拉扎勒斯·福克斯Lazarus Fuchs

拉扎勒斯·福克斯的父母是犹太教师Rafael Fuchs和他的妻子Caecilie Katz。我们没有福克斯父母的日期，但我们知道他写博士学位论文时，他的母亲已经去世，而他的父亲仍然在世。他最初在Moschin由父亲教导，但他有强烈的求知欲，所以1846年他去了Posen（今波兰的波兹南），在那里私下学习了两年古典语言，但主要是自学。然而，他的父母经济困难，无法给他任何支持。他确实从一个远房富有的亲戚那里得到了极少的支持，但他生活在极度贫困中。他找到了一个便宜的地方住，但条件极差。几个月里，他唯一的食物是咖啡和面包。他经常乞求较富裕的朋友给他蜡烛的最后一段，以便他能在夜间学习，实现进入当地文理中学的目标。1848年，他进入了Posen的弗里德里希·威廉文理中学的三年级，当时该校的校长是Gustav Kiessling。当他在文理中学学习时，亚伯拉罕·阿德里安·艾伯特 Gustav Heydemann（1808-1877）于1850年被任命为校长。

正是在文理中学，尽管他在开始学习之前对数学的准备非常差，但他的数学才能却非常清楚地显现出来。数学成为了一门学科，即使在早期阶段，福克斯就知道它将主导他余生。除了学校学习，福克斯还必须通过私人辅导赚钱，他培养学生隐藏技能和才能的能力很快得到认可，因此他成为了一位备受追捧的教师。他的一个朋友是莱奥·柯尼希斯贝格尔，他比福克斯小四岁，也是Posen的弗里德里希·威廉文理中学的学生。十八个月后，福克斯完成了中学，跳过了高中一年，他能够在1853年复活节参加成熟考试，并以“优秀”的成绩通过。

尽管在1853年有资格进入大学，但他无法开始学习，因为他完全一贫如洗。正是在这个阶段，莱奥·柯尼希斯贝格尔的父母伸出了援手。他们提出雇佣福克斯作为Leo的私人导师一年，在他们家中提供住宿，并支付他适度的费用。他欣然接受，在辅导莱奥·柯尼希斯贝格尔一年后，他于1854年复活节进入柏林大学。当时大学的校长是著名天文学家约翰·弗朗茨·恩克，福克斯被录取到哲学系，其系主任是哲学家Friedrich Adolf Trendelenburg（1802-1872）。从1857年复活节到1864年复活节，福克斯和莱奥·柯尼希斯贝格尔在柏林合住。他们住在许多不同的房子里，被迫过着非常简朴和谦逊的生活方式，因为尤其是在最初几年，他们靠福克斯通过私人课程赚取的收入生活。

在柏林大学，福克斯听了包括恩斯特·爱德华·库默尔和卡尔·魏尔斯特拉斯在内的许多著名数学家的讲座。他在博士论文中给出了在柏林大学四年期间教过他的所有教师的完整名单。这些教师是：数学家Peter Friedrich Arndt（1817-1866）、古典学者August Böckh（1785-1867）、卡尔·威廉·博尔夏特、鲁道夫·克劳修斯、约翰·彼得·古斯塔夫·勒热纳·狄利克雷、物理学家和气象学家Heinrich Wilhelm Dove（1803-1879）、天文学家约翰·弗朗茨·恩克、恩斯特·爱德华·库默尔、动物学家Martin Heinrich Carl Lichtenstein（1780-1857）、化学和物理学实验家Heinrich Gustav Magnus（1802-1870）、化学家和矿物学家Eilhard Mitscherlich（1794-1863）、数学家马丁·欧姆、物理学家Johann Christian Poggendorff（1796-1877）、历史学家和政治学家Friedrich Ludwig Georg von Raumer（1781-1873）、化学家Franz Leopold Sonnenschein（1817-1879）、哲学家Friedrich Adolf Trendelenburg（1802-1872）、矿物学家Christian Samuel Weiss（1780-1856）以及卡尔·魏尔斯特拉斯。他写道：-

我对所有这些人感激不尽，并将永远感激，他们为我提供了完美的杰出服务。

除了听上面列出的那些人的讲座之外，福克斯还阅读了卡尔·弗里德里希·高斯的Disquisitiones arithmeticae以及约瑟夫·傅里叶、皮埃尔·西蒙·拉普拉斯和奥古斯丁·路易·柯西的著作。对福克斯影响最大的教授是卡尔·魏尔斯特拉斯，他引导他进入函数论，以及恩斯特·爱德华·库默尔，后者后来指导了他的博士学位。他的学位论文De Superficierum lineis curvaturae Ⓣ（论曲面上的曲线）的答辩于1858年8月2日举行，考官是恩斯特·爱德华·库默尔和马丁·欧姆（格奥尔格·欧姆的兄弟），他的对手是尤利乌斯·外恩加滕、莱奥·柯尼希斯贝格尔和Eduard Fischer。福克斯于1858年由柏林大学授予学位。

在获得博士学位之前，福克斯开始担心是否应该从犹太教改信基督教。他知道，如果他继续信奉犹太教，就会遭受歧视，职业前景将非常有限。尽管在福克斯早年，犹太人曾受到官方歧视，但1850年普鲁士的新宪法赋予所有公民平等的权利，不论其宗教信仰如何。尽管有宪法，平等并未实现，歧视在普鲁士继续存在，对犹太人的许多限制和局限也是如此。福克斯花了三年时间深深担忧是否应该改信基督教。当然，这不仅仅是个人的事情，因为他非常担心如果自己改宗，他的犹太家庭会作何感想。恩斯特·爱德华·库默尔和卡尔·魏尔斯特拉斯都深知福克斯在这个问题上的痛苦煎熬，并鼓励他迈出这一步。他还得到了尤利乌斯·米伦西芬（1811-1893）的大力支持，后者是柏林圣玛利亚教堂的牧师。1860年，福克斯改信了福音派路德宗基督教。

在获得博士学位之前，福克斯于1859年3月19日被任命到柏林的一所文理中学任教。事实上，在转到弗里德里希·韦尔德贸易学校担任数学教师之前，他曾在几所不同的机构任教。在此期间，他从事研究，目标是成为大学教授。正是在这一时期，他在变系数齐次线性微分方程方面做出了杰出工作，发表在1866年奥古斯都·利奥波德·克雷勒杂志上的一篇重要的40页论文Zur Theorie der linearen Differentialgleichungen mit veränderlichen Coefficienten Ⓣ（《论变系数线性微分方程的理论》）中。这项工作将在下文更详细地描述。1865年8月，在提交了他的教授资格论文（Habilitation）学位论文Zur Theorie der linearen Differentialgleichungen mit veränderlichen Coeffizienten Ⓣ（《论变系数线性微分方程的理论》）后，他被任命为柏林大学的私人讲师，开始了他的大学教学生涯。1866年12月2日，他被提升为该校的编外教授，并在该大学任教至1868-69年冬季学期，之后他接受了格赖夫斯瓦尔德的正教授职位。福克斯还于1867年5月23日在柏林担任第二个职位，被任命为炮兵与工程学校的数学教授。

1868年，福克斯与玛丽·安德斯（1849年7月23日-1917年5月10日）结婚。他们有四个儿子和两个女儿，包括Clara Fuchs（1869-1954）和Richard Fuchs，后者成为数学家，在本档案中有传记。克拉拉嫁给了数学家Ludwig Schlesinger，后者在本档案中也有传记。我们注意到，Richard Fuchs和Ludwig Schlesinger共同编辑了福克斯的全集：Gesammelte mathematische Werke von L Fuchs，由Mayer & 埃米尔·穆勒于1904年、1906年和1909年在柏林分三卷出版。

福克斯于1869年2月3日在格赖夫斯瓦尔德就任教授职位，该职位因莱奥·柯尼希斯贝格尔（曾在格赖夫斯瓦尔德任教五年）被任命为海德堡的数学讲席而空缺。在格赖夫斯瓦尔德度过五年后，福克斯再次搬迁，这次去了哥廷根，于1874年1月23日就任常任教授。次年，他前往海德堡，在那里任教九年。在许多方面，这些年是他一生中最愉快的时期。他热爱城市周围的自然环境，这对福克斯来说很重要，因为他对自然有着深厚的感情。在海德堡，他与许多杰出的博士生关系特别好，并且与大学许多不同院系的其他教职员工相处得极为融洽。事实上，在海德堡的九年里，他指导了至少八名后来成为数学教授的学生的博士研究。我们将在下面稍微谈谈他在海德堡期间与儒勒·昂利·庞加莱的通信。1884年夏季学期，当他的老师退休后，他回到柏林接替恩斯特·爱德华·库默尔的讲席。福克斯余生一直担任这一职位。在他生命的最后十年，他还承担了重要的编辑职责，担任奥古斯都·利奥波德·克雷勒的期刊Journal für die reine und angewandte Mathematik的主编。福克斯在其职业生涯中在该期刊上发表了许多论文：参见THIS LINK上的论文。

Ernest Julius Wilczynski在柏林听了福克斯的讲座。他在[17]中写道，他：-

……记得柏林大学那间又小又拥挤的讲堂，通风很差，夏天闷热难耐，但对认真而善于思考的学生来说却充满了意义和启发。福克斯并不是一位才华横溢的讲师。他讲话平静而不张扬，但他说的话内容充实。对学生来说，看到一位最高阶的数学头脑全力工作是一种启发。因为福克斯讲课时就是在工作。他很少准备充分，而是当场产生他想说的内容。偶尔他会在复杂的计算中迷失。然后他会透过眼镜环顾听众，带着最迷人、孩子般的微笑。他总是确信自己论证的要点，但数值例子给他带来了很大麻烦。他完全意识到这个缺点，我清楚地记得有一次，在冗长的讨论之后，他相当强调这样一个事实：“在这种情况下，二乘二等于四。”

福克斯研究微分方程和函数论。George Mathews写道[12]：-

福克斯的数学论文读起来非常愉快，没有那种往往属于微分方程论著的沉重倾向。他有能力清晰地突出真正重要的要点，而不做过分细致的细节，他并不不屑于通过将方法应用于具体而明确的问题来展示其力量。

在[1]中，Jerome Manheim写道：-

福克斯 是一位有天赋的分析学家，其著作在 奥古斯丁·路易·柯西、波恩哈德·黎曼、尼尔斯·阿贝尔 和 卡尔·弗里德里希·高斯 的基础研究与 儒勒·昂利·庞加莱、保罗·潘勒韦 和 埃米尔·皮卡 所发现的现代微分方程理论之间架起了一座桥梁。

1865年，福克斯 研究了以复函数为系数的 n 阶线性常微分方程。Bölling 在 [3] 中对此作了描述：-

福克斯 以基础性的结果丰富了线性微分方程理论。他讨论了如下类型的问题：必须对微分方程的系数施加什么条件，才能使所有解具有预先指定的性质（例如是正则的或代数的）。这使他在 1865、1866 年引入了一类重要的、在复域中具有解析系数的线性微分方程（及方程组），这一类今天以他的名字命名（Fuchs 方程、Fuchs 类方程）。……他成功刻画了那些其解在扩充复平面上没有本质奇点的微分方程。福克斯 后来也研究了非线性微分方程与可去奇点。

福克斯（1876 年与 夏尔·埃尔米特 合作）把椭圆积分作为参数函数的研究，标志着向模函数理论（菲利克斯·克莱因、理查德·戴德金）迈出的重要一步。在一系列论文（1880—81）中，福克斯 以推广 卡尔·古斯塔夫·雅各布·雅可比 反演问题的方式，研究了通过反演二阶线性微分方程解的积分而得到的函数。

正是 福克斯 关于这一反函数的工作，促使 儒勒·昂利·庞加莱 引入他所称的 Fuchs 群，并将其用作发展自守函数理论的一个基本概念。儒勒·昂利·庞加莱 与 福克斯 之间的第一次接触，是 儒勒·昂利·庞加莱 于 1880 年 5 月 29 日写的一封信：-

我怀着极大的兴趣读了您最近发表在 奥古斯都·利奥波德·克雷勒 的杂志上一期的题为“Über die Verallgemeinerung des kehrungsproblems”的杰出论文。Ⓣ（论反演问题的一种推广。）亲爱的先生，我希望您允许我就此向您请求某些澄清。……亲爱的先生，我必须承认，这些想法使我对您所发表的结果的普遍性产生了一些疑问，我冒昧就此与您联系，希望这不会给您带来麻烦，能澄清此事。

尽管我们给出的是儒勒·昂利·庞加莱信件的英文摘录（依据[2]），实际上他用法文写作，而福克斯用德文回复。两人似乎都乐于如此安排。皮埃尔·弗朗索瓦·韦吕勒在[2]中写道：-

福克斯尽管比儒勒·昂利·庞加莱年长21岁，却始终保持着友好和关切的语气，即使当他开始清楚地看到这位年轻的法国通信者正在发展一种与他自己的方法相当不同且更为完备的进路时也是如此。

1880年6月12日，儒勒·昂利·庞加莱写信给福克斯：-

我发现这种情况下只有两个奇点，您所引入的函数具有非常显著的特征，由于我打算发表我所得到的结果，我请求您允许将它们称为富克斯函数；因为是您发现了它们。

1881年3月20日，儒勒·昂利·庞加莱写了他给福克斯的最后一封信：-

我继续研究以您命名的那些函数，并希望不久后发表我的结果。这些函数作为特例包含椭圆函数以及模函数。利用这些函数以及我称之为zeta富克斯函数的其他函数，可以求解：(1) 所有只有三个奇点（两个有限、一个无穷）的具有有理系数的线性微分方程。(2) 所有具有有理系数的二阶方程。(3) 大量具有有理系数的各阶方程。

福克斯还研究了如何找到连接微分方程在两个不同点附近的两个解系的矩阵。对福克斯工作的综述出现在[8]中，Gray在其中还描述了这项工作如何影响了菲利克斯·克莱因、卡米耶·若尔当、儒勒·昂利·庞加莱等人。在这篇有趣的论文中，Gray还讨论了福克斯的思想与其数学工具之间的关系，并说明了某些问题的解如何引导福克斯去研究进一步的问题。Gray在引言中写道：-

[福克斯']的工作可以被有益地视为试图将新兴的复函数理论的概念秩序强加于混沌的微分方程世界之上。他既是严格现代线性方程理论的建筑师，又提出了许多被其同时代人接手的问题，并为不变量理论和变换群理论的学派提供了一个有趣的战场。

在讨论了以福克斯命名的概念之后，Gray写道[8]：-

但争论以Fuchsian事物来纪念福克斯是否公正将是徒劳的。相反，福克斯的职业生涯仍然令人感兴趣，因为它清晰而戏剧性地表明数学思想是多么多面，以及解决一个问题可能需要多少新思想。有一种讽刺的真理存在于这样的评价中：福克斯开辟了一个新领域：福克斯一再指出分析中那些最好由群论解决的问题。由于拒绝了这一属于年轻一代的工具，可以说福克斯只能像摩西一样站着，眺望应许之地。

在[3]中，Bölling如下描述了福克斯的性格：-

……福克斯是柏林古典时期和后古典时期的代表。他的个性被描述为优柔寡断、胆怯，但同时又幽默且充满善意。

Lazarus Fuchs's parents were the Jewish teacher Rafael Fuchs and his wife Caecilie Katz. We do not have dates for Lazarus's parents but we know that at the time he wrote his doctoral thesis his mother was deceased but his father was still alive. He was first taught by his father in Moschin but he had a strong desire for learning so, in 1846, he went to Posen (now Poznań in Poland) where he was privately taught classical languages for two years but mainly studied on his own. However, his parents were in financial difficulties and could not give him any support. He did get very minimal support from a distant wealthy relative but he lived in extreme poverty. He found a place to live which was cheap but the conditions were extremely bad. For months his only food was coffee and bread. He often begged his better-off friends to give him the last end of a candle so that he could study during the night and achieve his aim of entering the local Gymnasium. In 1848 he entered the third class at the Friedrich Wilhelm Gymnasium in Posen which, at this time, had Gustav Kiessling as its director. While he was studying at the Gymnasium, Albert Gustav Heydemann (1808-1877) was appointed as director in 1850.

It was at the Gymnasium that his remarkable abilities at mathematics became very clear to his teachers despite the fact that he had a very poor preparation in the topic before beginning his studies at the Gymnasium. Mathematics became the subject which, even at this early stage, Fuchs knew was going to dominate the rest of his life. In addition to his school studies, Fuchs had to earn money by giving private tutoring and his abilities for developing hidden skills and talents in his pupils were quickly appreciated so that he became a much sought-after teacher. One of his friends was Leo Königsberger who was four years younger that Fuchs and was also a pupil at the Friedrich Wilhelm Gymnasium in Posen. After eighteen months, Fuchs had completed the Middle School and, missing out a year in the Upper School, he was able to take his maturity examination at Easter 1853 which he passed with grade "excellent".

Despite qualifying to enter university in 1853, he could not begin his studies since he was completely destitute. It was at this stage that Leo Königsberger's parents came to his rescue. They offered to employ Fuchs as a private tutor to Leo for a year, to give him accommodation in their home, and to pay him a modest fee. He happily accepted and, after tutoring Leo Königsberger for a year, he entered the University of Berlin at Easter 1854. The rector of the university at this time was the famous astronomer Johann Franz Encke and Fuchs was enrolled into the Faculty of Philosophy whose dean was the philosopher Friedrich Adolf Trendelenburg (1802-1872). From Easter 1857 to Easter 1864 Fuchs and Königsberger shared accommodation in Berlin. They lived in a great number of different houses, forced to live a very simple and modest life-style since, especially in the first few years, they lived off the income that Fuchs made through giving private lessons.

At the University of Berlin, Fuchs attended lectures by a number of famous mathematicians including Eduard Kummer and Karl Weierstrass. He gives the full list of those who taught him during his four years at the University of Berlin in his doctoral thesis. These teachers are: mathematician Peter Friedrich Arndt (1817-1866), classical scholar August Böckh (1785-1867), Carl Borchardt, Rudolf Clausius, Lejeune Dirichlet, physicist and meteorologist Heinrich Wilhelm Dove (1803-1879), astronomer Johann Franz Encke, Eduard Kummer, zoologist Martin Heinrich Carl Lichtenstein (1780-1857), experimentalist in chemistry and physics Heinrich Gustav Magnus (1802-1870), chemist and mineralogist Eilhard Mitscherlich (1794-1863), mathematician Martin Ohm, physicist Johann Christian Poggendorff (1796-1877), historian and political scientist Friedrich Ludwig Georg von Raumer (1781-1873), chemist Franz Leopold Sonnenschein (1817-1879), philosopher Friedrich Adolf Trendelenburg (1802-1872), mineralogist Christian Samuel Weiss (1780-1856), and Karl Weierstrass. He writes:-

I am, and will always be, grateful to all these men who perfectly rendered outstanding services to me.

As well as attending lectures by the people listed above, Fuchs read Gauss's Disquisitiones arithmeticae and works by Fourier, Laplace and Cauchy. The professors who had the greatest influence on Fuchs were Karl Weierstrass, who introduced him to function theory, and Eduard Kummer who went on to supervise his doctorate. The examiners for his doctoral dissertation, De Superficierum lineis curvaturae Ⓣ, held on 2 August 1858 were Kummer and Martin Ohm (the brother of Georg Simon Ohm) and his opponents were Julius Weingarten, Leo Königsberger and Eduard Fischer. Fuchs was awarded the degree by the University of Berlin 1858.

Before the award of his doctorate, Fuchs began to worry about whether he should convert from Judaism to Christianity. He knew that if he remained a follower of the Jewish faith he would suffer discrimination and his career prospects would be very limited. Although Jews had been officially discriminated against during the early years of Fuchs' life, a new constitution in Prussia in 1850 gave all citizens equal rights irrespective of their religion. Despite the constitution, equality did not happen and discrimination continued in Prussia as did many restrictions and limitations on Jews. Fuchs spent three years deeply worried about whether he should convert to Christianity. Of course, it was not simply a personal matter since he worried greatly about how his Jewish family would feel if he converted. Both Kummer and Weierstrass were well aware of Fuchs' painful suffering on the issue and encouraged him to take the step. He was also given great support from Julius Müllensiefen (1811-1893), who was a pastor at the church of St Marien in Berlin. In 1860, Fuchs converted to Evangelical-Lutheran Christianity.

Before obtaining his doctorate, Fuchs was appointed to a teaching post at a Gymnasium in Berlin on 19 March 1859. In fact he taught in several different institutions before moving on to a mathematics teaching position at the Friedrich Werder Trade School. During this time he was undertaking research with the aim of becoming a university professor. It was during this period that he did outstanding work on homogeneous linear differential equations with variable coefficients which was published in the important 40-page paper Zur Theorie der linearen Differentialgleichungen mit veränderlichen Coefficienten Ⓣ which appeared in Crelle's Journal in 1866. This work is described in more detail below. He began his university teaching career when he was appointed as a Privatdozen at the University of Berlin in August 1865 after submitting his habilitation thesis Zur Theorie der linearen Differentialgleichungen mit veränderlichen Coeffizienten Ⓣ. He was promoted to extraordinary professor there on 2 December 1866 and taught at the University until winter semester 1868-69 when he accepted an appointment as a full professor at Greifswald. Fuchs also held a second post in Berlin from 23 May 1867 when he was appointed as professor of mathematics at the Artillery and Engineering School.

In 1868 Fuchs married Marie Anders (23 July 1849 - 10 May 1917). They had four sons and two daughters including Clara Fuchs (1869-1954) and Richard Fuchs who became a mathematician and has a biography in this Archive. Clara married the mathematician Ludwig Schlesinger who also has a biography in this Archive. We note that Richard Fuchs and Ludwig Schlesinger jointly edited Lazarus Fuchs' complete works: Gesammelte mathematische Werke von L Fuchs which was published by Mayer & Müller, Berlin in three volumes in 1904, 1906 and 1909.

Fuchs took up the professorship in Greifswald on 3 February 1869, the position becoming vacant since Leo Königsberger, who had taught at Greifswald for five years, had been appointed to a chair of mathematics at Heidelberg. After spending five years in Greifswald, Fuchs moved again, this time to Göttingen where he took up an appointment as an ordinary professor on 23 January 1874. Then in the following year he went to Heidelberg and taught there for nine years. In many ways these years were the most enjoyable period of his life. He loved the natural environment around the city, something that was important to Fuchs who had deep feelings for nature. Also in Heidelberg he had a particularly good relationship with his many outstanding doctoral students, and he got on extremely well with the other members of staff in many different faculties of the university. In fact during nine years at Heidelberg he supervised the doctoral studies of at least eight students who went on to become professors of mathematics. We will say a little below about the correspondence he carried out with Henri Poincaré during his years at Heidelberg. In the summer semester of 1884 he returned to Berlin to fill Kummer's chair when his old teacher retired. Fuchs held this post for the rest of his life. He also undertook important editorial duties in the final ten years of his life when he was the editor of Crelle's journal, the Journal für die reine und angewandte Mathematik. Fuchs published many papers in this journal over his career: see the papers at THIS LINK.

Ernest Julius Wilczynski attended Fuchs' lectures in Berlin. He writes in [17] that he:-

... remembers the small and crowded lecture-room in the University of Berlin, poorly ventilated, stuffy and hot in the summer days, but so full of meaning and inspiration to the earnest and thoughtful student. Fuchs was not a brilliant lecturer. He spoke in a quiet, undemonstrative manner, but what he said was full of substance. To the student there was the inspiration of seeing a mathematical mind of the highest order full at work. For Fuchs worked when he lectured. He was rarely well prepared, but produced on the spot what he wished to say. Occasionally he would get lost in a complicated computation. Then he would look around at the audience over his glasses with a most winning and child-like smile. He was always certain of the essential points of his argument, but numerical examples gave him a great deal of trouble. He was fully conscious of this failing, and I remember well one occasion when, after a lengthy discussion, he laid considerable emphasis upon the fact that "in this case, two times two is four."

Fuchs worked on differential equations and the theory of functions. George Mathews writes [12]:-

Fuchs's mathematical papers are very pleasant to read and free from that tendency to heaviness which is apt to belong to memoirs on differential equations. He had the faculty of bringing out clearly the really important points without over-elaborate detail, and he did not disdain to show the power of his methods by applying them to specific and definite problems.

In [1] Jerome Manheim writes:-

Fuchs was a gifted analyst whose works form a bridge between the fundamental researches of Cauchy, Riemann, Abel, and Gauss and the modern theory of differential equations discovered by Poincaré, Painlevé, and Émile Picard.

In 1865 Fuchs studied nth order linear ordinary differential equations with complex functions as coefficients. This is described by Bölling in [3]:-

Fuchs enriched the theory of linear differential equations with fundamental results. He discussed problems of the following kind: What conditions must be placed on the coefficients of a differential equation so that all solutions have prescribed properties (e.g. to be regular or algebraic). This led him (1865, 1866) to introduce an important class of linear differential equations (and systems) in the complex domain with analytic coefficients, a class which today bears his name (Fuchsian equations, equations of the Fuchsian class). ... He succeeded in characterising those differential equations the solutions of which have no essential singularity in the extended complex plane. Fuchs later also studied non-linear differential equations and moveable singularities.

Fuchs' study (1876 with Hermite) of elliptic integrals as a function of a parameter marks an important step towards the theory of modular functions (Klein, Dedekind). In a series of papers (1880-81) Fuchs studied functions obtained by inverting the integrals of solutions to a second-order linear differential equation in a manner generalising Jacobi's inversion problem.

It was Fuchs' work on this inverse function which led Poincaré to introduce what he called a Fuchsian group, and use this as a fundamental concept in the development of the theory of automorphic functions. The first contact between Poincaré and Fuchs was a letter written by Poincaré on 29 May 1880:-

I have read with great interest the remarkable treatise that you had recently included in the last issue of Crelle's Journal with the title "Über die Verallgemeinerung des kehrungsproblems." Ⓣ I hope you will grant me, dear sir, to request from you certain clarifications on the subject. ... I have to confess, dear sir, that these thoughts have raised with me some doubts about the generality of the results that you have published and I have taken the freedom to approach you about it, hoping that it will not trouble you to clear this up.

Although we have given the extract from Poincaré's letter in English (following [2]) in fact he wrote in French while Fuchs replied in German. Both seemed happy with this arrangement. Verhulst writes in [2]:-

Fuchs, despite being 21 years Poincaré's senior, consistently maintained a tone of friendship and interest, even when it began to become clear that his young French correspondent was developing an approach that was quite different from his own and more complete.

On 12 June 1880 Poincaré wrote to Fuchs:-

I find in the case that there are two singular points only, that the function you have introduced has very remarkable characteristics, and because I intend to publish the results I have obtained, I am asking your permission to call them Fuchsian functions; for it was you who discovered them.

On 20 March 1881, Poincaré wrote his last letter to Fuchs:-

I have continued with the functions that I named after you and I hope to publish my results shortly. These functions contain as a special case the elliptic functions and also the modular function. With these and other functions that I have called zeta Fuchsian, one can solve: (1) All linear differential equations with rational coefficients that have three singularities only, two finite and one infinite. (2) All second order equations with rational coefficients. (3) A large number of equations of various orders with rational coefficients.

Fuchs also investigated how to find the matrix connecting two systems of solutions of differential equations near two different points. A survey of Fuchs work appears in [8] where Gray also describes how this work influenced Klein, Jordan, Poincaré and others. In this interesting paper Gray also discusses the relationships between Fuchs' ideas and his mathematical tools, and illustrates how solutions of certain problems led Fuchs to the study of further problems. Gray writes in his introduction:-

[Fuchs'] work can profitably be seen as an attempt to impose upon the inchoate world of differential equations the conceptual order of the emerging theory of complex functions. As well as being the architect of the rigorous modern theory of linear equations, he raised many questions which were taken up by his contemporaries and provided an interesting battleground for the schools of invariant theory and transformation group theory.

After discussing the concepts named for Fuchs, Gray writes [8]:-

But it would be idle to debate the justice of remembering Fuchs by things Fuchsian. Rather, Fuchs' career remains of interest because it shows clearly and dramatically how mathematical ideas are many-sided, and how many new ideas may be needed to solve a problem. There is an ironic truth in the assessment that Fuchs opened up a new province: repeatedly Fuchs pointed to problems in analysis that could best be solved by group theory. Denying himself this tool, which belonged to the younger generation, one might say that Fuchs could only stand like Moses and gaze upon the promised land.

In [3] Bölling describes Fuchs' character as follows:-

... Fuchs is a representative of both Berlin's classical and its post-classical era. His personality has been described as indecisive, timid, but at the same time humorous and full of kindness.