CHAPTER VIII - SINGULARITIES OF SINGLE-VALUED ANALYTIC FUNCTIONS

Summary

Isolated singularities. A point at which a single-valued function has not a derivative, or in every neighborhood of which there are points at which the function has not a derivative, is called a singular point of the function. An especially interesting class of such points is composed of those possessing a neighborhood throughout which the function is analytic but which, of course, does not include the point itself. A point answering to this description is an isolated singular point.

It is useful here, for purposes of comparison, to consider the kinds of isolated discontinuities which single-valued real functions of a real variable x may possess. Such a function may, first of all, be not defined at a point x = a; or its value at x = a may be given so that, although f(x) has a finite limit A as x approaches a, we do not have A = f(a). Thus if f(x) = 0 except at x = 0, then f(x) will be discontinuous for x = 0 unless we add to our definition that f(x) = 0 at x = 0. If f(x) has a finite limit A as x approaches a, a discontinuity due to no definition or an inappropriate definition of f(a) is called a removable or artificial discontinuity. It remains to consider discontinuities due to the fact that f(x) has no finite limit when x approaches a.