约斯塔·米塔格-莱弗勒Gösta Mittag-Leffler

约斯塔·米塔格-莱弗勒的父亲是Johan Olof 米塔格-莱弗勒，母亲是Gustava Wilhelmina Mittag。读者会注意到，米塔格-莱弗勒的姓氏结合了母亲的姓Mittag和父亲的姓米塔格-莱弗勒，因此可能会以为他的父母结婚时把两人的姓合在了一起；然而事实并非如此。米塔格-莱弗勒的母亲在结婚时改姓Gustava Wilhelmina 米塔格-莱弗勒，他们的第一个孩子米塔格-莱弗勒出生时，取名米塔格-莱弗勒 米塔格-莱弗勒。直到米塔格-莱弗勒 米塔格-莱弗勒二十岁做学生时，他才决定在自己的名字中加上“Mittag”。这清楚地表明了他对母亲以及母系祖父母的感情。

米塔格-莱弗勒的父亲是一名学校教师，后来成为斯德哥尔摩一所中学的校长。他还曾一度担任议会议员。正如我们提到的，米塔格-莱弗勒是家中长子，出生在他父亲任教学校的校舍里。过了一段时间，他的父母买下了自己的房子，在这个新家中，米塔格-莱弗勒的妹妹和两个弟弟出生了；他们都没有像米塔格-莱弗勒那样在自己的名字中加上Mittag。家族双方都源自德国，但已在瑞典生活了好几代。米塔格-莱弗勒的父系祖父是一名制帆匠，曾像他儿子后来那样一度担任议会议员。米塔格-莱弗勒的母系祖父是教会中的主任牧师，住在瑞典的乡村地区。米塔格-莱弗勒小时候每个暑假都和母系祖父母一起度过，他对母系家族怀有深厚的感情。

从米塔格-莱弗勒后来的生活可以看出，他在数学之外还有许多才能，而他的成长环境极大地促进了这些才能的发展。他的父母家中常常宾朋满座，为米塔格-莱弗勒及其兄弟姐妹提供了一个在学校教育之外还能学到许多东西的环境。米塔格-莱弗勒在求学过程中也展现出了多方面的才能，小学老师以及后来斯德哥尔摩文理中学的老师们都意识到他在数学方面具有非凡的能力，拉斯·戈丁写道[7]：-

从很小的时候起，米塔格-莱弗勒米塔格-莱弗勒就保持着写日记的习惯，但并不规律。他早期的日记显示出对文学的兴趣，后来，大约二十岁时，日记描绘出一个举止得体、受过良好教育、兴趣广泛的年轻人的形象。

米塔格-莱弗勒最初接受的是精算师训练，但后来转向了数学。他于1865年进入乌普萨拉大学学习。求学期间，他通过招收私人学生来维持生计。他提交并：-

...于1872年答辩了一篇关于辐角原理应用的并不出色的学位论文。

结果，米塔格-莱弗勒于1872年被任命为乌普萨拉大学的讲师（Docent）。也许对他一生影响最持久的事件，是他获得了一份薪水，这份薪水来自一笔捐赠，附带一个条件，即获得者必须在国外度过三年。1873年10月，米塔格-莱弗勒动身前往巴黎。

米塔格-莱弗勒在巴黎遇到了许多数学家，如让-克劳迪·波桂、夏尔·布里奥、米歇尔·沙勒、让·加斯东·达布和约瑟夫·刘维尔，此行的主要目的是向夏尔·埃尔米特学习。米塔格-莱弗勒听了夏尔·埃尔米特关于椭圆函数的讲座，但觉得很难懂。他在巴黎的逗留在他所记的日记中有详细描述。他的时间包括：-

……去看戏（莎拉·伯恩哈特）、讨论政治和宗教问题、上法语和英语课、参观工人区、对遇到的人进行思考等等。有时会为夏尔·埃尔米特的讲座而叹息，因为太难理解了。

当然，夏尔·埃尔米特热情洋溢地谈到了卡尔·魏尔斯特拉斯以及他和其他德国数学家所做的贡献，因此米塔格-莱弗勒决定在1875年春天去柏林。在那里，他听了卡尔·魏尔斯特拉斯的讲座，这些讲座对确定米塔格-莱弗勒后来的数学工作方向产生了极其重要的影响。晚年时，他这样描述自己在国外的三年：-

在访问这两个首都期间，我深刻感受到巴黎和柏林学术界之间的尖锐紧张关系。因此，我惊讶地发现，夏尔·埃尔米特和卡尔·魏尔斯特拉斯完全没有民族主义情绪或倾向。两人都是天生的天主教徒，而夏尔·埃尔米特，如同他之前的奥古斯丁·路易·柯西一样，是一位虔诚的信徒。卡尔·魏尔斯特拉斯有兴趣，或者更确切地说是觉得有趣，与博学的教士们谈论教会教义中最精微的要点。

米塔格-莱弗勒在柏林时得知，赫尔辛福斯（今称赫尔辛基）的数学教授爱德华·诺顿·劳仑次恩斯特·列奥纳德·林德勒夫，即恩斯特·恩斯特·列奥纳德·林德勒夫的父亲，即将离开讲席去担任行政职务。米塔格-莱弗勒写到了这些事件：-

1875年，我在柏林逗留期间，通过乌普萨拉间接得知赫尔辛福斯的数学教授职位空缺。正是L 林德勒夫，一位重要的数学家和大学校长，即将前往国家学校委员会……我决定申请他的职位。当我拜访我的老师、著名数学家卡尔·魏尔斯特拉斯，并告诉他我的计划时，他惊呼：不，请不要那样做！我已写信给文化部长，请他在柏林为你设立一个特别教授职位，我刚收到消息，我的申请已获批准！

从数学角度讲，我并非没有看到这样的职位相较于赫尔辛福斯职位的巨大优势。几乎从未有过如此杰出的杰出数学家聚集：卡尔·魏尔斯特拉斯、恩斯特·爱德华·库默尔、利奥波德·克罗内克、赫尔曼·冯·亥姆霍兹、古斯塔夫·基尔霍夫、卡尔·威廉·博尔夏特等。但对外国人来说，条件难以忍受。那是在德国战胜法国后不久，德国的傲慢达到了顶点。外国人受到傲慢的屈尊对待；伟大的皇帝、俾斯麦、毛奇等词语到处都能听到。人们理所当然地认为荷兰、瑞典等会成为德意志邦联的成员。对于非德国出生的人来说，在这种条件下生活是不可能的。至少我是这么想的。现在情况不同了，辉煌的胜利并未结出德国人想象的果实，他们采取了更现实的看法。然而，我寄往赫尔辛福斯的申请并未立即发出……

我们大量引用了米塔格-莱弗勒关于这次海外之行及其对民族竞争反应的文字。我们这样做是因为这对于确定米塔格-莱弗勒后来所扮演的国际角色很重要，因为他对数学国际合作的热情正是他在三年海外旅行中所见到的直接结果。

米塔格-莱弗勒于1876年被任命为赫尔辛基大学的讲席教授，五年后，他回到家乡斯德哥尔摩，在斯德哥尔摩大学担任讲席教授。他是斯德哥尔摩新大学（当时称为Stockholms Högskola）数学讲席的第一位持有者。上任后不久，他开始组织创办新的国际期刊Acta Mathematica。我们将在下面讨论他贡献的这一重要方面。1882年，他与出身富裕家庭的Signe af Lindfors结婚。他们是在米塔格-莱弗勒住在赫尔辛基时相识的，尽管Signe来自瑞典家庭，他们也住在芬兰。

米塔格-莱弗勒对数学分析做出了许多贡献，特别是在涉及极限的领域，包括微积分、解析几何和概率论。他研究函数的一般理论，研究自变量和因变量之间的关系。

他最著名的工作涉及单值函数的解析表示，这项工作最终导致了米塔格-莱弗勒定理。这项研究最初是为了推广卡尔·魏尔斯特拉斯讲座中的结果，他在讲座中描述了他关于存在一个具有指定零点和指定重数的整函数的定理。米塔格-莱弗勒在柏林学习期间试图将这个结果推广到亚纯函数。

他最终将关于将卡尔·魏尔斯特拉斯定理推广到亚纯函数的研究成果整理成一篇论文，并于1884年用法语发表在Acta Mathematica上。在这篇论文中，米塔格-莱弗勒基于格奥尔格·康托尔的新集合论[8]提出了一系列关于无限点集的一般拓扑概念：-

凭借这篇论文……米塔格-莱弗勒成为后来广为人知的一个定理的唯一所有者，并由此跻身国际知名数学家之列。

米塔格-莱弗勒是最早支持格奥尔格·康托尔集合论的数学家之一，但必须指出，其后果之一是利奥波德·克罗内克拒绝在Acta Mathematica上发表文章。

1900年至1905年间，米塔格-莱弗勒发表了一系列共五篇论文，他称之为关于发散级数求和的“札记”。这些札记的目的是构造幂级数在其收敛圆外的解析延拓。他能够做到这一点的区域现在被称为米塔格-莱弗勒星。

1882年，米塔格-莱弗勒创办了Acta Mathematica，并担任该期刊主编长达45年。创办这样一份期刊的最初想法来自Sophus 索菲斯·李在1881年的提议，但正是米塔格-莱弗勒对欧洲数学界的了解，加上他的政治手腕，才确保了它的成功。如前所述，他的妻子Signe极为富有，她的钱财帮助支持了该期刊的创办，但所需的大部分资金来自募捐和订阅。然而，像Acta Mathematica这样的期刊并非仅靠金钱就能成功。它需要一个国际基础，而米塔格-莱弗勒无疑完全明白这一点。他需要顶尖数学家向Acta Mathematica投稿，而格奥尔格·康托尔和儒勒·昂利·庞加莱为最初几卷贡献了许多论文。但它的成功还有另一个重要因素，正如戈弗雷·哈罗德·哈代所指出的[9]：-

……米塔格-莱弗勒对于提交给他发表的作品质量总是有很好的判断力。即使到了晚年，当大部分编辑工作都委托给他人时，他仍保留着那种奇特的感觉，使伟大的编辑能够感受到他几乎未曾一瞥的作品的价值……

1884年，即米塔格-莱弗勒发表其杰作的那一年，柯瓦列夫斯卡娅应他邀请来到斯德哥尔摩。1891年她去世时，米塔格-莱弗勒写道：-

她带着对她在柏林那位伟大导师——那位比其最喜爱的学生活得还久的可敬老人——的思想的信念和热情，从现代科学的中心来到我们这里。她的作品全都属于同一思想体系，通过新的发现展示了卡尔·魏尔斯特拉斯体系的力量。

到19世纪90年代初，米塔格-莱弗勒已在斯德哥尔摩郊区的Djursholm为家人建造了一座美妙的新居。此后，他的生活便分配在Djursholm的住宅和位于斯德哥尔摩以北约300公里的Tallberg乡间住所之间。在斯德哥尔摩花园郊区的家中，他拥有世界上最精美的数学图书馆。戈弗雷·哈罗德·哈代曾在米塔格-莱弗勒的图书馆中与他共度时光，并在[9]中描述道：-

所有书籍和期刊都在那里，……如果一个人累了，可以阅读世界上所有数学家的通信，或者从屋顶欣赏斯德哥尔摩的景色。

米塔格-莱弗勒的乡间宅邸戈弗雷·哈罗德·哈代写道：-

……在那里米塔格-莱弗勒展现出他最出色的一面，一位伟大的国际数学家与一位相当天真的乡绅的极为有趣的混合体。尽管他具有国际主义精神，但他是一个强烈的民族主义者，任何生活在如此美丽国家的人都可能如此；他热爱自己的房子、花园以及作为乡间地主的地位。

米塔格-莱弗勒及其妻子于1916年将他们在瑞典Djursholm的图书馆和遗产遗赠给瑞典科学院。第一次世界大战导致米塔格-莱弗勒Lefflers留下的资金贬值，这些资金本用于资助他们提议的数学基金会。然而，最终基于这座房子建立了米塔格-莱弗勒研究所，如今它已成为一个重要的数学研究中心。

戈弗雷·哈罗德·哈代在[9]中撰文，描述了米塔格-莱弗勒所受到的尊敬，尤其是在他自己的国家：-

我清楚地记得他最后一次向斯堪的纳维亚大会演讲的情景，那是1925年在哥本哈根，当他走进房间时，全场起立。对于一个来自不太讲究礼仪的国家的访客来说，这种接待起初相当令人惊讶；但这完全是一种普遍情感的自发表达，即过去五十年斯堪的纳维亚数学地位的重大提升，更多地归功于他，而非任何其他个人。

米塔格-莱弗勒获得了许多荣誉。他几乎是世界上每一个数学学会的荣誉会员，包括剑桥哲学学会、London Mathematical Society、皇家研究院、Royal Irish Academy和法兰西学院。他于1896年当选为伦敦皇家学会会士。他获得了牛津大学、剑桥大学、阿伯丁大学、圣安德鲁斯大学、博洛尼亚大学和克里斯蒂安尼亚大学（今奥斯陆）的荣誉学位。

他的贡献由戈弗雷·哈罗德·哈代 [9]很好地总结了：-

米塔格-莱弗勒在许多方面都是一个非凡的人。他是一位一流数学家，其对分析的贡献已成为经典，并在启发后来的研究方面发挥了重要作用；他是一个个性坚强的人，因对所选择的研究的强烈热爱而充满激情；而且他有毅力、地位和手段使他的热情产生影响。

Gösta Mittag-Leffler's father was Johan Olof Leffler while his mother was Gustava Wilhelmina Mittag. The reader will have noticed that Gösta Mittag-Leffler had a surname which combined both his mother's name Mittag and his father's name Leffler so might suppose that when his parents married they combined their names; however this was not so. Gösta's mother took the name Gustava Wilhelmina Leffler on her marriage and when their first child Gösta was born, he took the name Gösta Leffler. It was not until Gösta Leffler was a twenty year old student that he decided to add "Mittag" to his name. This was a clear indication of his feelings for his mother and for his grandparents on his mother's side.

Gösta's father was a school teacher who became a headmaster of a high school in Stockholm. He also served a spell as a member of parliament. As we mentioned, Gösta was the eldest of the family and he was born in the schoolhouse at the school where his father taught. After a while his parents bought a house of their own and in this new home Gösta's sister and two brothers were born; none of them would follow Gösta in adding Mittag to their names. Both sides of the family were of German origin but had lived for several generations in Sweden. Gösta's grandfather on his father's side was a sail maker who, like his son would do, served for a time as a member of parliament. Gösta's grandfather on his mother's side was a dean in the church, living in a country area of Sweden. As a young boy Gösta spent each summer holiday with his mother's parents and he had a great affection for his mother's family.

It was clear from his later life that Mittag-Leffler had many talents in addition to his mathematics and it was his upbringing that did much to foster these talents. His parents house was frequently filled with their friends and it provided an environment in which Gösta, together with his brothers and sister, learnt much in addition to their schooling. Gösta also showed his many talents as he progressed through school, and his teachers in elementary school and later those at the Gymnasium in Stockholm realised that he had an outstanding ability for mathematics Gårding writes [7]:-

From an early age Gösta Leffler kept an irregular diary. His early entries show an interest in literature and later, when he was around twenty, the diary gives the picture of a well-behaved and well-educated young man with general interests.

Mittag-Leffler trained as an actuary but later took up mathematics. He studied at Uppsala, entering in 1865. During his studies he supported himself by taking private pupils. He submitted and:-

... defended a less than remarkable thesis in 1872 on applications of the argument principle.

As a result Mittag-Leffler was appointed as a Docent at the University of Uppsala in 1872. Perhaps the event which would have the greatest lasting effect on his life was being awarded a salary which came through an endowment with a condition attached which said that the holder had to spend three years abroad. In October 1873 Mittag-Leffler set off for Paris.

Although Mittag-Leffler met many mathematicians in Paris, such as Bouquet, Briot, Chasles, Darboux, and Liouville, the main aim of the visit was to learn from Hermite. Mittag-Leffler attended Hermite's lectures on elliptic functions but found them hard going. His stay in Paris is described in detail in a diary he kept. His time consisted of:-

... a visit to the theatre (Sarah Bernhardt), discussions on political and religious matters, lessons in French and English, a visit to the workers' quarter, reflections on people met and so on. Sometimes there is a sigh over Hermite's lectures, so difficult to understand.

Certainly Hermite spoke in glowing terms about Weierstrass and the contributions he and other German mathematicians were making, so Mittag-Leffler made a decision to go to Berlin in the spring of 1875. There he attended Weierstrass's lectures which proved to be extremely influential in setting the direction of Mittag-Leffler's subsequent mathematical work. Much later in life he described his three years abroad:-

I got a vivid impression of the sharp tension between academic circles in Paris and Berlin during my visits to the two capitals. I was therefore struck by my experience that Hermite and Weierstrass were absolutely free of nationalistic feelings or leanings. Both were born Catholics and Hermite, as Cauchy before him, was a warm believer. Weierstrass was interested in, or rather amused by, talking to learned prelates about the finest points of the church dogmas.

While Mittag-Leffler was in Berlin he learnt that the professor of mathematics at Helsingfors (today called Helsinki), Lorenz Lindelöf, the father of Ernst Lindelöf, was leaving the chair to take up an administrative post. Mittag-Leffler wrote about these events:-

In 1875 during my stay in Berlin, I was informed in a roundabout way through Uppsala that the professorship in mathematics in Helsingfors was open. It was L Lindelöf, an important mathematician and rector of the university, who was going to the National School Board... and I decided to apply for his post. When I payed a visit to my teacher, the famous mathematician Weierstrass, and told him about my plans, he exclaimed: No, please, do not do that! I have written to the minister of culture and asked him to institute an extraordinary professorship for you here in Berlin and I just got the message that my application has been granted!

I was not blind to the great advantages, mathematically speaking, of such a position compared to the one in Helsingfors. Hardly ever was there such a brilliant collection of distinguished mathematicians: Weierstrass, Kummer, Kronecker, Helmholtz, Kirchhoff, Borchardt etc. But the conditions were not endurable for a foreigner. It was not so long after Germany's victorious war against France, and German arrogance was at a high point. Foreigners were treated with haughty condescension; der grosse Kaiser, Bismarck, Moltke etc. were words heard everywhere. It was taken as a matter of course that Holland, Sweden etc. would be members of a German Bund. For those who were not born Germans, it was impossible to live under such conditions. At least this is what I thought. Now things are different, the brilliant victory has not born the fruits that the Germans imagined and they have taken a more realistic view. However, my application to Helsingfors was not sent immediately ...

We have quoted extensively from Mittag-Leffler's writing about this overseas trip and his reaction to national rivalries. We have done so because it is important in determining the international role which Mittag-Leffler went on to play, for his passion for international cooperation in mathematics was a direct consequence of what he saw on his three year trip abroad.

Mittag-Leffler was appointed to a chair at the University of Helsinki in 1876 and, five years later, he returned to his home town of Stockholm to take up a chair at the University. He was the first holder of the mathematics chair in the new university of Stockholm (called Stockholms Högskola at this time). Soon after taking up the appointment, he began to organise the setting up of the new international journal Acta Mathematica. We will discuss this important aspect of his contribution below. In 1882 he married Signe af Lindfors who came from a wealthy family. They had met while Mittag-Leffler was living in Helsinki and, although Signe was from a Swedish family, they too were living in Finland.

Mittag-Leffler made numerous contributions to mathematical analysis particularly in areas concerned with limits and including calculus, analytic geometry and probability theory. He worked on the general theory of functions, studying relationships between independent and dependent variables.

His best known work concerned the analytic representation of a one-valued function, this work culminated in the Mittag-Leffler theorem. This study began as an attempt to generalise results in Weierstrass's lectures where he had described his theorem on the existence of an entire function with prescribed zeros each with a specified multiplicity. Mittag-Leffler tried to generalise this result to meromorphic function while he was studying in Berlin.

He eventually assembled his findings on generalising Weierstrass's theorem to meromorphic functions into a paper which he published (in French) in 1884 in Acta Mathematica . In this paper Mittag-Leffler proposed a series of general topological notions on infinite point sets based on Cantor's new set theory [8]:-

With this paper ... Mittag-Leffler became the sole proprietor of a theorem that later became widely known and with this he took his place in the circle of internationally known mathematicians.

Mittag-Leffler was one of the first mathematicians to support Cantor's theory of sets but, one has to remark, a consequence of this was that Kronecker refused to publish in Acta Mathematica .

Between 1900 and 1905 Mittag-Leffler published a series of five papers which he called "Notes" on the summation of divergent series. The aim of these notes was to construct the analytical continuation of a power series outside its circle of convergence. The region in which he was able to do this is now called Mittag-Leffler's star.

In 1882 Mittag-Leffler founded Acta Mathematica and served as chief editor of the journal for 45 years. The original idea for such a journal came from Sophus Lie in 1881, but it was Mittag-Leffler's understanding of the European mathematical scene, together with his political skills, which ensured its success. His wife Signe was, as we have mentioned, extremely wealthy and her money helped support the setting up of the Journal but most of the money needed came from an appeal and from subscriptions. However, periodicals like Acta Mathematica do not succeed just because of money. It required an international base and certainly Mittag-Leffler fully understood this. He needed leading mathematicians to send papers to Acta Mathematica and Cantor and Poincaré contributed many papers to the first few volumes. But there was another important factor in its success as Hardy points our [9]:-

.. Mittag-Leffler was always a good judge of the quality of the work submitted to him for publication. Even in his later years, when most of the editorial work was delegated to others, he retained that curious sense which enables the great editor to feel the value of the work at which he has hardly glanced...

In 1884, the year Mittag-Leffler published his masterpiece, Kovalevskaya arrived in Stockholm at his invitation. On her death in 1891 Mittag-Leffler wrote:-

She came to us from the centre of modern science full of faith and enthusiasm for the ideas of her great master in Berlin, the venerable old man who has outlived his favourite pupil. Her works, all of which belong to the same order of ideas, have shown by new discoveries the power of Weierstrass's system.

By the early 1890s Mittag-Leffler had built for his family a wonderful new home in Djursholm in the suburbs of Stockholm. He then divided his life between time spent at his home in Djursholm and at his country home at Tallberg, around 300 km north of Stockholm. In his home in the garden suburbs of Stockholm he had the finest mathematical library in the world. Hardy spent time with Mittag-Leffler in his library and describes it in [9]:-

All books and periodical were there, ... and if one got tired one could read the correspondence of all the mathematicians in the world, or enjoy the view of Stockholm from the roof.

Of Mittag-Leffler's country home Hardy writes:-

... there Mittag-Leffler appeared at his best, a most entertaining mixture of the great international mathematician and the rather naive country squire. He was a strong nationalist, in spite of his internationalism, as anyone who lived in so beautiful a country might be; and he loved his house and his garden and his position as the landowner of the countryside.

Mittag-Leffler and his wife bequeathed their library and estate at Djursholm in Sweden to the Swedish Academy of Sciences in 1916. World War I caused a loss in the value of money left by the Mittag-Lefflers to fund the mathematical foundation which they proposed. However, eventually the Mittag-Leffler Institute was set up based on the house and today is a major mathematical research centre.

Hardy, writing in [9], describes the regard that Mittag-Leffler was held in, particularly in his own country:-

I can remember well the occasion when he lectured for the last time to a Scandinavian Congress, at Copenhagen in 1925, and the whole audience rose and stood as he entered the room. It was a reception rather astonishing at first to a visitor from a less ceremonious country; but it was an entirely spontaneous expression of the universal feeling that to him, more than to any other single man, the great advance in the status of Scandinavian mathematics during the last fifty years was due.

Mittag-Leffler received many honours. He was an honorary member of almost every mathematical society in the world including the Cambridge Philosophical Society, the London Mathematical Society, the Royal Institution, the Royal Irish Academy, and the Institute of France. He was elected a Fellow of the Royal Society of London in 1896. He was awarded honorary degrees from the universities of Oxford, Cambridge, Aberdeen, St Andrews, Bologna and Christiania (now Oslo).

His contribution is nicely summed up by Hardy [9]:-

Mittag-Leffler was a remarkable man in many ways. He was a mathematician of the front rank, whose contributions to analysis had become classical, and had played a great part in the inspiration of later research; he was a man of strong personality, fired by an intense devotion to his chosen study; and he had the persistence, the position, and the means to make his enthusiasm count.