Hankel, Hermann (1839-1873)

German mathematician and mathematical historian who made significant contributions to the study of complex and hypercomplex numbers and the theory of functions. Much of his work was also in developing that of others. Hankel was born in Halle and studied at Leipzig, Göttingen, and Berlin. He lectured at Leipzig and Tübingen. Hankel's Theorie der complexen Zahlensysteme 1867 dealt with the real, complex, and hypercomplex number systems, and demonstrated that no hypercomplex number system can satisfy all the laws of ordinary arithmetic. In Untersuchungen über die unendlich oft oscillerenden und unstetigen Functionen, he presented a method for constructing functions with singularities at every rational point. He also explicitly stated that functions do not possess general properties; this work was an important advance towards modern integration theory. The Hankel functions provided a solution to the Bessel differential equation, which had originally occurred in connection with the theory of planetary motions. Today the equation is relevant in many fields. Hankel was also the first to suggest a method for assessing the magnitude, or 'measure', of absolutely discontinuous point sets (such as the set of only irrational numbers lying between 0 and 1). The 'measure' theory of point sets has now been extensively applied to probability, cybernetics, and electronics.