has the property that it is continuous everywhere but differentiable nowhere...
- Weierstrass M-test
- Weierstrass preparation theoremWeierstrass preparation theoremIn mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point P...
- Lindemann–Weierstrass theoremLindemann–Weierstrass theoremIn mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states that if 1, ..., are algebraic numbers which are linearly independent over the rational numbers ', then 1, ..., are algebraically...
- Weierstrass factorization theoremWeierstrass factorization theoremIn 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...
- Enneper–Weierstrass parameterization
- Sokhatsky–Weierstrass theorem

## 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, 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 HusserlEdmund HusserlEdmund Gustav Albrecht Husserl was a philosopher and mathematician and the founder of the 20th century philosophical school of phenomenology. He broke with the positivist orientation of the science and philosophy of his day, yet he elaborated critiques of historicism and of psychologism in logic...
- Sofia KovalevskayaSofia KovalevskayaSofia Vasilyevna Kovalevskaya , was the first major Russian female mathematician, responsible for important original contributions to analysis, differential equations and mechanics, and the first woman appointed to a full professorship in Northern Europe.She was also one of the first females to...
- Gösta Mittag-Leffler
- Hermann SchwarzHermann SchwarzKarl Hermann Amandus Schwarz was a German mathematician, known for his work in complex analysis. He was born in Hermsdorf, Silesia and died in Berlin...
- Carl Johannes ThomaeCarl Johannes ThomaeCarl Johannes Thomae was a German mathematician.-Biography:...

## Honours and awards

The lunar craterImpact crater

In the broadest sense, the term impact crater can be applied to any depression, natural or manmade, resulting from the high velocity impact of a projectile with a larger body...

Weierstrass

Weierstrass (crater)

Weierstrass is a small lunar crater that is attached to the northern rim of the walled plain Gilbert, in the eastern part of the Moon. It also lies very near the crater Van Vleck, a similar formation just to the southeast that is almost attached to the outer rim...

is named after him.

## External links

- Digitalized versions of Weierstrass's original publications are freely available online from the library of the
*Berlin Brandenburgische Akademie der Wissenschaften*.