ABSTRACT
In different fields of physics and engineering, we sometimes deal with sham functions1 that do not satisfy the axiom of functions in a rigorous sense. A typical example of a sham function is Dirac’s delta function2 δ (r). It is made up of, for instance, describing an electric point charge.3 In physics and engineering, the spatial
distribution of a point charge ρ(r) is usually expressed by
ρ(r) = ρ0δ (r), (8.1)
and δ (r) is defined by
δ (r) = 0 at r , 0, $
δ (r)dr = 1. (8.2)
However, the expression of δ (r) is not consistent with our familiar concept of functions (see section 8.2.1). In fact, the discrete and divergence properties inherent in δ (r) make it impossible to apply differentiation, integration, or Fourier transformation to δ (r). Hence, the sham function δ (r) should be of no use in describing the physical phenomena triggered by the presence or temporal movement of the point charge.4 If this is the case, then why is δ (r) repeatedly used in physics and engineering textbooks?
