Karl Theodor Wilhelm Weierstrass (/ˈvaɪərˌstrɑːs,-ˌʃtrɑːs/;[1] German:Weierstraß[ˈvaɪɐʃtʁaːs];[2] 31 October 1815 – 19 February 1897) was a Germanmathematician often cited as the "father of modernanalysis".[3] Despite leaving university without a degree, he studied mathematics and trained as a school teacher, eventually teachingmathematics,physics,botany andgymnastics.[4] He later received an honorary doctorate and became professor of mathematics in Berlin.
Karl Weierstrass was the son of Wilhelm Weierstrass and Theodora Vonderforst, the former of whom was a government official and both of whom were CatholicRhinelanders. His interest in mathematics began while he was agymnasium student at theTheodorianum inPaderborn. He was sent to theUniversity 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 theMünster Academy (an institution even then famous for mathematics), and his father was able to obtain a place for him in a teacher-training school inMünster; his efforts there did, eventually, lead to his certification as a teacher in that city. During this period of study, Weierstrass attended the lectures ofChristoph Gudermann and became interested inelliptic functions.
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. TheUniversity of Königsberg conferred anhonorary doctorate on him on 31 March 1854. In 1856 he took a chair at theGewerbeinstitut in Berlin (an institute to educate technical workers, which would later merge with theBauakademie to form theTechnische Hochschule in Charlottenburg; nowTechnische Universität Berlin). In 1864 he became professor at the Friedrich-Wilhelms-Universität Berlin, which later became theHumboldt Universität zu Berlin.
In 1870, at the age of fifty-five, Weierstrass metSofia 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. ProfessorReinhard Bölling [de] discovered the draft of the letter she wrote to Weierstrass when she arrived in Stockholm in 1883 upon her appointment asPrivatdocent atStockholm University.[8]
Weierstrass was immobile for the last three years of his life, and died in Berlin frompneumonia on the 19th of February, 1897.[9]
Weierstrass was interested in thesoundness 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. AlthoughBolzano had developed a reasonably rigorous definition of alimit 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 oflimits andcontinuity of functions.
The basic idea behindDelta-epsilon proofs is, arguably, first found in the works ofCauchy in the 1820s.[10][11]Cauchy did not clearly distinguish between continuity and uniform continuity on an interval. Notably, in his 1821Cours 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 theuniform limit of continuous functions is continuous (also, the uniform limit of uniformly continuous functions is uniformly continuous).This required the concept ofuniform 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:
is continuous at if such that for every in the domain of, In simple English, is continuous at a point if for each close enough to, the function value is very close to, where the "close enough" restriction typically depends on the desired closeness of toUsing this definition, he proved theIntermediate Value Theorem. He also proved theBolzano–Weierstrass theorem and used it to study the properties of continuous functions on closed and bounded intervals.
Weierstrass also made advances in the field ofcalculus 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 ofstrong extrema of variational problems. He also helped devise theWeierstrass–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.
^Weierstrass, Karl Theodor Wilhelm. (2018). In Helicon (Ed.),The Hutchinson unabridged encyclopedia with atlas and weather guide. [Online]. Abington: Helicon. Available from: http://libezproxy.open.ac.uk/login?url=Link Accessed 8 July 2018.
^abO'Connor, J. J.; Robertson, E. F. (October 1998)."Karl Theodor Wilhelm Weierstrass". School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved7 September 2014.
^Dictionary of scientific biography. Gillispie, Charles Coulston,, American Council of Learned Societies. New York. 1970. p. 223.ISBN978-0-684-12926-6.OCLC89822.{{cite book}}: CS1 maint: location missing publisher (link) CS1 maint: others (link)
