# Isolated Singularities

**Isolated Singularities:**

It is often important to consider a function that is holomorphic on a punctured open set U \ { P } C. Refer to Figure below. In this chapter we shall obtain a new kind of infinite series expansion which generalizes the idea of the power series expansion of a holomorphic function about a (nonsingular) point. We shall in the process completely classify the behavior of holomorphic functions near an isolated singular point.

**Holomorphic Function on a Punctured Domain:**

Let U C be an open set and P U. We call the domain U \ { P } a punctured domain. Suppose that f : U \ { P } → C is holomorphic. In this situation we say that f has an isolated singular point (or isolated singularity) at P. The implication of the phrase is usually just that f is defined and holomorphic on some such “deleted neighborhood” of P. The specification of the set U is of secondary interest; we wish to consider the behavior of f “near P.”

**Figure of an isolated singularity:
**

**Classification of Singularities:**** **

There are three possibilities for the behavior of f near P that are worth distinguishing:

**1**. | f (z) | is bounded on D(P, r) \ { P } for some r > 0 with D(P, r) U; i.e., there is some r > 0 and some M > 0 such that | f (z) | M for all z U D(P, r) \ { P }.

**2**. lim_{z→P} | f (z) | = .

**3**. Neither (1) nor (2).

Of course elementary logic tells us that these three conditions cover all possibilities. The description of is not very satisfying, but it turns out that that is the most subtle situation; there is no simple description of what goes on there.