# Hermann Hankel

**Hermann Hankel** (14 February 1839 – 29 August 1873) was a German mathematician who was born in Halle, Germany and died in Schramberg (Black Forest), Imperial Germany.

Hermann Hankel | |
---|---|

Born | |

Died | 29 August 1873 34) | (aged

Nationality | German |

Scientific career | |

Fields | Mathematical analysis Special functions |

He studied and worked with, among others, Möbius, Riemann, Weierstrass and Kronecker.

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."[1]
Furthermore, Hankel draws attention[2] 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.

## 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.

## See also

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

## Notes

- See (Fischbein 1987, p. 99).
- See (Hankel 1867, p. 16).

## References

- 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.

## External links

- 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