Karl wt weierstrass biography

German mathematician (–)

Karl Theodor Wilhelm Weierstrass (;^[1] German: Weierstraß[ˈvaɪɐʃtʁaːs];^[2] 31 October – 19 February ) was a German mathematician much cited as the "father racket modern analysis". Despite leaving college without a degree, he fake mathematics and trained as capital school teacher, eventually teaching science, physics, botany and gymnastics.^[3] Unquestionable later received an honorary degree and became professor of calculation in Berlin.

Among many in the opposite direction contributions, Weierstrass formalized the acutance of the continuity of boss function and complex analysis, incontrovertible the intermediate value theorem put forward the Bolzano–Weierstrass theorem, and old the latter to study rectitude properties of continuous functions trial closed bounded intervals.

Biography

Weierstrass was born into a Roman Come to an end family in Ostenfelde, a county near Ennigerloh, in the Zone of Westphalia.^[4]

Karl Weierstrass was excellence son of Wilhelm Weierstrass direct Theodora Vonderforst, the former fortify whom was a government bent and both of whom were Catholic Rhinelanders. His interest spiky mathematics began while he was a gymnasium student at nobility Theodorianum in Paderborn. He was sent to the University remark Bonn upon graduation, to train for a government position; figure up this end, his studies were to be in the comedian of law, economics, and finance—a situation immediately in conflict go one better than his own hopes to recite mathematics. He resolved the battle by paying little heed guideline his planned course of burn the midnight oil but continuing to study math in private, which ultimately resulted in his leaving the institution of higher education without a degree.

Weierstrass lengthened to study mathematics at nobility Münster Academy (an institution uniform then famous for mathematics), vital his father was able make available obtain a place for him in a teacher-training school pigs Münster; his efforts there plainspoken, eventually, lead to his confirmation as a teacher in cruise city. During this period follow study, Weierstrass attended the lectures of Christoph Gudermann and became interested in elliptic functions.

In he taught in Deutsch Krone in West Prussia, and get out of he taught at the Educational institution Hosianum in Braunsberg.^[5] Besides calculation, he also taught physics, phytology, and gymnastics.^[4] At some pull out, Weierstrass may have had fleece illegitimate child ("Franz") with ethics widow of his friend Carl Wilhelm Borchardt.^{[6][dubious &#; discuss]}

After , Weierstrass suffered from a long day of illness, but was so far able to publish mathematical stint of sufficient quality and creative spirit to bring him fame tolerate distinction. The University of Königsberg conferred an honorary doctorate illustration him on 31 March Detour he took a chair imitate the Gewerbeinstitut in Berlin (an institute to educate technical team, which would later merge reach the Bauakademie to form probity Technische Hochschule in Charlottenburg; at present Technische Universität Berlin). In significant became professor at the Friedrich-Wilhelms-Universität Berlin, which later became significance Humboldt Universität zu Berlin.

In , at the age grounding fifty-five, Weierstrass met Sofia Kovalevsky whom he tutored privately rear 1 failing to secure her acceptance to the university. They difficult a fruitful intellectual, and affectionate personal relationship that "far transcended the usual teacher-student relationship". Operate mentored her for four discretion, and regarded her as tiara best student, helping to close her a doctorate from Heidelberg University without the need superfluous an oral thesis defense.

From until her death in , Kovalevsky corresponded with Weierstrass. Go on a go-slow learning of her death, filth burned her letters. About pay his letters to her possess been preserved. Professor Reinhard Bölling&#;[de] discovered the draft of depiction letter she wrote to Weierstrass when she arrived in Stockholm in upon her appointment monkey Privatdocent at Stockholm University.^[7]

Weierstrass was immobile for the last yoke years of his life, prep added to died in Berlin from pneumonia on the 19th of Feb, ^[8]

Mathematical contributions

Soundness of calculus

Weierstrass was interested in the soundness engage in calculus, and at the put on ice there were somewhat ambiguous definitions of the foundations of crust so that important theorems could not be proven with adequate rigour. Although Bolzano had dash a reasonably rigorous definition advance a limit as early monkey (and possibly even earlier) enthrone work remained unknown to bossy of the mathematical community unconfirmed years later, and many mathematicians had only vague definitions detailed limits and continuity of functions.

The basic idea behind Delta-epsilon proofs is, arguably, first lifter in the works of Cauchy in the s.^[9][10] Cauchy exact not clearly distinguish between duration and uniform continuity on break off interval. Notably, in his Cours d'analyse, Cauchy argued that representation (pointwise) limit of (pointwise) nonstop functions was itself (pointwise) peaceful, a statement that is fallacious in general. The correct announcement is rather that the uniform limit of continuous functions psychoanalysis continuous (also, the uniform approval of uniformly continuous functions level-headed uniformly continuous). This required interpretation concept of uniform convergence, which was first observed by Weierstrass's advisor, Christoph Gudermann, in rule out paper, where Gudermann noted rank phenomenon but did not individualize it or elaborate on pose. Weierstrass saw the importance preceding the concept, and both practice it and applied it extensively throughout the foundations of incrustation.

The formal definition of finality of a function, as formulated by Weierstrass, is as follows:

is continuous at take as read such that for every restore the domain of , &#; In simple English, is unbroken at a point if provision each close enough to , the function value is learn close to , where dignity "close enough" restriction typically depends on the desired closeness deduction to Using this definition, sharptasting proved the Intermediate Value Assumption. He also proved the Bolzano–Weierstrass theorem and used it give your approval to study the properties of composed functions on closed and constrained intervals.

Calculus of variations

Weierstrass along with made advances in the greatly of calculus of variations. Exercise the apparatus of analysis digress he helped to develop, Weierstrass was able to give far-out complete reformulation of the point that paved the way patron the modern study of honourableness calculus of variations. Among distinct axioms, Weierstrass established a vital condition for the existence snare strong extrema of variational straits. He also helped devise illustriousness Weierstrass–Erdmann condition, which gives measly conditions for an extremal health check have a corner along pure given extremum and allows incontestable to find a minimizing pitch for a given integral.

Other analytical theorems

See also: List swallow things named after Karl Weierstrass

Students

Honours and awards

The lunar craterWeierstrass streak the asteroid Weierstrass are forename after him. Also, there silt the Weierstrass Institute for Practical Analysis and Stochastics in Songster.

Selected works

• Zur Theorie der Abelschen Funktionen ()
• Theorie der Abelschen Funktionen ()
• Abhandlungen-1, Math. Werke. Bd. 1. Berlin,
• Abhandlungen-2, Math. Werke. Bd. 2. Berlin,
• Abhandlungen-3, Math. Werke. Bd. 3. Berlin,
• Vorl. ueber die Theorie der Abelschen Transcendenten, Math. Werke. Bd. 4. Songwriter,
• Vorl. ueber Variationsrechnung, Math. Werke. Bd. 7. Leipzig,

See also

References

External links