Singularities/Residues/ Applications 653Definition 9.5 Let f: D --+ W, and suppose that a E D, the point a
being either finite or oo. If f is not regular at a, then it is said that f is
singular at a. The point a is called a singular point of a singularity off .t
In the case a= oo we may say, alternatively, that f(z) is singular at oo
iff g(z) = f(I/z) is singular at z = 0.
Examples1. z = 2 is a singular point of f(z) = z/(z - 2).
- z = 0 is a singular point of f(z) = e^1 fz.
- z = oo is a singular point of f(z) = z^2 + 3z + 1.
Definition 9.6 A point a (finite or oo) is said to be an isolated singular
point of f if the function f is singular at a yet analytic in some deleted
neighborhood of a.
Again, in the case a = oo we may say, equivalently, that oo is an isolated
singular point of f(z) iff z = 0 is an isolated singular point of g(z) = f(I/ z).
Definition 9. 7 A point a (finite or oo) is said to be a nonisolated singular
point of f if the function f is singular at a and has singularities in every
deleted neighborhood of this point.
As before, in the case a = oo an alternative condition for oo to be a
nonisolated singular point of f is that z = 0 be a nonisolated singular
point of g(z) = f(I/z).
Note If f is defined at a but at no other point of a certain neighborhood
of a, then a is called an isolated point of the domain of definition off. Such
a point should not be confused with an isolated singularity (Definition 9.6).
If f is neither defined at a nor in some neighborhood of a, then the
point a is exterior to the domain of definition off. By the process called
analytic continuation it may or may not be possible to extend the domain
of definition off so as to include a. This matter is discussed in Selected
Topics, Chapter 1.
Examples
- z = 0 is an isolated singular point of f(z) = e^1 fz
- z = oo is a nonisolated singularity of f ( z) = tan z.
- z = 2 is an exterior point of f(z) = :E~=l zn!. The domain of definition
of this function is the open unit disk JzJ < 1 (see Selected Topics,
Section 1.15).tThe term "singularity" appears for the first time in a note by A. Cayley (On
the singularity of surfaces, Cambridge and Dublin Math. J. 7 (1852), 166-171).
The older term was "discontinuity."