Hermann Hankel

Hankel was born on 14 February 1839 in Halle, Germany. His father, Wilhelm Gottlieb Hankel, was a physicist. Hankel studied at Nicolai Gymnasium in Leipzig before entering Leipzig University in 1857, where he studied with Moritz Drobisch, August Ferdinand Möbius and his father. In 1860, he started studying at University of Göttingen, where he acquired an interest in function theory under the tutelage of Bernhard Riemann. Following the publication of an award winning article, he proceeded to study under Karl Weierstrass and Leopold Kronecker in Berlin. He received his doctorate in 1862 at Leipzig University. Receiving his teaching qualifications a year after, he was promoted to an associate professor at Leipzig University in 1867. At the same year, he received his full professorship in University of Erlangen–Nuremberg and spent his last four years in University of Tübingen. He died on 29 August 1873 in Schramberg, near Tübingen. He was married to Marie Hankel.[1]

In 1867, he published Theorie der Complexen Zahlensysteme, a treatise on complex analysis. His works on the theory of functions include 1870's Untersuchungen über die unendlich oft oscillirenden und unstetigen functionen and his 1871 article “Grenze” for the Ersch-Gruber Encyklopädie. His work for Mathematische Annalen has highlighted the importance of Bessel functions of the third kind, which were later known as Hankel functions.[1]

His 1867 exposition on complex numbers and quaternions is particularly memorable. For example, Fischbein notes that he solved the problem of products of negative numbers by proving the following theorem: "The only multiplication in R which may be considered as an extension of the usual multiplication in R⁺ by respecting the law of distributivity to the left and the right is that which conforms to the rule of signs."[2] Furthermore, Hankel draws attention[3] to the linear algebra that Hermann Grassmann had developed in his Extension Theory in two publications. This was the first of many references later made to Grassmann's early insights on the nature of space.

• Hermann Hankel (1863) Die Euler'schen Integrale bei unbeschränkter Variabilität des Argumentes, Voss, Leipzig.
• Hermann Hankel (1867) Vorlesungen über die complexen Zahlen und ihre Functionen, Voss, Leipzig.
• Hermann Hankel (1869) Die Entwickelung der Mathematik in den letzten Jahrhunderten, Fues, Tübingen.
• Hermann Hankel (1870) Untersuchungen über die unendlich oft oscillirenden und unstetigen Functionen, Fues, Tübingen.
• Hermann Hankel (1874) Zur Geschichte der Mathematik in Alterthum und Mittelalter, Teubner, Leipzig.
• Hermann Hankel (1875) Die Elemente der projectivischen Geometrie in synthetischer Behandlung, Teubner, Leipzig.

• Hankel matrix/Hankel operator
• Hankel functions in the theory of Bessel functions
• Hankel contour
• Hankel transform

• Fischbein, Efraim (1987), Intuition in Science and Mathematics: An Educational Approach, Mathematics Education Library, Dordercht: Kluwer Academic Publishers, pp. xiv+225, ISBN 90-277-2506-3, MR 0921434.
• Letta, Giorgio (1994) [112°], "Le condizioni di Riemann per l'integrabilità e il loro influsso sulla nascita del concetto di misura" (PDF), Rendiconti della Accademia Nazionale delle Scienze Detta dei XL, Memorie di Matematica e Applicazioni (in Italian), XVIII (1): 143–169, MR 1327463, Zbl 0852.28001, archived from the original (PDF) on 2014-02-28. "Riemann's conditions for integrability and their influence on the birth of the concept of measure" (English translation of title) is an article on the history of measure theory, analyzing deeply and comprehensively every early contribution to the field, starting from Riemann's work and going to the works of Hermann Hankel, Gaston Darboux, Giulio Ascoli, Henry John Stephen Smith, Ulisse Dini, Vito Volterra, Paul David Gustav du Bois-Reymond and Carl Gustav Axel Harnack.

• Media related to Hermann Hankel at Wikimedia Commons
• O'Connor, John J.; Robertson, Edmund F., "Hermann Hankel", MacTutor History of Mathematics archive, University of St Andrews.
• Hermann Hankel at the Mathematics Genealogy Project