赫尔曼·汉克尔Hermann Hankel

赫尔曼·汉克尔的父亲是Wilhelm Gottlieb 汉克尔，雅各布·赫尔曼出生时他是哈勒的一位物理学家。赫尔曼在哈勒开始接受教育，但1849年Wilhelm被任命为莱比锡大学物理学讲席，于是全家搬到莱比锡，赫尔曼在那里就读于Nicolai 文理中学（Gymnasium）。在文理中学时，他[1]：-

……通过阅读古代数学家的原著提高了他的希腊语水平。

1857年，汉克尔进入莱比锡大学，师从奥古斯特·费迪南德·莫比乌斯学习数学，并跟随自己的父亲学习物理学。按照当时德国的传统，汉克尔并未在一所大学完成学业，而是在求学过程中转往多所不同的大学。1860年，他从莱比锡前往哥廷根，成为波恩哈德·黎曼的学生，次年又在柏林与卡尔·魏尔斯特拉斯和利奥波德·克罗内克共事。1862年，他凭借学位论文Über eine besondere Classe der symmetrischen Determinanten Ⓣ（《论一类特殊的对称行列式》）获得博士学位。

汉克尔的教授资格论文（Habilitation）于1863年被接受，他开始在莱比锡任教，并于1867年被任命为编外教授。编外教授的任命是在春季，但同年秋季，汉克尔已在埃尔朗根就任正教授。他在埃尔朗根与Marie Dippe结婚，但很快又再次搬迁，于1869年接受了蒂宾根的讲席。

他研究复数理论、函数论和数学史。然而，他在复分析方面的工作并不被认为属于第一流，在[8]中，他被归入那些有所贡献但其——

……对复分析基础的影响不如[波恩哈德·黎曼, 卡尔·魏尔斯特拉斯, 阿道夫·赫维兹, 路德维希·比贝尔巴赫 ...]中更详细讨论的那些数学家那样根本。

汉克尔在其Prinzip der Permanenz der formalen Gesetze Ⓣ（《形式定律的持久性原理》）（1867）中对算术规则作了系统研究，见[7]。他还写了另一部重要著作，也于1867年出版，即Theorie der complexen Zahlensysteme Ⓣ（《复数系理论》），这部著作对使赫尔曼·格拉斯曼的思想更为人所知起了很大作用。这部著作[1]：——

……对当时已知的实数系、复数系和超复数系作了长篇阐述。他从修订陈述乔治·皮科克的形式定律持久性原理开始，发展了复数以及诸如奥古斯特·费迪南德·莫比乌斯的重心演算、赫尔曼·格拉斯曼的某些代数、威廉·哈密顿的四元数等更高阶的代数系统。汉克尔是第一个认识到赫尔曼·格拉斯曼长期被忽视的著作之意义的人……

汉克尔研究了波恩哈德·黎曼的积分理论，并用测度论的概念重新表述了它。这项工作以及他在这一领域的其他工作，构成了通向当前积分理论的进展。他因汉克尔变换而被人们记住，该变换出现在研究仅依赖于到原点距离的函数时。他还在一系列发表于Mathematische Annalen的论文中研究了现在称为汉克尔函数或第三类Bessel functions的函数。

他的历史著作相当难以评价，因为其中包含许多错误，却又充满卓越的洞见。正如他看到了赫尔曼·格拉斯曼工作的重要性一样，汉克尔也必定因看到伯纳德·波尔查诺关于无穷级数工作的重要性而享有相当大的功劳。

Hermann Hankel's father was Wilhelm Gottlieb Hankel who was a physicist at Halle at the time Hermann was born. Hermann began his education in Halle but, in 1849 Wilhelm was appointed to the chair of physics at Leipzig so the family moved to Leipzig where Hermann attended the Nicolai Gymnasium. At the gymnasium he [1]:-

... improved his Greek by reading the ancient mathematicians in the original.

In 1857 Hankel entered the University of Leipzig where he studied mathematics with Möbius and physics with his own father. Following the tradition in Germany at that time Hankel did not complete his studies at one university, but moved to several different universities during the course of his studies. From Leipzig he went to Göttingen in 1860 where he became a student of Riemann and then, in the following year, he worked with Weierstrass and Kronecker in Berlin. He received his doctorate for a thesis Über eine besondere Classe der symmetrischen Determinanten Ⓣ in 1862.

Hankel's habilitation was accepted in 1863 and he began teaching at Leipzig where he was appointed extraordinary professor in 1867. The appointment as extraordinary professor had been in the spring but by the autumn of the same year Hankel was at Erlangen to take up an appointment as ordinary professor. He married Marie Dippe in Erlangen but again he would move fairly soon, accepting the chair at Tübingen in 1869.

He worked on the theory of complex numbers, the theory of functions and the history of mathematics. His work on complex analysis, however, is not considered of the first rank and in [8] he is included with those who contributed but whose:-

... influence on the foundations of complex analysis was not as essential as that of those mathematicians discussed in more detail [Riemann, Weierstrass, Hurwitz, Bieberbach ...]

Hankel made a systematic study of the rules of arithmetic with his Prinzip der Permanenz der formalen Gesetze Ⓣ (1867), see [7]. He wrote another important work which was also published in 1867 Theorie der complexen Zahlensysteme Ⓣ which did much to make Grassmann's ideas better known. This work [1]:-

... constitutes a lengthy presentation of much of what was then known of the real, complex, and hypercomplex number systems. Beginning with a revised statement of George Peacock's principle of permanence of formal laws, he developed complex numbers as well as such higher algebraic systems as Möbius's barycentric calculus, some of Hermann Grassmann's algebras, and W R Hamilton's quaternions. Hankel was the first to recognise the significance of Grassmann's long-neglected writings ...

Hankel looked at Riemann's integration theory and restated it in terms of measure theoretic concepts. This, and other work he did in this area, constitutes progress towards our current integration theories. He is remembered for the Hankel transformation which occurs in the study of functions which depend only on the distance from the origin. He also studied functions, now named Hankel functions or Bessel functions of the third kind, in a series of papers which appeared in Mathematische Annalen.

His historical writings are rather hard to evaluate since they contain many errors, yet they are filled with brilliant insight. In the same way that he saw the importance of Grassmann's work, Hankel also must have considerable credit for seeing the importance of Bolzano's work on infinite series.