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Karl Gustav Jacob Jacobi (Carl Gustav Jacob Jacobi) was born in Potsdam, Germany, on December 10, 1804. He is the second son of banker Simon Jacobi.

Young Jacobi's development in mathematics is in some ways the same as that of his rival, Abel. Jacobi also studied the works of the masters; from the works of Euler and Lagrange, he learned algebra, analysis, and number theory. Through self-study, Jacobi made an outstanding contribution to elliptic functions (described in more detail later). In terms of computing power, no one can match Euler and Jacobi except Ramanujin, an Indian mathematical genius. Compared with Jacobi, Abel's talent has more elements of philosophy and less form. Abel is closer to Gauss in insisting on rigor than Jacobi (it is not that Jacobi's work lacks rigour, but that its inspiration seems to be formalistic rather than rigorous).

Abel is two years older than Jacob. Jacobi did not know that Abel solved the general quintic equation problem in 1820. In the same year, he tried to find a solution, simplifying the general quintic equation to the following form.

It is pointed out that the solution of this equation can be deduced from the solution of some decadal equation. Although the attempt failed, Jacobi learned a lot about algebra, which he thought was an important step in his math education. But he didn't seem to think like Abel that the general quintic equation might not be solved by algebraic methods. This oversight, or lack of imagination, is a typical difference between Jacobi and Abel.

From April 1821 to May 1825, Jacobi attended university in Berlin. In the first two years, he spent his time evenly on philosophy, linguistics and mathematics. In mathematics, Jacobi continued to teach himself the works of the masters. He aptly described the university mathematics lecture as nonsense.

Abel wrote to Holmberg (Norwegian mathematician) on August 4, 1823, saying that he was busy studying elliptic functions.

This little work involved the inverse function of the elliptic transcendental function, and I proved something that seemed impossible; I asked Degen to go through it from beginning to end, but he couldn't find the wrong conclusion, nor did he know what was wrong; God knows how I can set myself free.

A strange coincidence is that when Jacobi finally made up his mind to devote himself to math, it was almost when Abel wrote this letter. Abel got off to an excellent start, but Jacobi soon caught up. Jacobi's first great work was on elliptic functions.

In his career in mathematics in August 1825, Jacobi received his Ph. D. After earning his degree, Jacobi taught the application of calculus to surfaces and spatial curves at the University of Berlin. The first few lectures made it clear that Jacobi was a very talented teacher. Later, he became the most popular math teacher at that time.

Jacobi of the University of Berlin seems to be the first math teacher to do so: he teaches his latest findings to train students to do research work by letting students see new disciplines created in front of them. He thinks it is right to throw young students into ice water and let them learn to swim or drown. Many students will not try to work independently until they have mastered everything that others have done and related to their problems. as a result, only a very small number of people have formed the habit of working independently. Jacobi said:

If your father insists on knowing all the girls in the world and then marrying one, he will never get married.

Jacobi won the position of lecturer at Konigsburg University in 1826 only half a year after getting the position of lecturer at the University of Berlin. A year later, Jacobi published some research results on number theory, which won Gauss's praise. Since Gauss is not an easily alarmed person, the Ministry of Education immediately noticed this and promoted Jacobi to associate professor (only 23 years old). Two years later (1829), when Kobe published his first masterpiece, the New Foundation of Elliptic function Theory.

In 1832, Jacobi's father died. The family went bankrupt in 1840. At the age of 36, Jacobi had nothing and had to support his mother because she was also bankrupt. But the economic woes had no effect on Jacobi's math. He continued to study as diligently as ever. In 1842 Jacobi and Bessel attended a conference in Manchester, where Jacobi of Germany met with Hamilton of Ireland. Jacobi learned about Hamilton's work on dynamics and promoted the development of dynamics, which is one of Jacobi's greatest glory.

Elliptic function Jacobi did his first great work was elliptic function. The elliptic function is only a detail of the function theory of a single complex variable. The function theory of single and complex variables is a main field of mathematics in the 19th century. Gauss once pointed out that every algebraic equation has a complex root.

Complex numbers first appear in the solutions of some equations, such as x ^ 2 + 1x 0. Plural problems are also encountered in factorization, such as decomposing x ^ 2 + y ^ 2 factors to get

Further, try to decompose x ^ 2 + y ^ 2 + z ^ 2 into two first-order factors. In this way, are positive, negative and imaginary numbers enough? In other words, is it necessary to invent some new "number" in order to solve this problem? It has been found that in order to get the necessary new "numbers", the general algebraic rule is disintegrated because of an important rule: the rule that the order in which "numbers" are multiplied together is "insignificant"; that is, for new numbers, a × b equals b × an is no longer true. This shows that the factorization of elementary algebra leads us to the field where complex numbers are not applicable.

If we insist that all the general laws of algebra hold true for these numbers, how far can we go? What is the most general number possible? In the second half of the 19th century, it was proved that the complex number x+iy is the most general number that makes ordinary algebra valid. In Cartesian geometry, the graph of the function f (x) gives us the graph of the function y of the real variable x. If the general algebra applied to these functions and the generalized calculus were applied to complex numbers, problems would arise in more than half of the many things discovered by early analysts. in particular, there are many puzzling irregularities in integral calculus that can be eliminated only when complex functions are adopted by Gauss and Cauchy.

In the theory of elliptic function, the complex number is inevitable. Through their extensive and detailed exposition of this theory, Gauss, Abel and Jacobi provide an experimental ground for discovering and improving the general theorem of single complex variable function theory. The two theories seem destined to complement and perfect each other-for a reason, the deep relationship between the elliptic function and the quadratic Gauss theorem. However, the consideration of space forces us to give up the theory of quadratic form. The special cases of the more extensive theorems in elliptic functions provide a lot of clues for the general theory. without these clues, the theory of single complex variable functions will develop much more slowly than the reality.

The history of elliptic functions is quite complex and is unlikely to arouse the interest of ordinary readers. Therefore, we briefly summarize the communications of Gauss, Jacobi and Legendre.

First of all, it is true that Gauss foresaw some of the most amazing discoveries by Abel and Jacobi 27 years ago. Gauss did say, "Abel followed the same path I took in 1798." Second, people seemed to agree that Abel was ahead of Jacobi in some important details, but Jacobi made his great discovery without knowing anything about Abel's work.

One of the important properties of elliptic functions is their biperiodicity (discovered by Abel in 1825): if E (x) is an elliptic function, then there are two special numbers, such as paired 1 and paired 2, so that

All values of the variable x are true.

Finally, on the historical side, Legendre worked on elliptic integrals (rather than elliptic functions) for 40 years, without noticing what Abel and Jacobi saw almost immediately, that is, as long as he reversed his point of view, the whole problem becomes extremely simple. The elliptic integral first appears in the problem of finding the arc length of an ellipse.

Let R (t) denote a polynomial of t, if R (t) is cubic or quartic, in the form of

The integral of R (t) is called an elliptic integral; if the degree of R (t) is more than four, the integral is called an Abelian integral. If R (t) is only quadratic, the integral can be easily calculated using an elementary function. Especially.

That is to say, if

Let's consider the upper limit x of the integral as a function of the integral itself (that is, y). This inversion of the problem solved most of the difficulties that Legendre had struggled with for 40 years. After removing this obstacle, the real theory of these important integrals almost emerged on its own.

Co-creating elliptic function theory with Abel is only a small part of Jacobi's huge workload, but it is a very important part. Let's briefly mention some of the other great work he has done.

Other achievements Jacobi was the first person to apply elliptic function theory to number theory. Number theory is a wonderful and esoteric subject, complex and ingenious algebra, in which number theory will unexpectedly reveal hitherto unexpected relationships between ordinary integers. It was in this way that Jacobi proved Fermat's famous conjecture: every integer is the sum of the squares of four integers (zero is also counted as an integer). Moreover, he knows how many ways any known integer can be expressed as such a sum.

In terms of dynamics, Jacobi made the first major progress beyond Lagrange and Hamilton, which is of fundamental importance in both applied science and mathematical physics. Readers familiar with quantum mechanics will recall the important role played by the Hamilton-Jacobi equation in that revolutionary theory.

In algebra, there is only one thing to mention, that is, Jacobi simplified the determinant theory to a simple form that every student in high school algebra is now familiar with.

For Newton-Laplace-Lagrange's theory of gravity, Jacobi excellently studied the repeated functions in this theory, and applied elliptic function and Abel function to the gravitation between ellipsoids, thus making a great contribution to the theory of gravity.

His great discovery in the Abel function has a higher degree of originality. Such a function is generated in the inversion of an Abelian integral, just as an elliptic function is generated in the inversion of an elliptic integral. He had no way to go here, and for a long time he lost his way in a clueless labyrinth. In the simplest case, the appropriate inverse function is a function with two variables with four periods. In general, these functions have n variables and 2n periods; an elliptic function is equivalent to niter1. This discovery is to 19th-century analysis, just as Columbus discovered America to 15th-century geography.

Fourier accused Abel and Jacobi of wasting time on elliptic functions instead of solving some unsolved problems in heat conduction. Jacobi said:

Mr. Fourier does have the view that the main purpose of mathematics is the needs of the public and the interpretation of natural phenomena; but a philosopher like him should know that the sole purpose of science is the glory of the human mind, and should know that, from this point of view, the problem of numbers is of the same value as the question of the cosmic system.

Today, as far as mathematical physics is concerned, Fourier's analysis is only a detail of the much broader boundary value theory, and the analytical method invented by Fourier, it is in the purest part of pure mathematics that finds its significance and its justification. Whether these modern researchers have added glory to the "human mind" may be left to experts to examine.

Jacobi died of smallpox at the age of 47 (February 18, 1851).

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

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