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

^{+} 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 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 (mathematics), space.

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.

German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
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**Ger ...

mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

. Having worked on mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...

during his career, he is best known for introducing the Hankel transform
In mathematics, the Hankel transform expresses any given function ''f''(''r'') as the weighted sum of an infinite number of Bessel functions, Bessel functions of the first kind . The Bessel functions in the sum are all of the same order ν, but di ...

and the Hankel matrix In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a square matrix in which each ascending skew-diagonal from left to right is constant, e.g.:
\qquad\begin
a & b & c & d & e \\
b & c & d & e & f \\
c & d & ...

.
Biography

Hankel was born on 14 February 1839 inHalle 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
** Hall ...

, Germany
)
, image_map =
, map_caption =
, map_width = 250px
, capital = Berlin
Berlin (; ) is the Capital city, capital and List of cities in Germany by population, largest city of Germany by both area and population. Its 3,769,495 inh ...

. 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 Germany, German States of Germany, state of Saxony. With a population of 605,407 inhabitants as of 2021 (1.1 million residents in the larger urban zone), it surpasses the Saxon c ...

before entering Leipzig University
Leipzig University (german: Universität Leipzig), in Leipzig
Leipzig (, also , ; Upper Saxon: ) is the most populous city in the Germany, German States of Germany, state of Saxony. With a population of 605,407 inhabitants as of 2021 (1.1 ...

in 1857, where he studied with , August Ferdinand Möbius
August Ferdinand Möbius (, ; ; 17 November 1790 – 26 September 1868) was a German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or nat ...

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 i ...

, 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
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics ...

. 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
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) incl ...

and Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German
German(s) may refer to:
Common uses
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* Germans, Germanic ethnic group, citizens of Germany or people of German ancestry
* For citizens of German ...

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

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
In public relations
Public relations (PR) is the prac ...

. He died on 29 August 1873 in Schramberg
Schramberg is a town in the Rottweil (district), 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, Heili ...

, 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
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Der ...

. 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 Allgemeine Encyclopädie der Wissenschaften und Künste, Ersch-Gruber Encyklopädie. His work for ''Mathematische Annalen'' has highlighted the importance of Bessel function#Hankel functions: H(1)α, H(2)α, Bessel functions of the third kind, which were later known as Hankel functions.
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 RSelected 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.

See also

*Hankel matrix In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a square matrix in which each ascending skew-diagonal from left to right is constant, e.g.:
\qquad\begin
a & b & c & d & e \\
b & c & d & e & f \\
c & d & ...

/Hankel operator
* Hankel functions in the theory of Bessel functions
* Hankel contour
* Hankel singular value
* Hankel transform
In mathematics, the Hankel transform expresses any given function ''f''(''r'') as the weighted sum of an infinite number of Bessel functions, Bessel functions of the first kind . The Bessel functions in the sum are all of the same order ν, but di ...

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, Henry John Stephen Smith, Ulisse Dini, Vito Volterra, Paul David Gustav du Bois-Reymond and Carl Gustav Axel Harnack.External links

* * * 1839 births 1873 deaths 19th-century German mathematicians 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