Hermann Hankel (14 February 1839 – 29 August 1873) was a German

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

. Having worked on

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...

during his career, he is best known for introducing the Hankel transform and the Hankel matrix.

Biography

Hankel was born on 14 February 1839 in

Halle Halle may refer to: Places Germany * Halle (Saale), also called Halle an der Saale, a city in Saxony-Anhalt ** Halle (region), a former administrative region in Saxony-Anhalt ** Bezirk Halle, a former administrative division of East Germany ** Hal ...

Germany Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated betwee ...

. His father, Wilhelm Gottlieb Hankel, was a physicist. Hankel studied at Nicolai Gymnasium in

Leipzig Leipzig ( , ; Upper Saxon: ) is the most populous city in the German state of Saxony. Leipzig's population of 605,407 inhabitants (1.1 million in the larger urban zone) as of 2021 places the city as Germany's eighth most populous, as ...

before entering

Leipzig University Leipzig University (german: Universität Leipzig), in Leipzig in Saxony, Germany, is one of the world's oldest universities and the second-oldest university (by consecutive years of existence) in Germany. The university was founded on 2 December ...

in 1857, where he studied with Moritz Drobisch, August Ferdinand Möbius and his father. In 1860, he started studying at

University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...

, where he acquired an interest in function theory under the tutelage of

Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...

. Following the publication of an award winning article, he proceeded to study under

Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...

and

Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers, ...

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 University of Erlangen–Nuremberg (german: Friedrich-Alexander-Universität Erlangen-Nürnberg, FAU) is a public research university in the cities of Erlangen and Nuremberg in Bavaria, Germany. The name Friedrich–Alexander comes from the univ ...

and spent his last four years in

University of Tübingen The University of Tübingen, officially the Eberhard Karl University of Tübingen (german: Eberhard Karls Universität Tübingen; la, Universitas Eberhardina Carolina), is a public research university located in the city of Tübingen, Baden-W� ...

. He died on 29 August 1873 in

Schramberg Schramberg is a town in the district of Rottweil, in Baden-Württemberg, Germany. It is situated in the eastern Black Forest, 25 km northwest of Rottweil. With all of its districts (Talstadt, Sulgen, Waldmössingen, Heiligenbronn, Schönbron ...

, near Tübingen. He was married to Marie Hankel. In 1867, he published ''Theorie der Complexen Zahlensysteme'', a treatise on

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

. 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 ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...

'' has highlighted the importance of Bessel functions of the third kind, which were later known as Hankel functions. His 1867 exposition on

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

s and

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...

s is particularly memorable. For example, Fischbein notes that he solved the problem of products of

negative number In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed ma ...

s 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." Furthermore, Hankel draws attentionSee . to the

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...

that

Hermann Grassmann Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mat ...

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 Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...

Selected publications

* 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.

Notes

References

*. *. "''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 Giulio Ascoli (20 January 1843, Trieste – 12 July 1896, Milan) was a Jewish-Italian mathematician. He was a student of the Scuola Normale di Pisa, where he graduated in 1868. In 1872 he became Professor of Algebra and Calculus of the Pol ...

Henry John Stephen Smith Prof Henry John Stephen Smith FRS FRSE FRAS LLD (2 November 1826 – 9 February 1883) was an Irish mathematician and amateur astronomer remembered for his work in elementary divisors, quadratic forms, and Smith–Minkowski–Siegel mass form ...

Ulisse Dini Ulisse Dini (14 November 1845 – 28 October 1918) was an Italian mathematician and politician, born in Pisa. He is known for his contribution to real analysis, partly collected in his book "''Fondamenti per la teorica delle funzioni di variabil ...

, Vito Volterra, Paul David Gustav du Bois-Reymond and Carl Gustav Axel Harnack.

External links

* * * 1839 births 1873 deaths 19th-century German mathematicians German historians of mathematics Mathematical analysts Complex analysts People from Halle (Saale) People from the Province of Saxony University of Göttingen alumni Humboldt University of Berlin alumni Leipzig University alumni Leipzig University faculty University of Erlangen-Nuremberg faculty University of Tübingen faculty {{Germany-mathematician-stub