Carl Theodore William Weierstrass

Born October 31, 1815, in Ostenfeld, Westphalia (present-day Germany).

Died February 19, 1897 in Berlin, Germany

Carl Theodore William Weierstrass

Karl Weierstrass is best known for constructing complex functions from power series.

Karl Weierstrass's father, Wilhelm Weierstrass, was the secretary of the mayor of Ostenfelde when Karl was born. Wilhelm Weierstrass is a well-educated man with a broad knowledge of the arts and sciences. He certainly has the ability to get a higher position than he did, and this attitude may be one of the reasons Karl Weierstrass's early career was far less than his excellent ability. Weierstrass's mother was Theodora Vonderforst, and Karl was the eldest of Theodora and Wilhelm's four children, none of whom were married.

Wilhelm Weierstrass became a tax inspector at the age of eight. The job allowed him to stay in either place for a short time, so as the family moved around Prussia, Karl often moved from school to school. Carl's mother, Theodora, died, and a year later his father William remarried. By 1829, Wilhelm Weierstrass had become an assistant to the main tax office in Paderborn, where Karl entered the Catholic Gymnasium. Despite having to work part-time as a bookkeeper to help with the family's finances, Weierstrass excelled at the gymnasium.

And in the gymnasium, Weierstrass's mathematical abilities certainly exceeded expectations. He regularly read Crelle's diary and tutored one of his brothers in math. However, Weierstrass's father wanted him to study finance after graduating from the gymnasium in 1834. After that, he entered the University of Bonn and planned a course for him, which included the study of law, finance and economics. The Prussian government career planned for him by his father was indeed an elaborate course. However, Weierstrass encountered a conflict of either obeying his father's wishes or learning his favorite subject, namely mathematics.

As a result of the internal conflict in Veerstrass, he did not attend either lectures in mathematics nor in lectures on his planned courses. He pretended not to care about his studies to deal with inner conflict, and he spent four years engaged in intensive fencing and alcoholism. As Biermann writes in [1]:-

...... Conflicts between duties and tendencies lead to physical and mental tension. He tried to overcome his problems by participating in a carefree student life, but in vain ...

He did teach himself mathematics, but he read Laplace's Celestial Mechanics (T) and then Jacobi's work on elliptic functions. By studying Goodman's speeches, he learned the necessary methods of elliptic function theory. In a letter to Lie nearly 50 years later, he explains how he made the clear decision to study mathematics at this time, despite his father's wishes (see [1]): -

... when I became aware of [a letter from Abel to Legendre] in Crelle's Journal during my student years, [it] was of the utmost importance. The immediate derivation of the form of the representation of the function given by Abel ..., from the differential equation defining this function, was the first mathematical task I set myself; and its fortunate solution made me determined to devote myself wholly to mathematics; I made this decision in my seventh semester ...

Weierstrass has decided to become a mathematician, but he should still take courses in public finance and management. After making his decision, he spent another semester at the University of Bonn, his eighth semester ended in 1838, leaving the university without taking the exams and not studying the subjects he had enrolled. Weierstrass's father was very frustrated that his son had given up school. He was persuaded by a family friend, the president of the Paderborn Court, to allow Karl to study at the Münster School of Theology and Philosophy so that he could take the necessary exams to become a secondary school teacher. 1839

On 22 May Weierstrass enrolled at the College of Münster. Goodman lectured in Münster, which is why Weierstrass was so keen to study there. Weierstrass attended Gudermann's lecture on elliptic functions, the first lecture on the subject, and Gudermann strongly encouraged Weierstrass to conduct mathematical research. Leaving Münster in the autumn of 1839, Weierstrass studied for the teacher examination he registered in March 1840. By this time, however, Vilstrasse's father had changed jobs and became the director of a salt factory in January 1840. The family now lives in Westerkoten, near the Lippe River west of Paderborn.

At the request of Weierstrass, he raised a question in a paper on elliptic function representations he received in May 1840, to which he presented his own important research as an answer. Gudermann evaluated the paper and evaluated Weierstrass' contribution:-

...... Equal status with the Discoverers crowned as glory.

Later in life, when Weierstrass learned of Goodman's comments, he said he would publish his results if he knew. Weierstrass also commented on how generous Goodman's praise was to him, especially since he was highly critical of Goodman's approach.

By April 1841, Weierstrass had taken the necessary oral examinations, and he began a one-year probationary period as a teacher at the Münster Gymnasium. Although he did not publish any mathematical papers at this time, he wrote three short papers in 1841 and 1842, described in [3]:-

The concepts on which Weierstrass based his theory of functions of a complex variable in later years after 1857 are found explicitly in his unpublished works written in Münster from 1841 through 1842, while still under the influence of Gudermann. The transformation of his conception of an analytic function from a differentiable function to a function expansible into a convergent power series was made during this early period of Weierstrass's mathematical activity.

Weierstrass began his career as a mathematics teacher in 1842 at Pro-Gymnasium in Deutsch Krone, West Prussia (now Poland), where he remained until 1848, when he moved to Collegium Hoseanum in Braunsberg. As a math teacher, he also had to teach other subjects, while Vilstrass taught physics, botany, geography, history, German, calligraphy and even gymnastics. In his later years, Weierstrass described the "endless dullness and boredom" of these tragic years, of which [ 1 ] :-

...... He had neither colleagues for mathematical discussions nor a mathematical library, and the exchange of scientific letters was a luxury he could not afford.

From around 1850, Weierstrass began to develop very severe dizziness and ended up with a severe illness about an hour later. Frequent seizures of about 12 years made it difficult for him to work, and it was thought that these problems were most likely caused by the psychological conflicts he suffered as a student and the pressure to devote himself to mathematics every time. While doing heavy teaching work, he can free up a minute of free time.

Not surprisingly, when Weierstrass published a paper on Abel's function in the Braunsberg school's prospectus, they went unnoticed by mathematicians. However, he published Zur Theorie der Abelschen Functionen (T) in Crelle's diary in 1854, which of course was noticed. The paper does not give a complete theory of the inversion of the superelliptive integral developed by Weierstrass, but rather a preliminary description of his methods, including the representation of the Abel function as a power series that is constantly converging.

With this paper, Weierstrass stood out from the shadows. On 31 March 1854, the University of Königsberg conferred on him an honorary doctorate. In 1855 the vacancy left in the chair of the Veerstrass application at the University of Breslau, Kummer moved to Berlin. Kumo, however, tried to influence things and let Weierstrass go to Berlin instead of Breslau, so Weierstrass was not appointed. A letter written by Dirichlet to the Prussian Minister of Culture in 1855 strongly supported Weierstrass's university appointment. Details are given in [ 10 ].

After being promoted to senior lecturer in Braunsburg, Veerstrass received a year's sabbatical to devote himself to advanced mathematical research. However, he had decided that he would never return to school to teach.

Weierstrass published a complete version of his theory of superelliptical integral inversion (T) in his next paper, Theorie der Abelschen Functionen, in Crelle's Journal in 1856. Many universities provided him with a chair. While Austrian universities were discussing this prospect, the Technical Institute of Berlin (later Technische Hochschule) offered a chairmanship. Although he preferred to go to the University of Berlin, Weierstrass certainly did not want to return to the Josen Academy in Braunsberg, so he accepted the invitation of the Institute on June 14, 1856. Continue to make offers to Weierstrass for his participation in the Congress of Vienna in September 1856

He was given a seat at any Austrian university of his choice. Before he decided what to do with the proposal, the University of Berlin offered him a professorship in October. It was a job he had always wanted, and he quickly accepted it, although he failed to officially become a professor at the University of Berlin for several years after accepting an invitation from the Industrial Institute earlier this year.

Weierstrass's successful lectures in mathematics attract students from all over the world. His lecture topics included: - The Application of Fourier Series and Integrals in Mathematical Physics (1856/57), An Introduction to Analytic Function Theory (he lists his 1841 but never published), elliptic function theory (his main research topic), and applications in geometric and mechanical problems.

In a lecture in 1859/60, Weierstrass introduced for the first time the foundations of analysis. In 1860/61 he taught integral science.

Above we describe the health problems that Weierstrass suffered from 1850 onwards. Although he had achieved his dream position, his health failed in December 1861 when he completely collapsed. It took him about a year to recover to being able to lecture again, and he would never be able to fully recover. Since then he sat down to give a lecture, and a student wrote for him on the blackboard. The attacks he suffered in 1850 stopped and were replaced by chest problems.

In his 1863/64 course on the general theory of analytic functions, Weierstrass began to expound his theory of real numbers. In his 1863 lecture, he proved that complex numbers are the only commutative algebraic extension of real numbers. Gauss promised to prove this in 1831, but failed to prove it.

In 1872 his emphasis on rigor led him to discover a function, though continuous, that must be at all times any derivative. Analysts who rely heavily on intuition for discoveries are rather frustrated by this counterintuitive function. Riemann suggested in 1861 that such a function could be found, but his example is non-trivial at all points.

Weierstrass' lectures developed into a four-semester course that he continued until 1890. The four courses are

Introduction to Analytic Function Theory,

ellipse function,

Abel function,

Variations or application calculus of elliptic functions.

Over the years, the course developed and published many editions, such as Killing's Notes in 1868 and Hurwitz's Notes in 1878. Weierstrass' approach still dominates instructional analysis today, which is clearly evident from the content and style of these lectures, especially the introductory courses. Its contents are: numbers, the concept of functions using the Weierstrass power series method, continuity and differentiation, analytic extensions, singularities, analytical functions for several variables, especially Weierstrass's "preparation theorem" and contour integrals.

In Berlin, Weierstrass had two colleagues, Kulmer and Kronecker, along with three, making Berlin a leading university for the study of mathematics. Crohneck had been a close friend of Weierstrass for many years, but in 1877 Crohneck's opposition to Cantor's work caused a rift between the two. This became so bad that at some point in 1885, Weierstrass decided to leave Berlin for Switzerland. However, he changed his mind and stayed in Berlin.

A large number of students benefit from the teaching of Weierstrass. We list some of the people mentioned elsewhere in our archives: Bachmann, Borza, Cantor, Engel, Frobenius, Geigenberg, Henzel, Hurd, Hurwitz, Killing, Klein, Kneser, Königsberger, Lech, Li Qun, Lüroth, Mertens, Minkowski, Mitag-Leffler, Neto, Schottky, Schwartz and Stolz. However, it is particularly worth mentioning a student.

In 1870 Kovalevskaya came to Berlin, and Weierstrass taught her privately because she was not allowed to enter the university. Apparently, as far as Weierstrass was concerned, she was a very special student, for he wrote to her and said:-

... Dream of and revel in the many riddles we have to solve, about finite and infinite space, about the stability of the world system, and all the other major questions about the future of mathematics and physics. ...... You've always been close... Throughout my life... I have never found anyone who could give me such an understanding of the highest goals of science as you did and so happily aligned with my intentions and basic principles.

It was through Weierstrass's efforts that Kovalevskaya received an honorary doctorate from the University of Göttingen, and he also used his influence to help her obtain the position in Stockholm in 1883. Weierstrass and Kovalevskaya corresponded for 20 years between 1871 and 1890. More than 160 letters were exchanged (see [5], [7], etc.), but Weierstrass burned her letters after Kovalevskaya's death. Strict standards set by Weierstrass, e.g. definitions,

Irrational numbers, as the limit of convergent series, strongly influenced the future of mathematics. He also studied the full function, the concept of uniform convergence, and the function defined by the infinite product. His efforts are summarized in [2] as follows:-

Known as the father of modern analytics, Weierstrass designed the series convergence test and contributed to theories such as periodic functions, real variable functions, elliptic functions, Abel functions, convergent infinite products, and variational calculus. He also proposed the theories of bilinear and quadratic forms.

Weierstrass published very little [1] :-

...... For his critical consciousness always compels him to base any analysis on a solid foundation, starting with new methods, constantly modifying and expanding.

However, he did edit the complete works of Steiner and Jacobi. He decided to supervise the publication of his own complete works, which in his case would involve a great deal of unpublished material in his lecture courses, and Veerstras realized that without his help it would be a daunting task. The first two volumes were published in 1894 and 1895, respectively, and were the only ones published before his death in 1897. His last years were tough [ 1 ] :-

During his last three years, he was confined to a wheelchair, unable to move and unable to take care of himself. He died of pneumonia.

The remaining volumes of his complete works appeared slowly; volume 3 in 1903, volume 4 in 1902, volumes 5 and 6 in 1915, and volume 7 in 1927. Volume 7 was republished in 1967. Today continues to publish more works, especially editions of his lecture courses, which are taken from notes taken by those who attended the lectures.