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Whether it is possible for a person to engage in elementary teaching for a long time and still maintain the vitality of mathematics. The life of Weierstras, the father of modern analysis, is the definite answer.

Before introducing Weierstrass in detail, we introduce his contemporary German mathematicians in chronological order, who put forward new ideas on at least one field of mathematics in the second half of the 19th century and the first 30 years of the 20th century. 1855 was a landmark time in mathematics, when Gauss's death marked the end of his final connection with the outstanding mathematicians of the previous century. In 1855, Verstras was 40; Cronek was 32; Riemann was 29; Dedeking was 24; and Cantor was a 10-year-old. Therefore, there is no shortage of new people in German mathematics to continue the great tradition of Gauss. Weierstras was just recognized; Cronek had just got off to a good start; Riemann had done some of the greatest work; and Dedkin was entering the field that made him famous (number theory). Of course, Cantor is still unknown.

These people come together on a central problem in mathematics, that is, the problem of irrational numbers. Weierstras and Dedkin resumed the discussion of irrational numbers and continuity after Odox; Cronek doubted and criticized Weierstras' amendment to Odox; and Cantor blazed a new path of his own, trying to understand the real infinity itself implied in the concept of continuity. From the work of Weierstras and Dedkin, it ushered in a modern era of analysis, that is, the strict logical accuracy of analysis (calculus, single complex function theory and real variable function theory). Gauss, Abel and Cauchy began the first stage of rigor; Weierstras and Dedkin pushed it to a higher level.

One of Weierstras's findings particularly shocked intuitive analysts: he made a continuous curve with no tangent at any point. Gauss once called mathematics "the science of the eye"; but it takes more than a pair of good eyes to "see" the curve of Weierstrass.

In mathematics, Weierstrass function is an example of a real-valued function which is continuous everywhere but not differentiable everywhere. The graph of Weierstrass function on the interval [− 2 and 2]. Like some other fractals, this function shows self-similarity, and each zoom (red circle) is similar to the global graph.

Cronecker attacked it violently, denying that it had any meaning, but left no impact on mathematical analysis. It was not until the second decade of the 20th century that his condemnation of the belief in continuous and irrational numbers currently accepted was seriously considered. Whether it makes sense or not, Cronecker's attack partly led to the third stage of rigor in modern mathematical reasoning.

Abel died in 1829, Galois in 1832 and Jacobi in 1851. In the era we are talking about, a prominent problem in mathematical analysis is to complete the work of Abel and Jacobi on multi-periodic functions (elliptic functions, Abelian functions). Weierstras and Riemann did what they were supposed to do from a completely different point of view (Weierstras thought he was to some extent the successor of Abel); Cronecker opened up new prospects in elliptic functions, but he did not compete with the other two in the field of Abelian functions. Cronecker is mainly an arithmetic and algebra mathematician; one of his best works is a detailed exposition and development of Galois's work in equation theory. In this way, Galois had a worthy successor shortly after his death.

Weir Strauss Karl William Theodor Weir Strauss (Karl Wilhelm Theodor Weier-strass) was born on October 31, 1815 in Ostenfield, Munster, Germany.

Mathematicians tend to like music, and an open-minded man like Weierstras can't stand music of any kind. But like Abel and many other first-rate mathematicians, he liked to "visit" math masters, and he was fascinated by Laplace's "celestial mechanics", which laid the foundation for the mechanics and simultaneous differential equations that interested him all his life.

On May 22, 1839, Weierstras began his career as a secondary school teacher in Munster. This was the most important ladder for his later outstanding achievements in mathematics. The biggest influence on Weierstrass was that Christopher Goodman, who was keen on elliptic functions, came to Munster as a professor of mathematics.

Jacobi published his New Foundation of Elliptic function Theory in 1829.

In Goodman's time, elliptic function theory could be developed in many different ways. Goodman's idea is that everything is based on the power series expansion of the function, but he has not achieved much. Importantly, Weierstras made the theory of power series (Goodman's inspiration) at the core of all his work in analysis.

When Goodman taught the first lecture of the elliptic function course, only 13 people listened. There is only one audience in the second lecture, that is, Weir Strasse. From then on, no third party dared to desecrate the divine friendship between the speaker and his only disciple.

Weierstras began his career of teaching in high school at the age of 26 and lasted for 15 years, which is usually considered to be the most creative 15 years of a mathematician's life. Weierstras's lecture is a perfect example, he can give students something untouchable, called inspiration. He created a large number of creative mathematicians.

Abel, a poor Norwegian, Weierstras often stays up late with his friend, Abel. When he became the world's first-class analyst and Europe's greatest math teacher, his first and last advice to many students was to "read Abel".

Most of Weierstras's creative ideas were conceived when he was an unknown high school teacher, where there were no advanced books. Because he could not afford to pay the postage, Weierstras could not communicate scientifically. Maybe that's a good thing for him. ∶, his originality could develop freely without being hampered by the ideas that were popular at the time. In his speeches, he always wants to start from scratch according to his own characteristics, and almost never mentions the work of others.

After a year as a trainee teacher in a high school in Munster, Weierstras wrote a paper on analytical functions. In this paper, among other things, he independently obtained Cauchy's integral theory, the so-called basic theorem of analysis. In 1842, at the age of 27, Weierstras applied the method he developed to the system of differential equations, which was mature and powerful. When he did this work, he didn't expect to publish it, just to lay the foundation for his lifelong career (on Abelian function).

The nameless village of Kroner, Germany, was fortunate enough to be the place where Weirstras first published his work in 1842. It stood out as the capital of a kingdom in the history of mathematics. Because it was here that Weierstras laid the foundation for the most important work of his life-"the completion of Abel's and Jacobi's lifelong career derived from the Abel theorem, and Jacobi's discovery of multivariable multi-periodic functions".

Abel died when he was young and strong, and had no chance to explore the significance of his amazing discovery, while Jacobi could not clearly see that the true meaning of his own work should be found in the Abel theorem. The consolidation and development of these achievements is one of the main problems in mathematics.

So Weirstras claimed that once he had a deep understanding of the problem and developed the necessary tools, he would go all out to work on it.

All of Weierstras's work on analysis can be regarded as a great beginning of many studies on him. He had long been convinced that in order to clearly understand what he wanted to do, the basic concept of mathematical analysis must be thoroughly revised; from this belief, he developed another belief that ∶ analysis must be based on ordinary integers. Irrational numbers give us concepts of limits and continuity, from which analysis is generated, while irrational numbers must be traced back to integers through inviolable reasoning; specious proofs must be abandoned or redone, blanks must be filled, and vague "self-evident reasoning" must be taken out to withstand strict questioning until everything is understood and stated clearly in a language that can be understood by integers. In a sense, Pythagorean ideal ∶ built all mathematics on the basis of integers, but Weierstras gave the plan a constructive and clear definition so that it could solve the problem.

This gave rise to the arithmetic movement of analysis in the 19th century, which was completely different from Cronecker's arithmetic method.

In 1853, Weir Strasse (then 38) spent his summer vacation at his father's home in Westcoten. Weierstras used this holiday to write a paper on Abelian functions. In 1854, this paper was published in Claire's Magazine and caused a sensation.

Konigsberg University at Konigsburg University, Jacobi once made his great discovery, and now Weirstras has entered the same field with a better masterpiece. Rishlo, a professor of mathematics at Konigsberg University, is a successor to Jacobi in the theory of multi-periodic functions. With a professional eye, he immediately saw the value of Weierstrass's paper. He immediately persuaded Konigsburg University to award Weilstrass an honorary doctorate and went to Braunsberg in person to hand over his degree. Borchardt, the editor of Krell's Magazine, hurried to Braunsberg to congratulate the world's greatest analyst and began their cordial friendship.

On July 1, 1856, Weierstras was appointed professor of mathematics at the Royal Comprehensive Engineering School in Berlin. In the fall of the same year, he became an assistant professor at the University of Berlin and was elected to the Berlin Academy of Sciences.

The motivation of the new working environment and the tension caused by too many lectures soon led to a nervous breakdown. Weierstras also worked too hard in his research work. In the summer of 1859, he had to give up his course to rest and treat. He went back to school in the autumn, continued to work, and his health obviously recovered. But in March of the following year, there was a sudden burst of dizziness, and he fell down in a lecture.

Later, Weierstras's popularity spread all over Europe (and later to America), and the classes taught by Weierstras began to be too large to control. He gathered around him a group of extremely capable young mathematicians who had done a lot of work to publicize his ideas. Since Weierstras has never been active in publishing his works, his influence on 19th-century mathematical thought would have been greatly hindered if his disciples had not taken the initiative to spread his speech.

Friendship with Kovalevskaya

When Sonia Kovalevskaya Verstras was a professor of mathematics in Berlin (1864 murmur1897), the career of the world's recognized first-rate analyst was full of scientific and human interest. There is one thing that cannot be satisfied once and for all. This is his friendship with his beloved student Sonia Kovalevskaya.

Kovalevskaya was born in Moscow on January 15, 1850 and died in Stockholm, Sweden, on February 10, 1891.

Sonia began to study mathematics at the age of 15. By the age of 18, she had made rapid progress, was able to do advanced work, and was addicted to the subject. Because she came from a wealthy aristocratic family, her ambition to study abroad was fulfilled and was admitted to the University of Heidelberg. This talented girl became not only a first-rate female mathematician in modern times, but also famous as a leader of the women's liberation movement. Besides, Sonia is extremely beautiful.

In 1870, the Franco-Prussian War caused Weir Strauss to give up his usual summer trip, and he stayed in Berlin to teach elliptic functions. Sonia, then a dazzling 19-year-old woman, has been studying elliptic functions from Leo Koenig Berger at the University of Heidelberg since the fall of 1869. Koenigsberger was one of the earliest students of Weierstras and a first-rate propagandist of Verstras. Sonia decided to go directly to the master himself for inspiration and inspiration.

In the 1870s, the situation of unmarried female college students was something special. In order to prevent gossip, Sonia entered into an engagement at the age of 18, nominally married, left her husband in Russia and left for Germany. In her relationship with Weierstrass, it was thoughtless of her not to tell Weierstras that she was married at the beginning.

Having decided to learn from the master himself, Sonya ventured to visit Verstras in Berlin. She was 20 years old, very enthusiastic, very sincere and very determined; Weir Strasse, 55, sympathetically understood the aspirations of young people. Sonya wore a big fluffy hat to hide her panic, so that Verstras could not see her astonishing eyes, which no one could resist if she wanted to.

Sonia's apparent enthusiasm on her first visit made a good impression on Verstras. So he wrote to Koni Hisberger, asking about her math talent. He also asked if he could provide the necessary guarantee for the lady's character. After receiving an affirmative answer, Verstras tried to ask the university council to allow Sonia to attend his math lecture. The request was rudely refused, so he took care of her himself in his spare time. He lectured Sonia at his place every Sunday afternoon, and Weirstras paid her a return visit once a week. After the first few classes, Sonya took off her hat. The class began in the fall of 1870 and lasted until the fall of 1874, only slightly interrupted by holidays or illness. When the two friends can't meet for some reason, they write to each other. After Sonia's death in 1891, Verstras burned all the letters she wrote to him, along with many of his other letters and perhaps more than one math paper.

The correspondence between Weierstras and his lovely young friend was very human, even though most of it was about mathematics. There is no doubt that most communications are of great importance in science, but unfortunately, Sonia is a disorganized woman when it comes to documents, and most of what she leaves behind is fragmented or disorganized.

Weierstras himself is not a perfect person in this respect. Without taking notes, he casually lent his unpublished manuscripts to the students, and they did not always return what they had borrowed. Some people even unscrupulously rewrite some of their teachers' works and publish the results as their own. Although Verstras complained in his letter to Sonia about this intolerable practice, he was annoyed not by the despicable plagiarism of his ideas, but that his ideas were shoddy in the hands of the incompetent, resulting in damage to mathematics. Sonya would never have done such a thing, of course, but on the other hand, she was not entirely without fault. Verstras sent Sonia an unpublished work that he attached great importance to, and he never saw it again. She obviously lost it, for whenever he mentioned it, she was careful not to talk about it.

To make up for this mistake, Sonya tried to make Verstras a little more cautious about the rest of his unpublished works. He is used to taking a large white wooden box with him when he travels frequently, containing all his work notes and various manuscripts of his unfinished papers. His habit is to revise a theory many times until he finds the best "natural" way to develop it. As a result, his work was published very slowly, and it was only when he thoroughly studied a topic from a consistent point of view that he signed his name and published a work. In 1880, Weir Strass was on a holiday trip when the box was lost. I haven't heard from it since.

After receiving a degree from the University of Gottingen in 1874, Sonia returned to Russia to rest. Her "rest" plunged into the frenzied frivolity of St. Petersburg's busy social season, while Wellstrass returned to Berlin and traveled all over Europe, trying to find a place for his beloved student commensurate with her talents. His futile efforts disgusted him with the narrowness of orthodox academic thinking.

In October 1875, Verstras received the news of her father's death from Sonia. She apparently did not respond to his kind condolences. For nearly three years, she completely disappeared from his life. In August 1878, he wrote to ask her if she had received a letter he had written to her a long time ago, the date of which he had forgotten.

Didn't you get my letter? Or something can stop you, as you used to do, free to me, you often call your best friend, reveal your secret this is a mystery, only you can tell me the answer.

She didn't reply to her old friend's letter for two whole years, though she knew he was unhappy and in poor health. When the reply came, it was quite disappointing. Sonia's sexual desire outweighed her ambition, and she and her husband lived happily ever after. Her misfortune at this time was to become the center of flattery and foolish surprise of a group of shallow artists, journalists and half-bottle jealous literati, who chatted endlessly about her incomparable genius, and her superficial flattery excited her. If she had been in frequent contact with intellectuals like her, she could have lived a normal life and kept her enthusiasm, and she would not have treated the man who shaped her mind as despicably as she had done.

Sophia and her daughter were born in October 1878, when Sonia's daughter Fofi was born. She had to be quiet, which once again aroused the mother's potential interest in mathematics, and she wrote to Verstras asking for technical advice. He replied that he had to look up the relevant literature before giving his opinion. Although she had snubbed him, he was still ready to give her generous encouragement. Her only regret was that her long silence deprived him of the opportunity to help her.

The material tribulations awakened Sonia to the truth. She is a born mathematician, can no longer leave mathematics, like a duck can not do without water. So in October 1880 she wrote again asking Weirstras to give her advice, and without waiting for his answer, she packed her bags and left Moscow for Berlin. However, when the distraught Sonya arrived unexpectedly, he spent the whole day carefully examining her difficulties. He must have spoken frankly to her; for when she returned to Moscow three months later, she was so devoted to her math that her dissolute friends and foolish flatterers could no longer recognize her. At the suggestion of Weierstras, she set out to solve the problem of light propagation in a certain crystalline medium.

There are two new aspects of the correspondence of 1882, one is about the interest in mathematics, the other is the frank opinion of Weierstrass that Sonia and her husband are not suitable for each other, especially when her husband can't really value her intellectual achievements. Mathematics involved Poincare, who was at the beginning of his career. With his reliable instinct to identify young geniuses, Verstras called Poincare a promising man, hoping that he would give up the habit of publishing too quickly, make his research mature and not spread them over a wide range of fields. Speaking of Poincare's flood of papers, he said ∶ "publishes a really successful article every week-that's impossible."

Sonia's family problems were soon solved by her husband's sudden death in March 1883. She was in Paris and her husband was in Moscow. The blow overwhelmed her. For four whole days, she shut herself in her room alone, refused to eat, lost consciousness on the fifth day, recovered on the sixth day, asked for paper and pen, and filled the paper with mathematical formulas. In the fall, she returned to her original state and attended a science conference in Odessa.

She lectured at Stockholm University in the autumn of 1884 and was appointed tenured professor in 1889. Shortly afterwards, the Italian mathematician Volterra pointed out that she had made a serious mistake in her work on the refraction of light in a crystalline medium, and that she had suffered a rather embarrassing setback. Weierstras did not see the mistake. He was overwhelmed by official business. Apart from these official duties, he had only time to eat, drink and sleep. He was nearly 70 now, but with the increase of physical illness, his intelligence was still as strong and agile as ever.

The master's 70th birthday became a day of public respect for him, with his disciples and former students coming together from all over Europe. After that, he gave fewer and fewer public speeches and spent 10 years in his own home receiving a small number of students. On his 80th birthday, he held a more unforgettable celebration than his 70th birthday, and he became a national hero of the German people to some extent.

One of the greatest pleasures that Weierstras experienced in his twilight years was that his beloved student finally won recognition. On Christmas Eve 1888, Sonia won the Bourdan Prize of the French Academy of Sciences for her paper on the rotation of a solid around a fixed point.

Weir Strasse was ecstatic. He wrote that ∶

I don't need to tell you how happy I and my sisters (and your friends here) are with your achievements. In particular, I feel a real satisfaction; the qualified judges have now made their verdict that my loyal student, the person I love, is not really a frivolous liar.

Two years later (February 10, 1891), Sonia contracted the prevailing influenza at the age of 41 and died soon after in Stockholm. Weierstras lived for another six years after her death. On February 19, 1897, after a long illness and influenza, he died peacefully at his home in Berlin at the age of 82. Sonia is buried in Stockholm, and Verstras and his two sisters are buried in a Catholic cemetery in Berlin.

Two basic concepts

We now give hints on the two basic concepts on which Weir Strass laid the foundation for his work in analysis. A power series is in the form

The expression of the coefficient, where the coefficient is 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, , axin,... Is a constant, z is a variable, the number involved can be a real number or a complex number.

The sum of the first n terms of a series is called a partial sum. If for a special value of z, these parts and a series of numbers given converge to a certain limit, we say that the power series converges to the same limit for this z value.

The convergence domain of the series is formed by making the power series converge to all z values of a limit; for any value of the variable z in the convergence domain, the series converges; for other values of z, the series diverges.

If the series converges to a certain z value, then as long as the number of terms is sufficiently large, the series value of the z value can be calculated to achieve any desired degree of approximation.

Nowadays, in most mathematical problems that are useful to science, the "answer" given by people is often the series solution of a differential equation (group). This solution can rarely be obtained from the finite expressions of usual mathematical functions (such as logarithmic functions, trigonometric functions, elliptic functions, etc.). Then, in such a problem, two things must be done, ∶.

Prove the convergence of series

If it converges, calculate its value until the required precision.

If the series does not converge, it is usually a signal that the problem is either misstated or missolved. A large number of functions that appear in pure mathematics are treated in the same way, regardless of whether they may have scientific applications or not.

Finally, all of these (pure and applied aspects) have been extended to multivariable power series. For example, the power series of two variables:

It can be said that without the theory of power series, most of the mathematical physics as we know it (including most astronomy and astrophysics) would not exist.

The difficulties together with the concepts of limit, continuity and convergence prompted Weirstras to create his theory of irrational numbers.

Suppose we find the square root of 2 as we did in school, and calculate to a lot of decimal places, we get the number sequence 1, 1, 4, 1, 4, 1, 4, 1, 1, 1, 4, 1, 4, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 4, 1, 4, 4, 1, 4, As a gradual approach to the square root. According to the usual rules, continue according to clear steps, as long as enough efforts are made, if necessary, we can give the first thousand or the first million rational numbers of this approximation sequence. We find that when we go far enough, we completely determine the rational number, which contains as many decimal places as we want, and this rational number is different from any rational numbers that appear later in the sequence.

This shows what the convergent sequence of numbers means. The rational number of ∶ constituting sequence provides us with a more and more approximate value of the irrational number that we call the square root of 2. We assume that this irrational number is defined by the sequence of convergent rational numbers. The significance of this definition is that ∶ 's method of calculating any special members of the sequence with finite steps has been specified.

Although it is impossible to actually show the whole sequence, we regard the process of constructing any member of the sequence as a definite object that we can discuss. In doing so, we have an operable method in mathematical analysis that uses the square root of 2 and similarly uses any irrational number.

Rightly or wrongly, however, Weierstras and his school made this theory work. In mathematics, as in everything else, perfection is a fantasy, and in Krell's words, we can only hope to get closer and closer to the mathematical truth, just as Weilstras defines irrational numbers with a sequence of convergent rational numbers.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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