约瑟夫·刘维尔Joseph Liouville

约瑟夫·刘维尔的父亲是拿破仑军队中的一名上尉，因此刘维尔不得不在他生命的最初几年与他的叔叔一起度过。他的父亲确实幸运地在战争中幸存下来，拿破仑战败后，他退休与家人同住。全家随后定居在土尔，刘维尔在那里上学。从土尔，他去了巴黎的圣路易学院，在那里他学习了最高水平的数学。在阅读了约瑟夫·热尔岗杂志上的文章后，他证明了一些几何结果，并将其写成论文，尽管这些论文从未发表。

刘维尔于1825年进入巴黎综合理工学院，并在1825-26学年参加了安德烈-马里·安培的Cours d'analyse et de mécaniqueⓉ（分析与力学教程）。他还参加了弗朗索瓦·阿拉戈在巴黎综合理工学院的课程，以及安德烈-马里·安培在法兰西学院的第二门课程。尽管刘维尔似乎没有参加过奥古斯丁·路易·柯西的任何课程，但很明显奥古斯丁·路易·柯西对他产生了强烈影响。刘维尔于1827年毕业，de Prony和西莫恩·德尼·泊松是他的考官。

从巴黎综合理工学院毕业后，刘维尔进入了桥梁与道路学院。然而，当他不得不承担工程项目时，他的健康受到了损害，他在土尔的家中休养了一段时间。此时，刘维尔已决心从事学术生涯，他发现无法离开巴黎学习。在多次请假后，其中一次让他得以结婚并度了几天蜜月，他清楚地意识到必须从桥梁与道路学院辞职。他在1830年10月这样做了，但即使在这个阶段，他已经写了许多论文，提交给巴黎科学院，内容涉及电动力学、偏微分方程和热理论。

1831年，刘维尔被任命为他的第一个学术职位，作为Claude Mathieu的助手，后者已被任命为巴黎综合理工学院安德烈-马里·安培的讲席。他还被任命到一些私立学校和中央学院。值得注意的是，在他生命的这一时期，刘维尔在不同机构每周教授35到40小时。也许课程安排如此繁重，某些课程进展不特别顺利也就不足为奇了，看来他对一些能力较差的学生讲课水平过高。

1836年，刘维尔创办了一份数学期刊Journal de Mathématiques Pures et Appliquées。这份期刊，有时被称为Journal de Liouville，在整个19世纪为法国的数学做出了很大贡献。刘维尔已经通过在奥古斯都·利奥波德·克雷勒的期刊上发表论文赢得了国际声誉，但与此同时，奥古斯都·利奥波德·克雷勒的期刊的质量让他意识到法国数学出版途径的不足。当然，他对巴黎期刊的风格不满意，因为他在1836年写道：-

……一种奇特的移民精神抓住了一些批评家，我们看到他们一个接一个地辱骂那些在各个科学领域以极大的尊严为法国增光的人。……这种尖锐而专横的风格……永远不会是我的，因为它既损害了采用它的人的性格，也损害了他们的才能。

刘维尔成为填补巴黎综合理工学院讲席的热门人选，该讲席因克洛德-路易·纳维于1836年去世而空缺。然而，经过激烈的竞争，让-马里·迪阿梅尔获得了任命。1837年，刘维尔被任命为法兰西公学院的讲师，作为让-巴蒂斯特·毕奥的替代者。1838年，刘维尔被任命为巴黎综合理工学院分析与力学教授。

第二年，他当选为Académie des Sciences的天文学部成员，但这只是在遭到Libri的强烈反对之后。事实上，刘维尔和Libri之间的争吵在他当选Académie之后加剧了。1840年，在西莫恩·德尼·泊松去世造成空缺后，刘维尔当选为经度局成员。

在许多方面，1840年是刘维尔职业生涯的转折点。正如Lützen在[4]中所写：-

1840年之前，刘维尔曾沿着一些明确的道路来确保自己的职业生涯；在接下来的二十年里，推广和发展其他数学家的思想成为了核心议题。

刘维尔的生活演变成一年分为两个截然不同的部分。在漫长的夏季期间，他在图勒度过，进行研究、撰写论文并履行编辑职责。从11月到7月，他住在巴黎，履行教学和行政职责。

然而，并非一切都如刘维尔所愿。当西尔维斯特·佛朗索瓦·拉克鲁瓦于1843年去世时，刘维尔申请了他在法兰西公学院的讲席，在那里他仅作为让-巴蒂斯特·毕奥的替代者授课。然而，经过激烈的选举，Libri获得了任命。刘维尔立即从法兰西公学院辞职，在辞职信中写道：-

昨天在法兰西公学院发生的事件使我作为一个人和作为一个几何学家深感屈辱。从这一刻起，我不可能在这个机构授课了。

刘维尔一生的另一个方面是他参与政治。他的朋友兼数学同行之一是弗朗索瓦·阿拉戈，后者于1831年进入众议院，并成为共和党领袖。其他数学同行也卷入了当时的政治事件，例如乌惹内·查尔斯·卡塔兰，其政治观点与刘维尔的共和观点相似，损害了他的数学事业。刘维尔在推进自己的数学事业时，肯定从未让政治观点阻碍自己，不像奥古斯丁·路易·柯西，后者拒绝向国王宣誓效忠，而刘维尔甚至弗朗索瓦·阿拉戈都准备这样做。

在弗朗索瓦·阿拉戈的鼓励下，刘维尔于1848年竞选制宪议会。在推荐刘维尔为候选人时，弗朗索瓦·阿拉戈写道：-

……刘维尔先生是我最好的朋友之一。他是一位非常杰出的人，一位爱国者，一位经验丰富的共和主义者。愿上帝让国民议会中有许多这样水准的成员。

刘维尔于1848年4月23日当选，在温和共和派多数中占据一席。然而巴黎发生了动荡，因为工人们觉得他们的革命已被资产阶级接管。当弗朗索瓦·阿拉戈试图向人群讲话时，一名诘问者喊出了意味深长的评论：-

弗朗索瓦·阿拉戈先生，你从未挨过饿。

刘维尔继续其政治生涯，于1849年被再次提名为议会选举候选人，但潮流已转向反对温和共和派，他未能当选。

选举失败被证明是刘维尔一生中的另一个转折点。正如4在[4]中所写：-

政治上的失败改变了刘维尔的性格。在早先的信件中，他常因疾病而沮丧，并会向Libri这样的敌人发泄怒火，但他始终为自己认为正确的事而奋斗。1849年选举之后，他辞职了，变得 bitter，甚至对老朋友也是如此。当他坐到书桌前时，他不仅工作，……他还思索自己不幸的命运。……他的数学笔记中夹杂着诗人和哲学家的引语……

Libri在1848年革命期间逃离法国，并非出于政治原因，而是为了逃避因盗窃珍贵书籍和手稿而面临的监禁。他在法兰西公学院的讲席于1850年被宣布空缺，奥古斯丁·路易·柯西和刘维尔竞争这一职位。在一场势均力敌的竞争中，刘维尔胜出，并于1851年开始在法兰西公学院授课。

刘维尔在参与政治期间数学产出大幅减少，尽管健康有问题，但在19世纪50年代又恢复了。事实上，1856年和1857年是刘维尔最多产的两年。然而，在1857年被任命为理学院力学讲席后，他的教学负担开始对他造成损害。他不仅教学负担重，而且刘维尔是个完美主义者，这意味着当他觉得自己无法投入所有必要的时间来尽可能上好课时，他开始感到痛苦。他继续发表论文，但大多是他在1856年这个高产年份发现的成果，而且这些成果没有证明，因为他现在找不到时间来润色。

对刘维尔的另一个打击是约翰·彼得·古斯塔夫·勒热纳·狄利克雷在1859年去世。这在数学上对他是一个严重打击，因为除了失去一位密友外，他还失去了主要的数学通信者。

刘维尔的数学工作范围极其广泛，从数学物理到天文学再到纯数学。他最早研究的主题之一，源于他早期关于电磁学的工作，是一个新主题，现在称为分数阶微积分。他定义了任意阶$D^{t}$的微分算子。通常$t$是整数，但在刘维尔于1832年至1837年间发表的论文中发展的这一理论里，$t$可以是有理的、无理数，或者最一般地，是一个复数。

刘维尔在1832-33年间研究了代数函数的积分成为代数的准则。在四篇论文中确立了这一点后，刘维尔继续研究以有限项积分代数函数的一般问题。他的工作起初独立于尼尔斯·阿贝尔的工作，但后来他了解到尼尔斯·阿贝尔的工作，并将几个想法纳入自己的工作中。

刘维尔今天仍被人们铭记的另一个重要领域是超越的数。刘维尔对这一领域的兴趣源于阅读克里斯蒂安·哥德巴赫与丹尼尔·伯努利之间的通信。刘维尔确实旨在证明$e$是超越数，但他没有成功。然而他的贡献巨大，并促使他在1844年证明了超越数的存在，当时他利用连分数构造了一类无穷的此类数。1851年，他发表了关于超越数的结果，摆脱了对连分数的依赖。特别是他给出了一个超越数的例子，这个数现在被称为刘维尔数

0.1100010000000000000000010000...

其中在$n!$处为1，其他地方为0。

他在边值问题上关于微分方程的工作被记住，是因为今天所谓的雅克·夏尔·弗朗索瓦·施图姆-刘维尔理论，该理论用于求解积分方程。这个理论在数学物理中具有重大重要性，是在1829年至1837年间发展起来的。雅克·夏尔·弗朗索瓦·施图姆和刘维尔研究了广义线性二阶微分方程，并考察了其eigenvalues的性质、本征函数的行为以及任意函数按这些本征函数的级数展开。

刘维尔为研究conformal transformations的微分几何做出了贡献。他证明了一个关于哈密顿动力学保测性质的重要定理。这一结果在统计力学和测度论中具有根本重要性。

1842年，刘维尔开始阅读埃瓦里斯特·伽罗瓦未发表的论文。1843年9月，他向巴黎科学院宣布，他在埃瓦里斯特·伽罗瓦的工作中发现了深刻的结果，并承诺将埃瓦里斯特·伽罗瓦的论文连同他自己的评注一起出版。因此，刘维尔在使埃瓦里斯特·伽罗瓦的工作引起普遍注意方面产生了重大影响，他于1846年在其期刊上发表了这一工作。然而，他在发表这些论文之前等待了三年，而且相当奇怪的是，他从未发表自己的评注，尽管他确实写了一篇填补埃瓦里斯特·伽罗瓦证明中空缺的评注。刘维尔还就埃瓦里斯特·伽罗瓦的工作和约瑟夫·阿尔弗雷德·塞雷进行了讲授，可能还有约瑟·伯特兰和夏尔·埃尔米特一起听了这门课。

在数论中，刘维尔写了大约200篇论文，研究二次互反律和许多其他主题。他总共写了400多篇论文。

Joseph Liouville's father was an army captain in Napoleon's army so Joseph had to spend the first few years of his life with his uncle. His father was certainly fortunate to survive the wars and after Napoleon was defeated he retired to live with his family. The family then settled in Toul where Joseph attended school. From Toul he went to the Collège St Louis in Paris where he studied mathematics at the highest levels. After reading articles in Gergonne's Journal he proved some geometrical results which he wrote up as papers although they were never published.

Liouville entered the École Polytechnique in 1825 and attended Ampère's Cours d'analyse et de mécanique Ⓣ in session 1825-26. He also attended courses by Arago at the École Polytechnique as well as a second course by Ampère at the Collège de France. Although Liouville does not seem to have attended any of Cauchy's courses, it is clear that Cauchy must have had a strong influence on him. Liouville graduated in 1827 with de Prony and Poisson among his examiners.

After graduating from the École Polytechnique Liouville entered the École des Ponts et Chaussées. However his health suffered when he had to undertake engineering projects and he spent some time at his home in Toul recovering. By now Liouville was set on an academic career and he found it impossible to study away from Paris. After a number of periods of leave, one of which allowed him to marry and have a few days honeymoon, it became clear to him that he must resign from the École des Ponts et Chaussées. This he did in October of 1830 but even at this stage he had written a number of papers which he had submitted to the Paris Academy on electrodynamics, partial differential equations and the theory of heat.

In 1831 Liouville was appointed to his first academic post, as assistant to Claude Mathieu who had been appointed to Ampère's chair at the École Polytechnique. He was also appointed to a number of private schools and to the École Centrale. It is remarkable that during this period of his life Liouville taught between 35 and 40 hours a week at the different institutions. Perhaps with a schedule this heavy it is not surprising that some courses would not go particularly well and it appears that he lectured at too high a level for some of the less able students.

In 1836 Liouville founded a mathematics journal Journal de Mathématiques Pures et Appliquées. This journal, sometimes known as Journal de Liouville, did much for mathematics in France throughout the 19th century. Liouville had already gained an international reputation with papers published in Crelle's Journal but at the same time the quality of Crelle's Journal made him aware of deficiencies in the avenues for mathematical publications which there were in France. Certainly he was unhappy with the style of the Paris Journals for he wrote in 1836:-

.. a peculiar spirit of emigration has seized some critics and we have seen them heap abuse on one after the other of the men who in various fields of science have honoured France with great dignity. ... this sharp and peremptory style ... will never be mine, for it dishonours both the character and talent of those who adopt it.

Liouville became favourite to fill the chair at the École Polytechnique which fell vacant when Navier died in 1836. However, after a close competition, Duhamel was appointed. In 1837 Liouville was appointed to lecture at the Collège de France as a substitute for Biot. In 1838 Liouville was appointed Professor of Analysis and Mechanics at the École Polytechnique.

The following year he was elected to the astronomy section of the Académie des Sciences but this was only after strong opposition from Libri. In fact the quarrel between Liouville and Libri intensified after his election to the Académie. In 1840, after a vacancy resulting from the death of Poisson, Liouville was elected to the Bureau des Longitudes.

In many ways 1840 was a turning point in Liouville's career. As Lützen writes in [4]:-

Before 1840, Liouville had pursued some clear paths to secure his own career; during the following twenty years, the promotion and development of other mathematicians' ideas became the central issue.

Life for Liouville developed into a year with two distinct parts. During the long summer period, spent at Toul, he undertook research, wrote papers and carried out editing duties. From November to July he lived in Paris and carried out his teaching and administrative duties.

Not everything went Liouville's way however. When Lacroix died in 1843, Liouville applied for his chair at the Collège de France where he lectured only as a substitute for Biot. However after a close election Libri was appointed. Liouville immediately resigned from the Collège de France, writing in his resignation letter:-

I am profoundly humiliated as a person and as a geometer by the events that took place yesterday at the Collège de France. From this moment it is impossible for me to lecture at this institution.

Another aspect of Liouville's life was his involvement in politics. One of his friends, and mathematical colleagues, was Arago who entered the Chamber of Deputies in 1831 and became leader of the Republican Party. Other mathematical colleagues had also become involved with the political events of the time, for example Catalan, whose political views were similar to the republican views of Liouville, had damaged his mathematical career. Liouville certainly never let his political views hold him back as he advanced his mathematical career, unlike Cauchy who had refused to swear the oaths of allegiance to the King that Liouville and even Arago had been prepared to do.

Encouraged by Arago, Liouville stood for election to the Constituting Assembly in 1848. In his recommendation of Liouville as a candidate Arago wrote:-

... Mr Liouville is one of my best friends. He is a very eminent man, a patriot, an experienced republican. God grant that the National Assembly will contain many members of this calibre.

Elected on 23 April 1848, Liouville took his seat among the moderate republican majority. However there was unrest in Paris as workers felt that their revolution had been taken over by the bourgeoisie. When Arago tried to address the crowds a heckler shouted the telling comment:-

Mr Arago, you have never been hungry.

Liouville continued his political career by being renominated for the Assembly elections in 1849 but the tide had turned against the moderate republicans and he was not elected.

The election defeat proved another turning point in Liouville's life. As Lützen writes in [4]:-

The political defeat changed Liouville's personality. In earlier letters, he was often depressed because of illness, and could vent his anger towards his enemies such as Libri, but he always fought for what he believed was right. After the election in 1849, he resigned and became bitter, even towards his old friends. When he sat down at his desk, he did not only work, ... he also pondered his ill fate. ... his mathematical notes were interrupted with quotes from poets and philosophers...

Libri escaped from France during the 1848 revolution, not for political reasons, but to avoid a prison sentence for stealing precious books and manuscripts. His chair at the Collège de France was declared vacant in 1850 and Cauchy and Liouville competed for the post. In a close contest Liouville triumphed and began his lectures at the Collège de France in 1851.

Although Liouville's mathematical output had been greatly reduced while he was involved with politics, it picked up again in the 1850s despite health problems. In fact 1856 and 1857 were two of Liouville's most productive years. However after being appointed to the chair of mechanics at the Faculté des Sciences in 1857 his teaching load began to take its toll on him. Not only did he have a high teaching load but Liouville was a perfectionist which meant that when he felt that he could not devote all the time necessary to give the best possible lectures he began to suffer. He continued to publish but mostly results he had discovered during his highly productive year of 1856 and these without the proofs which now he could not find time to polish.

Another blow to Liouville was the death of Dirichlet in 1859. This was a serious blow to him mathematically for, as well as losing a close friend, he lost his main mathematical correspondent.

Liouville's mathematical work was extremely wide ranging, from mathematical physics to astronomy to pure mathematics. One of the first topics he studied, which developed from his early work on electromagnetism, was a new topic, now called the fractional calculus. He defined differential operators of arbitrary order $D^{t}$. Usually $t$ is an integer but in this theory developed by Liouville in papers between 1832 and 1837, $t$ could be a rational, an irrational or most generally of all a complex number.

Liouville investigated criteria for integrals of algebraic functions to be algebraic during the period 1832-33. Having established this in four papers, Liouville went on to investigate the general problem of integration of algebraic functions in finite terms. His work at first was independent of that of Abel, but later he learnt of Abel's work and included several ideas into his own work.

Another important area which Liouville is remembered for today is that of transcendental numbers. Liouville's interest in this stemmed from reading a correspondence between Goldbach and Daniel Bernoulli. Liouville certainly aimed to prove that $e$ is transcendental but he did not succeed. However his contributions were great and led him to prove the existence of a transcendental number in 1844 when he constructed an infinite class of such numbers using continued fractions. In 1851 he published results on transcendental numbers removing the dependence on continued fractions. In particular he gave an example of a transcendental number, the number now named the Liouvillian number

0.1100010000000000000000010000...

where there is a 1 in place $n!$ and 0 elsewhere.

His work on boundary value problems on differential equations is remembered because of what is called today Sturm-Liouville theory which is used in solving integral equations. This theory, which has major importance in mathematical physics, was developed between 1829 and 1837. Sturm and Liouville examined general linear second order differential equations and examined properties of their eigenvalues, the behaviour of the eigenfunctions and the series expansion of arbitrary functions in terms of these eigenfunctions.

Liouville contributed to differential geometry studying conformal transformations. He proved a major theorem concerning the measure preserving property of Hamiltonian dynamics. The result is of fundamental importance in statistical mechanics and measure theory.

In 1842 Liouville began to read Galois's unpublished papers. In September of 1843 he announced to the Paris Academy that he had found deep results in Galois's work and promised to publish Galois's papers together with his own commentary. Liouville was therefore a major influence in bringing Galois's work to general notice when he published this work in 1846 in his Journal. However he had waited three years before publishing the papers and, rather strangely, he never published his commentary although he certainly wrote a commentary which filled in the gaps in Galois's proofs. Liouville also lectured on Galois's work and Serret, possibly together with Bertrand and Hermite, attended the course.

In number theory Liouville wrote around 200 papers, working on quadratic reciprocity and many other topics. He wrote over 400 papers in total.