Joseph Liouville (March 24, 1809 to September 8, 1882), a French mathematician, devoted his whole life to the research of mathematics, mechanics and astronomy, with extensive involvement and rich achievements, especially for the double cycleelliptic function 、differential equation Boundary value problemAnd in number theoryTranscendental numberThe problem has been studied in depth.
Liu Weir studied what was later called“Liouville number”And proved its transcendence, which is the first confirmationTranscendental numberPeople who exist.He has made important academic contributions to mathematical research.[1]
Liu Weir, born on March 24, 1809FranceCalaisSaint Aumert died in Paris on September 8, 1882.
Liu Wei's father, Claud Joseph Liouville, was aArmy CaptainHis mother's name is Th é r é se Bal land.Liu Wei is their second son, who learned from Comesi and Tuer when he was young.He came in 1825Paris Polytechnic SchoolStudy, A.M. Amp è re served as the teacher of the analysis and mechanics class, and the two discussed togetherelectrodynamicsQuestion.He was transferred to the Bridge and Highway School in November 1827 and received his bachelor's degree in 1831.
Shortly after graduation, he resigned from theIzelI hope to get a teaching position so that I can concentrate on academic work.In November 1831, he was enrolled in a comprehensive engineering courseSchoolingThe committee was elected as the teaching assistant of the analysis and mechanics course of L. Mathieu, which started its ownscientific researchCareer.
From 1833 to 1838, Liu Weir taught mathematics and mechanics in the newly established Central Higher Technological Manufacturing School, but the content was elementary.In order to keep his teaching work at the university level, he obtained a doctor's degree in 1836. His thesis was entitled "On the sine andcosineSurle d é velop pement des functions ou parts de functions en s é ries dc sinuset de cosinusFourier series The book was published in Paris in the same year.
In order to meet the needs of French mathematical research, Liu Weir founded《Journal of Pure and Applied Mathematics》(Journal de math é matiques pures et appliqu é es), and personally presided over the editing and publishing of the first 39 volumes (Volume 1, Volume 1-20, 1836-1855; Volume 2, Volume 1-19, 1856-1874).The magazine publishes pureapplied mathematicsThe papers of all branches of the field recorded part of the important contents of mathematical activities in the 40 years in the middle of the 19th century and were called "Liu ville's Journal" by later generations.
Liu Weir not only kept close contact with some important mathematicians at that time and regularly published their achievements, but also enthusiastically guided young scholars to provide them with opportunities to publish books.What is worth mentioning most is that he edited and published the article of E. Galois.In May 1832, Galois was killed in a duel. Liu Wei sorted out some of his manuscripts and published them in the Journal of Pure and Applied Mathematics in 1846. His original work in algebra was known to the world.
In 1838, Liu Weir succeeded Mathieu as the chair of analysis and mechanics in the comprehensive engineering school, and worked until 1851 when he transferred toCollege de France Until he holds the chair of mathematics.In June 1839 and 1840, he was elected as a member and standard of the Astronomy Department of the Paris Academy of Sciencesmeasurement bureauMembers, regularly participate in these two activities.
Liu Weir's academic activities were slightly interrupted during the French Revolution.On April 23, 1848, he was electedConstitutional AssemblyHe was one of the representatives of the Mert Administrative Region. In May of the next year, his election for parliament failed, and his political activities ended.
After arriving at the French Academy in 1851, Liu Weir's teaching work was quite free, and he had more time to carry out his own research work and discuss with others extensively.He worked in this position until 1879.However, after he quit the editing work of the Journal of Pure and Applied Mathematics in 1874, he no longer published books and rarely participated in the activities of the French academic community.Liu Weir devoted his whole life to academic work and lifeCalm and quietEvery year, they return to their old home in Turkey for vacation.In 1830, he worked withCousinMarie Louise Balland married and had three daughters and one son.
Academic achievements
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Function theory
Liu Wei carefully studied G.W. Leibniz and John·Bernoulli(Johann Bernoulli) and LEuler(Euler）Works of.In his early work, he expanded the concept of differential and integral as much as possible, especially the theory of establishing derivatives of any order.
The algebraic function of a finite number of [url] complex variables is an elementary function of type 0;Ez and logz are elementary functions of the first kind;The two are collectively referred to as at most the first kind of elementary functions.If at most the n-1st elementary function has been defined, then its combination with at most the 1st elementary function is called at most the n-th elementary function.The elementary function of the nth class is called the nth elementary function.
The integral of elementary function is still elementary function under any conditions, which is also the problem he focuses on.When Liu Wei set foot in the field of science, Abel and CJacobian (Jacobi)elliptic function Theory is in a period of vigorous development.In December 1844, in a letter to the Paris Academy of Sciences, Liu Weir explained how to use the Jacobi's theorem (single variableSingle valueMeromorphic functionThe number of cycles of is not more than 2, and the cycle ratio isNon real number）Set out to establish a complete set of double periodic elliptic functionsTheoretical system。This is a great contribution to elliptic function theory.Around the double periodicity, Liu Weir showed the essential properties of elliptic functions and proposed the following theorem:
Liouville's first theorem
Having no poles in a periodic parallelogramelliptic function Is a constant;
Nth orderelliptic function Take any value n times in a period parallelogram;
Liouville's fourth theorem
The difference between the sum of zeros and the sum of poles in a period parallelogram is equal to one period.
Later, two German mathematicians C.W. who visited ParisBorchardtBor chardt and F. Joachimsthal asked Liu Weir about his work in detailCollege de France LecturedBiperiodic functionThe course is also systematically introduced in the book Th é orie des functions doublementp é riodiques (1859) by C.A. Briot and J.C. Bou quet.Therefore, although Liu Weir's relevant conclusions were rarely published, they could still spread rapidly and have an impact inside and outside France. The lecture notes on the double periodic function were later published in the German Journal of Pure and Applied Mathematics, Volume 88, 1880.
Liu Weir's interest in the problem of logarithm theory is due toFermat's big theoremStarted.In 1840, he willFermatThe problem is transformed and the equation un+vn=wn is provedUnsolvableIt means that x2n-y2n=2xn is insoluble.From 1856, Liu Weir gave up almost all mathematical research in other aspects, and put his energy into the field of number theory.
In 1836, Liu Wei andsturm They jointly give information aboutalgebraic equationimaginary rootNumberCauchy theoremCertification of;The next year, he solved the duality in a different way from AbelAlgebraic equationsOfElimination problem。These are all J.ASere(Serret) was included in the fourth edition (1877) of Cours d'Alg è bre superieure, which he wrote, and was widely spread in French schools.
In order to publish Galois's works, Liu Weir conducted a thorough study of his manuscripts from 1843 to 1846.In his introduction to the publication of Galois' works, he spoke highly of Galois' work.He also invited some friends, including Sere, to take part in a series of speeches on Galois's work.Therefore, Liu Wei indirectly promoted the development of modern mathematics and group theory.
staygeometryIn 1841 and 1844, Liu Weir proved and popularized theMetric property, also found a new method to determine whether anyElliptic surfaceOfGeodesic line, this isJacobian Studying hyperbolaTranscendental numberProblems arising from.In 1850, he was responsible for publishing GMengri(Monge)'s book Application de l ′ anal yse à la g é om é trie, the fifth edition, has attached C.F. Gauss's famous book "Discositions general circles su perficiencies curvas" and seven notes written by him at the end of the book.These notes refer to curves and their relativecurvatureandGeodesic curvature、Geodesic lineEquationTotal curvatureConcept, etc.Liu Weir also wrote a few articles about thermal theory, electricityCelestial mechanicsandTheoretical mechanicsAnd so on.
