The ordinary differential equations and automatic differentiation unified

Abstract

Analytical continuation of a function represented with an arbitrary Taylor expansion at only one point is not constructive (in the sense further explained). What makes it constructive is an equation (algebraic or differential of certain class) defining this function and enabling its continuation by means of integration of the ordinary differential equations (ODEs). Such functions comprise an important sub-class of holomorphic functions – generalized elementary functions widening the class of conventional elementary functions so that it becomes closed. In terms of the generalizing definition, the solutions of elementary ODEs are elementary too. In a frame of the unifying view based on the generalized elementary functions, automatic differentiation merges with the theory of holomorphic ODE's. It is showed, that the continuation of (generalized) elementary functions via integration of its ODEs does not necessarily expand them into each and every point where these functions exist and are holomorphic. Some entire functions are suspects for being elementary everywhere except an isolated point: the point of their ‘removable’ or ‘regular’ singularity. Thus the unifying view uncovers a new meaning of the notion ‘removable singularity’ as a new type of special point (which is rather ‘unmovable’, being proper to a particular holomorphic function).