VIII. A memoir on internal functions

1. Since the fundamental discoveries of Weierstrass, much progress has been made with regard to uniform transcendental functions; but the advances of modern mathematics appear to have included no attempt formally to classify and investigate the properties of natural groups of such functions. Consider, for instance, the case of transcendental integral functions which admit one possible essential singularity at infinity. They form the most simple class of uniform functions of a single variable, and yet of them we know, broadly speaking, the nature of but four types:- (1) The exponential function, with which are associated circular and (rectangular) hyperbolic functions; (2) The gamma functions; (3) The elliptic functions and functions derived therefrom, such as the theta functions and Appell’s generalisation of the Eulerian functions; (4) Certain functions which arise in physical problems (such as x-s J„ (x )) whose properties have been extensively investigated for physical purposes.