Cauchy (Augustine Louis 1789- 1857) was born in Paris. His father, Louis Fran? ois Cauchy, was an official of the French Bourbon dynasty and had been holding public office in the turbulent political vortex of France. Due to family reasons, Cauchy himself belongs to the orthodox school that supports the Bourbon dynasty and is a devout Catholic. And in the field of mathematics, he has made great achievements and accomplishments. Many mathematical theorems and formulas are also named after him, such as Cauchy inequality and Cauchy integral formula.

Chinese name: augustin louis cauchy.

Augustine Luis Kochi

Nationality: France

Place of birth: Paris

Date of birth: 1789 August 2 1.

Date of death:1May 23, 857

Occupation: mathematician, physicist, astronomer

Graduate school: Paris School of Bridges and Roads.

Faith: the orthodox faction that supported the Bourbon dynasty.

Main results: Cauchy limit existence criterion

Cauchy sequence

cauchy inequality

cauchy integral formula

Representative works: Analysis Course, Introduction to Differential Analysis Course, Application of Calculus in Geometry.

outline

Cauchy (1789— 1857) is a French mathematician, physicist and astronomer. /kloc-At the beginning of the 9th century, calculus has developed into a huge branch with rich contents and wide applications. At the same time, its weaknesses are increasingly exposed, and the theoretical basis of calculus is not strict. In order to solve new problems and clarify the concept of calculus, mathematicians have carried out rigorous work of mathematical analysis. In the foundation work of analytical foundation, Cauchy, a great mathematician, was the first to make outstanding contributions.

Cauchy was born in Paris on August 2 1789. My father is a lawyer who is proficient in classical literature and has close contacts with the great French mathematicians Lagrange and Laplace at that time. Cauchy's mathematical talent as a teenager was highly praised by two mathematicians, and he predicted that Cauchy would become a great man in the future. Lagrange suggested to his father to "give Cauchy a solid literary education as soon as possible" so that his hobbies would not lead him astray. Therefore, his father strengthened Cauchy's literary education and made him show great talent in poetry.

From 1807 to 18 10, Cauchy studied in the Institute of Technology and worked as a traffic road engineer. Due to poor health, he accepted the advice of Lagrange and Laplace, gave up the engineer and devoted himself to the study of pure mathematics. Cauchy's greatest contribution to mathematics is to introduce the concept of limit in calculus and establish a logical and clear analysis system based on limit. This is the essence of the development history of calculus and Cauchy's great contribution to the development of human science.

In 182 1, Cauchy put forward the method of limit definition, describing the limit process as inequality, which was improved by Weierstrass and became the current Cauchy limit definition or definition. Today, all calculus textbooks still follow Cauchy's definitions of limit, continuity, derivative and convergence (at least in essence). His explanation of calculus was widely adopted by later generations. Cauchy has done the most systematic and pioneering work in definite integral, which he defined as the "limit" of sum. It is emphasized that the existence of integral must be established before definite integral operation. He first strictly proved the basic theorem of calculus by using the mean value theorem. Through the efforts of Cauchy and later Wilstrass, the basic concepts of mathematical analysis were strictly discussed. So as to end the ideological confusion of calculus in the past two hundred years, liberate calculus and its popularization from the complete dependence on geometric concepts, movements and intuitive understanding, and make calculus develop into the most basic and huge mathematical subject in modern mathematics.

The rigorous work of mathematical analysis has had a great influence from the beginning. Cauchy put forward the theory of series convergence at an academic conference. After the meeting, Laplace hurried home and checked whether the series used in his masterpiece "Celestial Mechanics" were all convergent according to Cauchy's strict discriminant method.

Cauchy's research achievements in other fields are also very rich. He founded the calculus theory of complex variable function. He also made outstanding contributions in algebra, theoretical physics, optics and elasticity theory. Cauchy's mathematical achievements are not only brilliant, but also amazing in number. Cauchy, with 27 volumes and more than 800 works, is a prolific mathematician in the history of mathematics, second only to Euler. His glorious name is remembered by many textbooks today along with many theorems and standards.

As a scholar, he has quick thinking and outstanding achievements. From Cauchy's voluminous works and achievements, it is not difficult to imagine how he worked tirelessly all his life. But Cauchy is a complicated person. He is a loyal royalist, an enthusiastic Catholic and a lonely scholar. Especially as a prestigious master of science, he often ignores the creation of young scholars. For example, because Cauchy "lost" the pioneering manuscripts of the papers of talented young mathematicians Abel and Galois, it took about half a century for the group theory to come out.

1857 On May 23rd, Cauchy died in Paris. His last famous saying, "People will die, but achievements will last forever." Knocked on the hearts of generations of students for a long time.

Cauchy's skill in pure mathematics and applied mathematics is quite profound. In mathematics writing, he is considered to be second only to Euler. He wrote 789 papers and several books in his life, including many classics, but not all his creations were of high quality, so he was criticized as prolific and reckless, contrary to the prince of mathematics. It is said that the "periodical" of the French Academy of Sciences. Because there are too many works by Cauchy, the Academy of Sciences has to pay a lot of printing expenses, which is beyond the budget of the Academy of Sciences. So later, the Academy of Sciences stipulated that the longest paper could only have four pages, so Cauchy's longer paper had to be submitted elsewhere.

When Cauchy was young, his father often took him to the office of the French Senate, where he instructed him in his studies, so he had the opportunity to meet two great mathematicians, Senator Laplace and Senator Lagrange. They appreciate his talent very much; Lagrange thought he would be a great mathematician in the future, but advised his father not to study mathematics before learning liberal arts well.

The life of the character

18 1 1 and 18 12 years.

Cauchy entered middle school on 1802. In middle school, he got excellent grades in Latin and Greek and won many competitions. Math scores are also highly praised by teachers. 1805 was admitted to a comprehensive engineering school, mainly studying mathematics and mechanics. 1807 was admitted to Daqiao Highway School, 18 10 graduated with honors and went to Cherbourg to participate in the harbor construction project.

Cauchy went to Cherbourg with Lagrange's analytic function theory and Laplace's celestial mechanics, and later received some math books sent from Paris or borrowed locally. In his spare time, he carefully studied books in various branches of mathematics, from number theory to astronomy. According to Lagrange's suggestion, he studied polyhedron and submitted two papers to the Academy of Sciences at181and 18 12. The main results are as follows:

(1) It is proved that there are only five convex regular polyhedrons (the number of faces is 4, 6, 8, 12, 20 respectively) and four star regular polyhedrons (the number of faces is 12, and the number of faces is 20).

(2) Another proof about the Euler relation of the number of vertices, faces and edges of polyhedron is obtained and generalized.

(3) It is proved that a polyhedron with a fixed surface must be fixed, from which a theorem of Euclid that has never been proved can be derived.

These two papers have had a great influence in mathematics. Cauchy fell ill at work in Cherbourg and returned to her parents' home in Paris on 18 12.

18 13 years

Cauchy was appointed as the Paris Canal Engineering Engineer on 18 13. During his rest and work as an engineer in Paris, he continued to devote himself to studying mathematics and participating in academic activities. His main contributions during this period are:

(1) has studied substitution theory and published basic papers on substitution theory and group theory in history.

(2) Prove Fermat's conjecture about polygon number, that is, any positive integer is the sum of angles. This speculation has been put forward for more than a hundred years at that time, and it has not been solved after many mathematicians' research. The above two studies began when Cauchy was in Cherbourg.

(3) Calculating the real integral with the integral of complex variable function is the starting point of Cauchy's integral theorem in the theory of complex variable function.

(4) Studied the propagation of liquid surface waves, obtained some classical results in fluid mechanics, and won the 18 15 mathematics prize of French Academy of Sciences.

The publication of the above outstanding achievements brought Cauchy a high reputation, and he became an internationally famous young mathematician at that time.

18 15- 182 1

Napoleon failed in France, Bourbon was restored, and Louis Stanislas Xavier became king of France. On 18 16, Cauchy was hired as an academician of French Academy of Sciences and a professor of comprehensive engineering. 182 1 was appointed as a professor of mechanics at the University of Paris and also taught at the French Academy. His main contributions during this period are:

(1) Teaching analysis courses in comprehensive engineering schools, establishing the basic limit theory of calculus and expounding the limit theory. Before that, the concepts of calculus and series were vague. Because Cauchy's speech was different from the traditional way, the teachers and students of the school put forward many criticisms to him at that time.

Cauchy's works published in this period include Algebraic Analysis Course, Infinitesimal Analysis Course Outline and Application Course of Calculus in Geometry. These works laid the foundation of calculus, promoted the development of mathematics and became a model of mathematics curriculum.

(2) Cauchy studied continuum mechanics again after being a professor of mechanics at the University of Paris. In a paper from 65438 to 0822, he established the foundation of elasticity theory.

(3) Continue to study the calculation of integrals and residues on the complex plane, and apply related results to study partial differential equations in mathematical physics.

A large number of his papers have been published in the Journal of French Academy of Sciences and his own periodical Mathematical Exercises.

After 1830

1830, the revolution that overthrew the Bourbon dynasty broke out in France. King Charles of France fled hastily, and Louis Philippe, Duke of Orleans, succeeded him. At that time, it was stipulated that he must swear allegiance to the new king when he held public office in France. Because Cauchy belonged to the orthodox school that supported the Bourbon dynasty, he refused to swear allegiance and left France by himself. I went to Switzerland first, and then worked as a professor of mathematical physics at the University of Turin in Italy on 1832- 1833, and participated in the academic activities of the local academy of sciences. At that time, he studied the series expansion of complex variable function and differential equation (strong series method), and made important contributions to this.

From 1833 to 1838, Cauchy first worked in Prague, then worked as a teacher of the Bourbon Crown Prince and the Duke of Bordeaux in Golz, and was finally awarded the title of Baron. During this period, his research work was less.

Cauchy returned to Paris on 1838. Because he didn't swear allegiance to the king of France, he could only participate in academic activities of the Academy of Sciences, but could not engage in teaching. He published a large number of important papers on complex variable function, celestial mechanics, elasticity and so on in the report of French Academy of Sciences "and his own periodic analysis and mathematical physics exercises".

1848, the French revolution broke out again, Louis Philippe fell, the Republic was re-established, and the oath of allegiance of public officials to the French king was abolished. Cauchy became a professor of mathematical astronomy at the University of Paris in 1848, and resumed his teaching work in French institutions of higher learning, which was interrupted in 18.

1852, Napoleon staged a coup for the third time, and France changed from a republic to an imperial system, restoring public officials to swear allegiance to the new regime. Cauchy immediately resigned from the University of Paris. Later, Napoleon granted a third exemption from the oath of loyalty of himself and physicist arago. So Cauchy was able to continue his teaching work until/kloc-0 died in the suburbs of Paris in 857. Cauchy continued to participate in academic activities and published scientific papers until his death.

1857 died suddenly on May 23rd at the age of 68. He died of a fever. Before he died, he was still talking to the Archbishop of Paris. The last thing he said was:

"People will die, but achievements will last forever."

Personal anecdote

nickname

Cauchy had a nickname "Bitter Melon" when he was a student, because he was usually taciturn, like a bitter gourd. If he said anything, it was also very brief and confusing. It is very painful to communicate with such people. Cauchy has no friends around him, only a group of people who are jealous of his cleverness. At that time, social philosophy was popular in France, but Cauchy often read books after work, but only Lagrange's math book (1736- 18 13) and the spiritual book "Imitating Christ", which earned him another nickname "the man with a split brain", which means mental derangement.

Cauchy's mother heard the rumor and wrote to ask him the truth. Cauchy wrote back: "If Christians become mental patients, the madhouse will be full of philosophers. Dear mother, your child is like a leaf on a windmill. Mathematics and faith are his wings. As soon as the wind blows, the windmill will rotate in a balanced way, generating the power to help others. 』

18 16, Cauchy returned to Paris and became a professor of mathematics at her alma mater. Cauchy wrote: "I am as excited as a salmon that has found its own river. Soon he got married, and a happy married life helped him communicate with others.

famous

Bernoulli, a master of mathematics, once said, "Only mathematics can explore infinity, and infinity is one of God's attributes". Physics, chemistry and biology are all limited disciplines, and "infinity" can represent the limit that can never be measured. The concept of infinity makes philosophers crazy, makes theologians sigh and makes many people deeply afraid. On the other hand, Cauchy uses infinity to define a more precise mathematical meaning. He regarded the differential of mathematics as "the change of infinite hours" and expressed the integral as "the sum of infinite infinitesimals". Cauchy redefined calculus with infinity, which is still the beginning of every calculus textbook.

182 1 year, Cauchy's reputation spread far and wide. Students from as far away as Berlin, Madrid and St. Petersburg came to his classroom to attend classes. He also published a very famous "eigenvalue" theory and wrote: "In the field of pure mathematics, it seems that there is no actual physical phenomenon to prove it, and there is nothing in nature to explain it, but it is a paradise that mathematicians can see from a distance. Theoretical mathematicians are not discoverers, but reporters of promised land.

old age

After forty, Cauchy was unwilling to be loyal to the new government. He believes that academics should not be influenced by politics. He gave up his job and his motherland and took his wife to teach in Switzerland and Italy. He is welcomed by universities all over the world. But he wrote: "the stimulus of mathematics is that the body can't bear the load for a long time and is very tired!" After Cauchy was forty, she stopped doing research after class.

His health is gradually weakening. 1838, he returned to teach at the University of Paris, but left again because of political loyalty. Because of his insistence, 1848 the academic freedom of French university professors is based on personal conscience and is not subject to political restrictions. Since then, universities all over the world have followed this system, and universities have become places of academic freedom.

Paris Gui Zhi

It is said that Cauchy contributed to the Journal of the Paris Academy of Sciences when he was young, which made the printing house snap up the stocks of all paper shops in Paris in order to print these papers, which made the market short of paper, the price of paper increased greatly and the cost of printing house increased. So the Academy of Sciences passed a resolution that each paper published in the future should not exceed 4 pages. Many of Cauchy's long papers are not allowed to be published in China, but only in other countries.

Personal realization

Cauchy is a famous prolific mathematician. His complete works were published from 1882 to 1974, and the last volume was published, with a total of 28 volumes. His main contributions are as follows:

Simple complex variable function

Cauchy's most important and creative work is about the theory of simple complex function. /kloc-mathematicians in the 0/8th century adopted definite integrals with imaginary upper and lower bounds. But there is no clear definition. Cauchy first clarified the related concepts, and used this kind of integral to study various problems such as the calculation of definite integral, the expansion of series and infinite product, and the solution of differential equation expressed by parametric variable integral.

Analytical basis

Cauchy's analysis courses and related textbooks in comprehensive engineering schools have had a great influence on mathematics. Since Newton and Leibniz invented calculus (infinitesimal analysis for short), the theoretical basis of this subject is vague. In order to further develop, we must establish a strict theory. Cauchy first successfully established the limit theory.

The role of limit theory

Let the function f(x) be defined in the centripetal neighborhood of point X. If there is a constant a, there is always a positive number δ for any given positive number ε (no matter how small it is), so that when x satisfies the inequality 0|f(x)-A|, the constant a is called the function f(x) when x→ x. Time limit.

"Strictly speaking, there is no such thing as mathematical proof. In the end, we will do nothing but do something; It proves that this is what Litowood and I call God blowing. It is a touching rhetoric, enough pictures on the blackboard in class, and a way to stimulate students' imagination. "-Hardy.

Mathematics is so important that it has the same status as China literature in China. The reason is that mathematics itself is a language and a universal world language. Therefore, it is very necessary to strictly distinguish the parts of speech of mathematical concepts, which is not only the requirement of mathematics itself, but also the requirement of language science.

Speaking of language and part of speech, it is necessary to know some basic knowledge of Chinese.

1. Noun: a word indicating the name of a person or thing, place, position, etc.

2. Verbs: words expressing actions, development and changes, psychological activities, etc.

Calculus has never left contradiction and refutation since the first day of its birth. For example, Becker refutation (infinitesimal refutation), Zeno paradox and so on. If, through these arguments, we can find that they are actually just discussing the final form in disguise! Just as Leibniz cares about the ultimate fate of particles. Some people say that Cauchy-Wilstrass's definition of limit has the phenomenon of "limit avoidance". This statement is one-sided and not objective, but it still points out some problems (it should be said that it is the ultimate form of avoidance). Cauchy-Wilstrass's definition of limit was very classic when it was translated into China. Cauchy-Weierstrass's definition of limit not only defines the limit, but also describes a movement phenomenon-the movement approaching the limit (the final form). Finally, make the finishing point and call the final form a (if it exists, it is not clear how it came from) limit.

Grammatically speaking, this statement essentially gives the "final form" a title (name)-restriction. Therefore, in Cauchy-Wilstrass's definition of limit, limit is a noun, not a verb.

Therefore, the movement close to the limit is called the limit phenomenon. Many people understand Cauchy-Wilstrass's definition of limit, confuse limit phenomenon and limit, and generally call "limit phenomenon" and "limit" limit.

About the final form of learning, I once briefly talked about it in the secret report of Calculus 4. Because the modern definition of function limit does not explain the final form (avoidance)! So, what is the story of the limit definition of function? What is the relevant mathematical proof?

In fact, it is saying one thing: if there is a limit (final form), there must be a limit phenomenon; On the contrary, if there is a limit phenomenon, there must be a limit! Simply put, limit phenomenon is a necessary and sufficient condition for limit (final form). Therefore, to prove the existence of limit (without studying how it came from) is enough to prove the existence of limit phenomenon, which is indeed suspected of opportunism!

Because of this, the modern definition of limit can not tell you where the limit comes from, but can only tell you that the limit exists (and can be proved). Limit phenomenon is essentially a movement phenomenon. What is the ideal tool to describe the motion phenomenon-function? Therefore, in the modern definition of function (professional) limit, it is not surprising that some functions have flavor (one-to-one correspondence, there is always ε and δ correspondence).

Some people are also quite outrageous, saying that limit is a verb. The reason is that the essence of limit is: "a variable quantity is infinitely close to a fixed quantity." This is the essence of extreme phenomena, not extremes.

However, to describe the limit phenomenon. Must there be Cauchy-Wilstrass model? Of course not, the model can be changed, and elementary calculus has changed this model. It simplifies some complicated mathematical proofs, such as uniqueness of limit and monotonicity of function.

In Cauchy's works, there is no common language, and his statements seem inaccurate, which sometimes leads to errors, such as those caused by the failure to establish the concepts of uniform continuity and uniform convergence. But regarding the principle of calculus, his concept is mainly correct, and its clarity is unprecedented. For example, his definition of continuous function and its integral is accurate. He first proved Taylor formula accurately, and he gave the definition of convergence and divergence of series and some discrimination methods.

ordinary differential equation

Cauchy's most profound contribution to analysis is in the field of ordinary differential equations. He first proved the existence and uniqueness of the solution of the equation. No one had asked such a question before him. Generally speaking, Cauchy's three main methods, namely Cauchy-Lipschitz method, step-by-step approximation method and strong series method, were used to approximate and estimate solutions in the past. Cauchy's greatest contribution is to see that by calculating strong series, it can be proved that the approximation step converges, and its limit is the solution of the equation.

Mathematical theory of elasticity

Cauchy is the founder of elastic mathematics theory in mechanics. In his paper 1823 "Research on the Balance and Motion of Elastomer and Fluid (Elastic or Inelastic)", he put forward the general equation of the balance and motion of (isotropic) elastomer (later he extended this equation to the case of anisotropy), gave the strict definitions of stress and strain, and proposed that they can be expressed by six components respectively. This paper is also meaningful to the fluid motion equation, which is later than the result obtained by C.-L.-M.-H. Naville in 182 1, but it adopts the continuum model, and the result is more general than that obtained by Naville. The fluid equation he proposed in 1828 is only one less static pressure term than the Naville-Stokes equation (1848).

other

Although Cauchy mainly studies analysis, he has made contributions in all fields of mathematics. As for other disciplines applying mathematics, his achievements in astronomy and optics are secondary, but he is one of the founders of mathematical elasticity theory. In addition to the above, his other contributions to mathematics are as follows:

1. Analysis: the basic concept of traveling characteristic line in the theory of first-order partial differential equation; Realize the function of Fourier transform in solving differential equations.

2. Geometry: The integral geometry is established, and the formula for expressing the length of plane convex curve by some orthogonal projections on a plane straight line is obtained.

3. Algebra: First, it is proved that the matrix with order exceeding has eigenvalue; Firstly, the concept of permutation group is clearly put forward, and some unconventional results in group theory are obtained. Independent discovery of the so-called "algebraic essence", that is, grassmann's external algebraic principle.