Editing Hermann Hankel (section)

==Biography==
Hankel was born on 14 February 1839 in [[Halle, Saxony-Anhalt|Halle]], [[German Confederation|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 Wilhelm Drobisch|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 from an illness on 29 August 1873 in [[Schramberg]], near Tübingen. He was married to [[Marie Hankel]].<ref name="encyc">{{cite web |url=https://mathshistory.st-andrews.ac.uk/DSB/Hankel.pdf |title=Hankel, Hermann |last=Crowe |first=Michael J. |website=[[Encyclopedia.com]] }}</ref>

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 [[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.<ref name="encyc"/>

His 1867 exposition on [[complex number]]s and [[quaternion]]s is particularly memorable. For example, Fischbein notes that he solved the problem of products of [[negative number]]s by proving the following theorem: "The only multiplication in R which may be considered as an extension of the usual multiplication in R<sup>+</sup> by ''respecting the law of distributivity'' to the left and the right is that which conforms to the rule of signs."<ref>See {{harv|Fischbein|1987|p=99}}.</ref>
Furthermore, Hankel draws attention<ref>See {{harvtxt|Hankel|1867|p=16}}.</ref> 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]].