World of Mathematics on Karl Theodor Wilhelm Weierstrass
Karl Wilhelm Theodor Weierstrass was considered one of the greatest mathematical analysts of 19th century Europe. He is well known as a cofounder of the theory of analytic functions and their representation as power series. Weierstrass made crucial contributions to the arithematization of analysis and to the theory of real numbers. He showed the importance of uniform convergence, furthered the understanding of elliptic functions, and made contributions to the field of differential equations. Weierstrass' reputation for high standards of proof and definition is reflected in the modern development of calculusand analysis.
The eldest child of Wilhelm Weierstrass, a customs officer, and Theodora (nee Forst), Weierstrass was born in Ostenfeld, Germany, on October 31, 1815. The family soon moved to Westernkotten in Westphalia, where Weierstrass' three siblings, Peter, Klara, and Elise, were born. Shortly after Elise's birth in 1826, Weierstrass' mother died, and a year later his father remarried. His father, who had once been a teacher and had a reputation for righteousness and uncompromising authority, wished for his son to receive a solid education and pursue a bureaucratic career that might afford him a comfortable lifestyle. Since there was no school in the village of Westernkotten, Weierstrass was sent to study in nearby Muenster. At the age of 14 he entered the Gymnasium in Paderborn, from which he graduated in 1834. Earning awards in almost all of his subjects and finishing among the top in his class, Weierstrass appeared bound for the success his father had envisioned. It would not be long, however, before ambitions of a comfortable bureaucratic career were derailed. Weierstrass was directed by his father to attend the University of Bonn, where it was expected he would acquire an education in business and law. He enrolled at the university, but rather than taking his studies seriously, he embraced the sport of fencing and the habit of socializing in German pubs. After four years Weierstrass returned home without a degree. His father and siblings, reportedly shamed by his failure, decided he might salvage the family name by obtaining a teaching certificate. In 1839 he began studying toward this end at the Academy of Muenster.
Although Weierstrass had read Celestial Mechanicsby Pierre Simon Laplace during his time in Bonn, he had, till 1839, shown no extraordinary interest in mathematics. At the Academy of Muenster, however, he came under the tutelage of Christof Gudermann, who taught a course in elliptic functions. Gudermann's special interest was in attempting to represent elliptic functions with power series, an approach to analysis that was to have a profound influence on Weierstrass' career. Weierstrass proved, in Muenster, to be an able and dedicated student. In 1841, when he took his examination for the teaching certificate, he asked Gudermann to provide a mathematical problem that might normally be presented to a doctoral candidate. Gudermann obliged, and proposed that Weierstrass find the power series developments of the elliptic functions. The examination results were so exemplary that Gudermann recommended Weierstrass be granted a university post, but the Academy was only able to grant him a teaching certificate.
Conducts Research in Isolation
Weierstrass began his teaching career at the Gymnasium in Muenster in 1841. In 1842 he took a position as a teacher of mathematics and physics in Deutsche-Krone, West Prussia. That same year his first mathematical work, "Remarks on Analytical Factorials ," was printed in a school publication. The narrow circulation of the journal prevented Weierstrass' significant original contribution to the field from being immediately recognized. It would be more than a decade before the paper came to the attention of a wider, more scholarly audience. During this period Weierstrass was forced to confine his mathematical research to his spare hours, devoting most of his energies to the daily demands of his secondary school teaching. Occasionally, he would work until dawn on a mathematical problem of interest, unaware that the night had passed. Although he worked in isolation and had not yet established personal contact with the prominent European mathematicians of his day, he did manage to remain somewhat current in his readings. In 1848, Weierstrass moved to Braunsberg to teach at the Royal Catholic Gymnasium. By now he had taken an interest in the work of Niels Henrik Abel. Shortly after assuming his new teaching post, he published a paper in the gymnasium's journal entitled "Contributions to the Theory of Abelian Integrals ." Once again, limited circulation meant his important discoveries would receive no immediate notice.
Achieves Sudden Fame with Treatise on Abelian Functions
During the summer of 1853, Weierstrass spent his vacation in his father's home in Westernkotten, writing another paper on Abelian functions. This time, he chose to submit the work to the Journal fur die reine and angewandte Mathematik, a prestigious mathematical research publication begun in 1826 by August Leopold Crelle. The paper was accepted for publication and appeared in print in 1854, bringing Weierstrass instant acclaim. He had successfully solved a problem of hyper-elliptic integrals, and established himself as one of the truly great mathematical analysts. The University of Königsberg bestowed on Weierstrass an honorary doctorate, and the gymnasium in Braunsberg granted him a leave so that he could pursue mathematics full time. In 1856, he was appointed professor at the Royal Polytechnic School in Berlin. He received a joint appointment as an assistant professor at the University of Berlin and was awarded membership in the Berlin Academy. That same year his first paper, "Remarks on Analytical Factorials," was reprinted in Crelle's journal, finally receiving an appropriate audience.
In Berlin, Weierstrass rapidly took on a heavy research and teaching schedule, and his lectures on Abelian functions and transcendents were widely attended. Although he wrote very few manuscripts, his ideas on analytic functions became widely known through the dissertations and other writings of his students. Eventually, in 1886, G.H. Halphen, a student of Weierstrass, published a comprehensive discussion of Weierstrass' theory of elliptic functions, entitled Theorie Des Fonctions Elliptiques et des leurs Applications.
At the core of Weierstrass' mathematical research was his work on the theory of analytic functions based on power seriesand the process of analytic continuation. Weierstrass demonstrated that the integral of an infinite series is equal to the sum of the integrals of the separate terms when the series converges uniformly within a given region. In 1861, Weierstrass demonstrated a function that is continuous over an interval but does not possess a derivative at any point on this interval. Before this, it had been assumed that a continuous function must have a derivative at most points. In 1863 he provided a proof of a Gaussian theoremthat complex numbers are the only commutative algebraic extensions of the real numbers. In the field of the calculus of variations, he brought clarity and rigor to necessary and sufficient conditions for elliptic functions. Toward the end of his career Weierstrass became interested in the astronomical problem of the stability of the solar system.
In addition to his numerous contributions to the field of mathematics, Weierstrass also earned a reputation as an excellent teacher and lecturer. He gathered about him a band of young students, many of whom themselves became successful mathematicians and propagators of Weierstrass' ideas. His students and the auditors in his seminars read like a who's who of 19th century mathematics. But undoubtedly, his most favorite pupil was the Russian mathematician Sonya Kovalevskaya. The two first met in 1870 when he was 55 and she was 20. Because the University of Berlin would not allow a woman to officially attend Weierstrass' lectures, he taught her privately for four years. Through his efforts, Kovalevskaya finally received her doctorate in absentia from Göttingen in 1874. They continued to correspond after her return to Russia and eventual appointment as professor of mathematics at the University of Stockholm until her untimely death in 1891.
Weierstrass remained at the University of Berlin for 30 years. His activities as a mathematician and lecturer in Berlin were interrupted from time to time because of chronic illness. He developed vertigo in the 1860s and later suffered from chronic bronchitis and phlebitis. In 1894 he became confined to a wheelchair. Weierstrass died of influenza, following a long illness, on February 19, 1897, in Berlin. A lifelong bachelor, he was buried in a Catholic cemetery alongside his two sisters, with whom he had lived for much of his adult life.
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