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 Ersch-Gruber Encyklopädie. His work for Mathematische Annalen has highlighted the importance of Bessel functions of the third kind, which were later known as Hankel functions.[1]
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."[2]
Furthermore, Hankel draws attention[3] 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.
