Topic #8 ------------ OP-SF NET 5.6 ----------- November 15, 1998 ~~~~~~~~~~~~~ From: OP-SF NET Editors Subject: Article on Weierstrass by U. Skornik [The following item appeared in our Activity Group's Newsletter, October 1998; it was reprinted, with permission, from the Russian Newsletter "Integral Transforms and Special Functions" ] Karl Theodor W. Weierstrass - Life and Work On the 100th anniversary of his death by U. Skornik Institute of Mathematics, Polish Academy of Sciences, Staromiejska 8/6, 40-013 Katowice, Poland E-mail: skornik@impan.gov.pl 19 February 1997 marked the hundredth anniversary of the death of the great German mathematician, the father of classical mathematical analysis and theory of Abelian and special functions, Karl Theodor Wilhelm Weierstrass. He was born on 31 October 1815, the first child of Wilhelm Weierstrass, secretary of the mayor of Ostenfelde and Theodora Vonderforst. Nothing in his early life indicated that he would become a famous mathematician. Karl attended the Catholic Gymnasium in Paderborn from 1829 to 1834 when he entered the University of Bonn in order to follow a course in public finance, economics and administration. This choice, far from his own interests, was dictated by his father with the result that, after four years spent on fencing, drinking and mathematics, Karl returned home without having taken any examinations. The years in Bonn, however, were not entirely wasted. It was during the stay there that Weierstrass attended lectures of the famous geometer Plucker, studied "Mecanique Celeste" by Laplace, "Fundamenta nova" by Jacobi and extended his knowledge by an accidentally found transcript of lectures on elliptic functions by Gudermann. As a last resort Karl was sent in 1839 to the Theological and Philosophical Academy at Munster where he was to prepare for a career as a secondary school teacher. He attended lectures on elliptic functions given by Cristof Gudermann. The theory of these functions was initiated by Gauss and developed by Abel and Jacobi. In the early XIXth century Abel considered the elliptic integral of the form \alpha = int(0 to x) {dx}\over \sqrt{(1-c^2x^2)(1+e^2x^2)} and the function x=\varphi(\alpha), inverse to the integral. The function \varphi(\alpha) is called an elliptic function and, when extended to the whole complex plane, gives a doubly periodic function. Jacobi based his theory on the integral u =int( 0 to \varphi {d\varphi}\over \sqrt{1-k^2\sin^2\varphi}}, with a parameter k, between 0 and 1, called the modulus of the elliptic integral. The elliptic functions obtained from that integral were called modular functions by Gudermann. It was a new theory and Gudermann was the first after Jacobi to give lectures on the subject. On 2 May 1840 Weierstrass was given problems for his final examinations; one of them was posed by Gudermann in response to a special wish of his student and concerned elliptic functions. In autumn 1840 Weierstrass presented the results of his research on the decomposition of modular functions. He expressed Jacobi modular functions as ratios of entire functions whose power series coefficients are polynomials in k^2. In memory of Abel, he called them Al functions. Next, Weierstrass introduced his famous \sigma-functions, which differ from Al functions by a multiplier. His dissertation contained significant new material and could ensure an academic position for him in Germany or elsewhere. It is not known why this work, highly prized by Gudermann, was not published until 54 years later in the first volume of Weierstrass's Collected Papers. Instead of gaining mathematical fame, Weierstrass, after passing the second oral part of his examinations in spring 1841, worked for 14 years as a secondary school teacher. After a probationary year in Munster, Weierstrass worked at the Catholic gymnasium (a high level high school) in Deutsch-Krone (West Prussia) from 1842 to 1848 and then in a similar school in Braunsberg (East Prussia) from 1848 to 1855. He taught not only mathematics and physics but also German, botany, geography, history, gymnastics and calligraphy. In 1844 Weierstrass took part in a course for gymnastics teachers in Berlin. During this time he visited the famous geometer Steiner. This, however, did not change his situation. In Deutsch-Krone Weierstrass had neither access to the mathematical literature, nor the possibility of exchanging ideas with any other mathematician. He felt isolated and filled his life with work. During those years he developed the theory of Abelian functions which form a larger class than the elliptic functions. Abelian integrals are defined like elliptic integrals by u = \int(0 to v) R(t,\sqrt{f(t)})dt = I(v) where R(x,y) is a rational function of x and y, except that the function f is of a very general type which includes all polynomials. Inversion then yields Abelian functions, just as elliptic functions arise from the inversion of elliptic integrals, i.e. the integrals where f(t) is a polynomial. During Weierstrass' probationary year at the gymnasium in Munster he worked on three papers. The first "Darstellung einer analytischen Funktion, deren absoluter Betrag zwischen zwei gegebenen Grenzen liegt" contained a proof of Cauchy's integral theorem without the use of double or surface integrals. He also introduced Laurent series and arrived at the Laurent theorem two years before it was officially published in Comptes Rendus by Cauchy. In that work Weierstrass expressed complex numbers in the form r{1+\lambda i}\over {1-\lambda i}, where r is the absolute value and \lambda is a real number given by \lambda = \tan(\theta/2), where \thet is an amplitude of the complex number. The paper "Zur Theorie der Potenzreihen" dates from autumn 1841. In that work Weierstrass introduced the notion of uniform convergence and examined series in several variables. In the next paper "Definition analytischer Funktionen einer Veranderlichen vermittelst algebraischer Differentialgleichungen", dating from 1842, he proved the theorem of Cauchy concerning the solution of systems of differential equations dx/dt = G_i(x_1,...,x_n) with the initial conditions x_i(0) =a_i, where the G_i are polynomials. In that paper Weierstrass described the process of analytic continuation of power series. Those papers were published only in the first volume of his Collected Papers in 1894 but they make it clear that, already in 1842, he was in full possession of all the methods and ideas which allowed him to construct his theory of functions. Unfortunately, his first published paper "Bemerkung uber die analytischen Fakultaten" appeared in a supplement to the school report of the year 1842 in Deutsch-Krone and received little attention. The same happened to another paper on Abelian functions which was published in 1848 in the Braunsberg school prospectus. Weierstrass would have remained unnoticed but for the fact that in summer 1853 during his stay in Munster he read Gudermann's opinion of his dissertation for the first time. Already in 1840 Cristof Gudermann not only recognised Karl's rare talent but also placed him among famous discoverers and suggested that his student should work at a university rather than as a secondary school teacher. This note encouraged Weierstrass to publish his paper ``Zur Theorie der Abelschen Funktionen". Its appearance in 1854 in Crelle's Journal caused a sensation in the mathematical world. The consequences were amazing. The first recognition was the award of an honorary doctorate by the University of Konigsberg. Then the Prussian ministry of education gave him a year's paid leave from the Braunsberg gymnasium to enable him to concentrate on his research. Weierstrass gained enough confidence to apply for the post at the University in Breslau which was vacated by Kummer's appointment as professor in Berlin. It may sound strange that Weierstrass's application was rejected because of Kummer. The reason was that Kummer, who spent 13 years teaching in Breslau, intended to take Weierstrass to Berlin. He applied on 12 June 1856 to the university in Berlin with a request for a post for Weierstrass. He was not successful on that occasion but his eventual success gave to mathematics in Berlin three great names: Weierstrass, Kummer and Kronecker. Weierstrass's "Theorie der Abelschen Funktionen" published in Crelle's Journal in 1856 contained results of his dissertation. D. Hilbert considered these results concerning the solution of the Jacobi inversion problem for the hyperelliptic integral the greatest achievement of analysis. This publication was a turning point in the life of Karl Weierstrass. He became famous abroad and the Austrian government made inquiries about him through Alexander von Humboldt. This pushed von Humboldt to action and on 1 July 1856 Weierstrass was appointed professor at the Industry Institute in Berlin. In September 1856 Weierstrass and Kummer went to Vienna. There, Graf Thun offered Weierstrass 2000 Gulden and a professorship at an Austrian university of his choice. Weierstrass declined that offer but it became clear to Kummer that if they wanted to keep the great mathematician in Germany he had to take action again. Shortly afterwards as a result of Kummer's efforts, Weierstrass was appointed Associate Professor at the University in Berlin. In November 1856 he became a member of the Berlin Academy. From then until 1890 he lectured on a great variety of topics, including periodic lectures on elliptic functions, lectures on geometry, and on mechanics. The famous mathematical seminar which he initiated together with Kummer in 1861 attracted international interest. From 1862 it was customary to make awards to the best participants. It is worth mentioning that the first mathematical-physical seminar was founded in Konigsberg in 1834 by Jacobi, Neumann and Sohnke, the second in Halle in 1838 and the third in Gottingen in 1850. Weierstrass' heavy work load resulted in a breakdown in his health in December 1861. He returned to teaching after a year but he lectured from a sitting position with a student writing the necessary text on the blackboard. For the rest of his life he suffered recurring bouts of bronchitis and phlebitis, but his determination kept him teaching and pursuing his research. In his lectures Weierstrass initiated the logical and rigorous development of analysis starting with his own construction of the real number system. He established the epsilon-delta notions in the concept of continuity and convergence, uniform convergence, absolute value, neighbourhood (a-\delta, a+\delta) of a point $a$, and many others. He rejected intuitive arguments which were still prevalent among many contemporary mathematicians. It was Weierstrass who in his lectures of 1862 gave the famous example of a continuous nowhere differentiable function. In that year he first developed in his lectures the theory of the \gamma(u) and \sigma(u) functions. His famous approximation theorem appeared in connection with the heat equation and was published in July 1885 in the Proceedings of the Meetings of the Berlin Academy of Sciences. He also contributed to the development of the Calculus of Variations on which he lectured in 1879. His main lectures however concentrated on Abelian functions. His periodically presented lectures included "Introduction to the Theory of Analytic Functions", "The Theory of Elliptic Functions", once approached from the point of view of differential equations, another time from the point of view of the theory of functions, "Application of Elliptic Functions to Geometry and Mechanics", "Application of Abelian Functions to Geometric Problems and the Calculus of Variations". His lectures drew audience of up to 250, among them over 100 future professors, including S. Kovalevsky, Schwarz, Fuchs, G. Mittag-Leffler, L. Koenigsberger, H. Minkowski, and Cantor. Weierstrass became the leading influence on the mathematical world. He obtained high recognition throughout Europe. The Swedish mathematician Mittag-Leffler mentions a nice anecdote. When he got a scholarship to study abroad in 1873 he went to Paris where Hermite greeted him with the words "Vous avez fait erreur, Monsieur, vous auriez du suivre les cours de Weierstrass a Berlin". And of course Mittag-Leffler followed Hermite's advice and went to Berlin. The most remarkable among Weierstrass's students was Sonia Kovalevsky, the daughter of a Russian artillery general. Women were not allowed to study in Russia so in 1868 she contracted a marriage of convenience to a young paleontologist, Vladimir Kovalevsky, and the couple went to study abroad. They first enrolled at the University of Heidelberg but after two years they separated. Vladimir went to Jena and Sonia travelled to Berlin hoping to attend Weierstrass's lectures. Unfortunately, there was a ban on women students at the University in Berlin and her application was rejected. So she went straight to Weierstrass with her Heidelberg references. He gave her some problems to solve and her solutions and enthusiasm impressed him so he decided to teach her privately. During the four years that she spent in Berlin, she became his close friend and an irreplaceable partner for scientific discussions. We do not know much about Weierstrass' private life. Sonia, however, the second woman (after Maria Sklodowska Curie) to hold a university post, a novelist and a revolutionary, became a wonderful topic for many biographies and novels. Weierstrass regarded her as his most talented student. In fact during her stay in Berlin she produced three outstanding papers; on differential equations, on Abelian integrals and on Saturn's rings, and managed to obtain a doctorate from the University of Gottingen. However, Weierstrass, much to his regret, could not secure a job for her in Germany and she had to return to Russia. Their friendship, although based on purely scientific interaction, became the subject of rumour. Weierstrass felt deeply hurt by persistent insinuations surrounding his student and her mathematical achievements. On return to Russia her interest in mathematics ceased. She couldn't work at the University in Petersburg because her qualifications were not recognised. She returned to family life, gave birth to a daughter in 1878 and together with her husband, Vladimir, tried to make money as an estate entrepreneur, but this venture ended in bankruptcy. Weierstrass and Sonia Kovalevsky corresponded fairly regularly from the time they met till her death in 1891. His letters and encouragement to take up mathematics again inspired her and helped her through financial difficulties and her husband's suicide in 1883. In 1884 Mittag-Leffler succeeded in getting her a post as lecturer at the University in Stockholm. In 1888 she achieved great success, when her famous paper {\it "On the rotation of a solid body about a fixed point"} won the French Academy's Bordin prize. Sadly, Weierstrass burned all Sonia's letters after her early death. His letters, however, survived and are preserved at the Mittag-Leffler Institute in Djursholm, Sweden. The vast correspondence to Mittag-Leffler, H. Schwarz, Paul du Bois-Reymond, L. Koenigsberger, Riemann, L. Fuchs and Sonia, which contains mostly mathematical problems, also illuminates Weierstrass's life. He was considered successful but, in fact, his life was full of sufferings and personal problems. He did not marry after an engagement in Deutsch-Krone was broken due to the unfaithfulness of the fiancee, according to his brother, Peter. His career in Berlin had hardly started when his health collapsed. Moreover, towards the end of his life, rather than enjoying fame and appreciation, he felt isolated. This was because of a conflict with the mathematician and philosopher Leopold Kronecker. Weierstrass and Kronecker were friends for more than twenty years, sharing many fruitful mathematical discussions and ideas. Unfortunately, at the end of 1870's their views on mathematics, especially the foundations of mathematics, diverged gradually. Weierstrass's work on limits and convergence led him to develop a theory of irrational numbers based on convergent sequences of rationals which he called ``aggregates". His student, Georg Cantor, founded the theory of transfinite numbers which caused a revolution in mathematical thought. Before Cantor, mathematicians had accepted the notion of the infinite in the situation of a sequence "tending to infinity", but were not prepared to accept an actual infinity per se. Cantor's great achievement was to introduce this precise concept, in fact a whole class of different infinities -- that corresponding to the countable sets such as the natural numbers, that of the continuum of real numbers, and so on. For Kronecker, the type of non-constructive reasoning used by Weierstrass and Cantor was deeply suspect. His famous dictum, "God made the integers, all the rest is the work of man", pronounced at the Berlin Congress in 1886, encapsulated his philosophy, and he envisaged the inclusion of essentially the whole of mathematics within the terms of arithmetic. Kronecker did not hide his objections to the work of Weierstrass and Cantor and criticized them openly in front of students. No wonder that this mathematical conflict turned into personal quarrels. Towards the end of the 1880's Weierstrass evidently admitted that his long friendship with Kronecker was over though Kronecker himself seemed unaware of it. Weierstrass even considered the possibility of leaving Berlin for Switzerland to avoid the continuing conflict; but, since he did not wish his successor at the university to be chosen by Kronecker, he decided to stay. It became clear to him, however, that if he did not publish his lectures and works his achievements might fall into oblivion. His worry was probably exaggerated because his students were spread all over Europe and continued his research. His greatest successor was H. Poincare in Paris. Nevertheless the situation in Berlin was tense. In 1885 a special commission responsible for editing his works was set up. Weierstrass himself was no longer able to supervise the whole process so his students undertook the task of gathering and polishing up his lectures based on their own notes or transcripts. Weierstrass intended to give to mathematics an extensive treatment of analysis, clear and complete. So Knoblauch and Hettner were to prepare the theory of Abelian functions, his greatest aim in life. His dream was not fulfilled, however, and the draft was highly unsatisfying in his opinion and not up to the standard he expected. The lack of precision and printing errors worried him too. The first volume appeared in 1894 and contained his collected papers. The second was printed in 1895 but the next five volumes were published between 1902 and 1927, after his death. It is worth mentioning that it is to Weierstrass that we owe the publication of the collected papers of Jacobi, Dirichlet, Steiner and the letters of Gauss. In his old age he worked as an editor to supplement his salary which was not sufficient to maintain his family. He lived in Berlin with his sisters Elise and Klara who kept house, his father who died in 1869 and his uncle's grandson whom he took custody of as a two-year-old in 1884. He also took care of Borchardt's six children after his closest friend's death in 1880. In 1892 Weierstrass received the Helmholtz medal and in 1895 he was awarded the Copley Medal, the highest honour of the Royal Society of London. In the same year he celebrated his 80th birthday among his students. He spent his last three years in a wheelchair and died on 19 February 1897, after an inflammation of the lungs. References 1. Ahrens, W.: Skizzen aus dem Leben Weierstrass'. Math.-naturwiss. Blatter 4 (1907), 41-47. 2. ---: Gudermanns Urteil uber die Staatsexamensarbeit von Weierstrass. Math.-naturwiss Blatter 13 (1916), 44-46. 3. Baker, A. C.: Karl Theodor Wilhelm Weierstrass, the father of modern analysis. Math. Spectrum 1996/7, vol.29, No.2, 25-29. 4. Biermann, K. R.: Karl Weierstrass, Ausgewahlte Aspekte seiner Biographie. Journal fur die reine und angewandte Mathematik, 223 (1966), 191-220. 5. Bolling, R.: Karl Weierstrass - Stationen eines Lebens. Jahresber. Dt. Math.-Vereinigung, 96 (1994), 56-75. 6. Dorofeeva, A. & Chernova, M.: Karl Weierstrass. Mathematics and Cybernetics 7 (1985), Moscow (in Russian). 7. Flaskamp, F.: Herkunft und Lebensweg des Mathematikers Karl Weierstrass. Forsch. u. Fortschr. 35 (1961), 236-239. 8. Hensel, K.: Gedachtnisrede auf Ernst Eduard Kummer. Festschr. zur Feier des 100. Geburststages Eduard Kummers. Leipzig und Berlin 1910, 1-37. 9. Hilbert, D.: Zum Gedachtnis an Karl Weierstrass. Nachrichten v. d. Kgl. Ges. d. Wissensch. zu Gottingen 1897, 60-69. 10. ---: Uber das Unendliche. Mathematische Annalen 95 (1926), 161-185. 11. Kiepert, L.: Personliche Erinnerungen an Karl Weierstrass. Jahresber. d. Dt. Math.-Vereinigung 35 (1926), 56-65. 12. Kochina, P.: Karl Weierstrass. Moscow "Nauka" 1985 (in Russian). 13. Lampe, E.: Karl Weierstrass. Jahresber. d. Dt. Math.-Vereinigung 6 (1899), 27-44. 14. ---: Zur hundertsten Wiederkehr des Geburtstages von Karl Weierstrass. Jahresber. d. Dt. Math.-Vereinigung 34 (1915), 416-438. 15. von Lilienthal, R.: Karl Weierstrass. Westfalische Lebensbilder 2 (1931), 164-179. 16. Lorey, W.: Die padagogischen Ansichten des Mathematikers Karl Weierstrass. Blatter fur hoheres Schulwesen 32 (1915), 626-629. 17. ---: Karl Weierstrass zum Gedachtnis. Zeitschr. f. math. u. naturwiss. Unterricht 46 (1915), 597-607. 18. ---: Das Studium der Mathematik an den deutschen Universitaten seit Anfang des 19. Jahrhunderts. IMUK-Abhandlungen, Bd. 3, H. 9. Leipzig und Berlin 1916. 19. Mittag-Leffler, G.L: Une page de la vie de Weierstrass. Compte Rendu du deuxieme international des mathematiciens. Paris, 1902, 131-153. 20. ---: Zur Biographie von Weierstrass. Acta Mathematica 35 (1912), 29-65. 21. ---: Die ersten 40 Jahre des Lebens von Weierstrass. Acta Mathematica 39 (1923), 1-57. 22. ---: Sophie Kovalevsky: Notice biographique. Acta Mathematica 16 (1892/93), 385-395. 23. ---: Weierstrass et Sonja Kovalevsky. Acta Mathematica 39 (1923), 133-198. 24. Rachmanowa, A.: Sonja Kovalevski. Zurich 1953. 25. Rothe, R.: Bericht uber den gegenwartigen Stand der Herausgabe der Mathematischen Werke von Karl Weierstrass. Jahresber. d. Dt. Math.-Vereinigung 15 (1916), 59-62. 26. ---: Bericht uber die Herausgabe des 7. Bands der Mathematischen Werke von Karl Weierstrass. Jahresber. d. Dt. Math.-Vereinigung 37 (1928), 198-208. 27. Runge, C.: Personliche Erinnerungen an Karl Weierstrass. Jahresber. d. Dt. Math.-Vereiningung 35 (1926), 175-179. 28. Schubert, H.: Zum Andenken an Karl Weierstrass. Zeitschr. f. math. u. naturwiss. Unterricht 28 (1897), 228-231. 29. Siegmund-Schultze, R.: Der Beweis des Weierstrassschen Approximationssatzes 1885 vor dem Hintergrund der Entwicklung der Fourieranalysis. Historia Mathematica 15 (1988). 30. Voit, K. & Lindemann, F.: Karl Theodor Wilhelm Weierstrass. Sitzungsberichte d. math.-physikal. Klasse der k. b. Akademie der Wissenschaften zu Munchen 27, 1897 (1898), 402-409.

Back to Home Page of

SIAM AG on Orthogonal Polynomials and Special Functions

Page maintained by Martin Muldoon