ABSTRACT
The exposition of the theory of generalized functions given in this chapter is tailored to the needs of theoretical and mathematical physics.
1.1. Introduction. A generalized function is a generalization of the classical notion of a function. On the one hand, this generalization permits expressing in a mathematically proper form such idealized concepts as the density of a material point, the density of a point charge or dipole, the spatial density of a simple or double layer, the intensity of an instantaneous point source, the magnitude of an instantaneous force applied to a point, and so forth. On the other hand, the notion of a generalized function can reflect the fact that in reality one cannot measure the value of a physical quantity at a point but can only measure the mean values within sufficiently small neighbourhoods of the point and then proclaim the limit of the sequence of those mean values as the value of the physical quantity at the given point.
