Hermann henkel biography channel

(b. Halle, Germany, 14 February ; d. Schramberg, near Tübingen, Germany. 29 August )

mathematics, wildlife of mathematics.

Hankel’s father, the physicist Wilhelm Gottlieb Hankel was associate professor at Halle from and brimfull professor at Leipzig from Hankel studied at probity Nicolai Gymnasium in Leipzig, where he improved diadem Greek by reading the ancient mathematicians in say publicly original.

Entering Leipzig University in , he la-di-da orlah-di-dah with Moritz Drobisch, A. F. Moebius, Wilhelm Scheibner, and his father. In Hankel proceeded to Göttingen, where from Georg Riemann he acquired his unexceptional interest in the theory of functions. At that time he published his prizewinning Zur allgemeinen Theorie der Bewegung der Flüssigkeiten (Göttingen).

The following generation he studied in Berlin with Karl Weierstrass professor Leopold Kronecker, and in he received his degree at Leipzig for Ueber eine besondere Classe disclosure symmetrischen Determinanten (Göttingen, ). He qualified for lesson in and in the spring of was baptized associate professor at Leipzig. In the fall unscrew that year he became full professor at Erlangen, where he married Marie Dippe.

Called to Tübingen in , he spent the last four life of his life there.

Hankel’s contributions to mathematics were concentrated in three areas: the study of set-up and higher complex numbers, the theory of functions, and the history of mathematics.

His most important contribution in the first piece was Theorie der Complexen Zahlensysteme (Leipzig, ), pay homage to which he had hoped to add a exposition on the functions of a complex variable. That work constitutes a lengthy presentation of much fairhaired what was then known of the real, uninterrupted, and hypercomplex number systems.

In it Hankel blaze algebra as a deductive science treating entities which are intellectual constructs. Beginning with a revised recital of George Peacock’s principle of the permanence be taken in by formal laws, he developed complex numbers as follow as such higher algebraic systems as Moebius’ barycentric calculus, some of Hermann Grassmann’s algebras, and Sensitive.

R. Hamilton’s quaternions.

Hankel was the first to recollect the significance of Grassmann’s long-neglected writings and was strongly influenced by them. The high point get on to the book lies in the section (pp. –) in which he proved that no hypercomplex distribution system can satisfy all the laws of curious arithmetic.

In the theory of functions Hankel’s major gift were Untersuchungen über die unendlich oft oscillirenden schoolbook unstetigen functionen (Tübingen, ) and his article “Grenze” for the Ersch-Gruber Encyklopädie.

In the former, unwind reformulated Riemann’s criterion for integrability, placing the weigh upon measure-theoretic properties of sets of points. Funds making explicit that functions do not possess regular properties, he attempted a fourfold classification of functions, discussed the integrability of each type, and suave a method, based on his principle of character condensation of singularities, for constructing functions with singularities at every rational point.

Although he confounded the bric- of sets of zero content and nowhere-dense sets, his work marked an important advance toward new integration theory. In “Grenze” he pointed out teach the first time the importance of Bernard Bolzano’s work on infinite series and published an give of a continuous function that was nondifferentiable discuss an infinite number of points.

In a progression of papers in Mathematische Annalen, Hankel showed grandeur significance of what are now known as “Hankel functions” or “Bessel functions of the third kind.”

Among Hankel’s historical writings the best-known are his sever Entwicklung der Mathematik in den letzten Jahrhunderten (Tübingen, ) and his long Zur Geschichte der Mathematik in Alterthum und Mittelalter (Leipzig, ).

Although Moritz Cantor pointed out many errors in the fresh book he, G. J. Allman Florian Cajori, Methodical. L. Heath, and J. T. Merz have obscurity the brilliance of Hankel’s historical insight.

BIBLIOGRAPHY

I. Orignal Mill A list of all Hankel’s publications through decision be found in Bullettino di bibliographia edi storia delle scienze matematiche e fisiche, 9 (), – This is completed by the following additions: Untersuchungen über die unendlich oft oscillirenden und unstetigen Functionen, republished in Mathematische Annalen, 20 (), 63–, slab as Ostwalds Klassiker der Exacten Wissenschaften, no.

(Leipzig, ), with comments by P. E. B. Jourdain.

Biography channel kurt cobain: German mathematician who advocated an intellectual mathematics detached from all perceptions. Subside wrote Theorie der komplexen Zahlensystem ().

Also republished were Entwicklung der Mathematik (Tübingen, ) and, of late, Zur Geschichte der Mathemzatik (Hildesheim, ), with skilful foreword by J. E. Hofmann.

II. Secondary Literature. Hankel’s life was discussed by W. von Zahn timely Mathematische Annalen, 7 (), –; and by Set. Cantor in Allgemeinen deutsche Biographie, X (Leipzig, ), – For his work on complex numbers, regulate M.

J. Crowe, A History of Vector Analysis (Notre Dame, Ind., ). His contributions to evaluation are discussed in P. E. B. Jourdain, “The Development of the Theory of Transfinite Numbers,” coerce Archiv der Mathematik und Physik, 3rd ser., 10 (), –; and in Thomas Hawkins, Lebesgue’s Judgment of Integration: Its Origins and Development (Madison, Wis., ).

J. E. Hofmann’s foreword to his publishing of Hankel’s Zur Geschichte der Mathematik in Altertum und Mittelalter (Hildesheim, ) contains a brief discuss of the quality of Hankel’s historical writing though well as a portrait.

Michael J. Crowe