The delta-distribution ^[*] or delta function, as it was originally called, is a singular function beyond the realm of classical analysis. As we observed in the text, in physical contexts the delta distribution arises in the consideration of point charges, and similarly in that of mass points. The density of a unit charge or unit mass at (say) x = 0 which is written ?( x) is everywhere zero except at x = 0, where it is so large that the total charge, i.e. its integral over all space, becomes 1, i.e.

No function of classical analysis has such properties since for any function which is everywhere zero except at one point, the integral must vanish (irrespective of the concept of the integral). As a further example, which leads to a singular function, we consider the case of two charges of opposite signs with intensities 1/ ? and located at the points x = 0, x = ?. The density distributions of the charges are ?( x)/ ?, ??( x ? ?)/ ?. In the limit ? ? 0 the charges approach each other, with the product of intensity 1/ ? and mutual separation ?, i.e. the dipole moment, remaining constant. One thus obtains a dipole with density

This limiting value is undefined and does not exist in the context of classical analysis. However, in the theory of distributions developed by L.