4.2 Isolated Singularities

An isolated singularity of a complex function is a point where a function is holomorphic on some open punctured neighborhood of (namely, for some , the punctured disk ), but not necessarily defined or holomorphic at itself. The nature of this isolated singularity is characterized by the principal part (let be the holomorphic part) of the Laurent series of at the point . Specifically, we can analyze the behavior of as .

• If exists and is finite, then is a removable singularity and can be analytically continued to by Theorem 3.2.6. Consequently, has a convergent Taylor expansion and the principal part of its Laurent expansion vanishes, and .

• If , then is a pole of (from the stereographic projection and the Riemann sphere, the is a single point in , and approaching does not distinguish between different directions, unlike the use of and ).

  Theorem 4.2.1.

  The condition is equivalent to there being a finite number of nonzero ‘s, where .

  In other words the principal part of is equal to

  for some . Therefore,

  on the punctured disk , where

  is holomorphic on and does not attain a zero at . Then has a pole at with order . If , the pole is also called a simple pole.

  Proof.

  Obviously, under the assumption of a finite, nonempty number of nonzero terms in the principal part of the Laurent expansion coefficients, . Now we will prove the converse. Let

  Then . There exists a such that is nonzero on . Then is holomorphic on and has a removable singularity at . By Theorem 3.2.6, can be analytically continued to . Let the multiplicity of the zero at be . Then

  where is holomorphic and nonzero at . Then there exists a such that is nonzero on . It follows that is holomorphic and nonzero on . We can then write its Taylor expansion as

  where . It follows that

  By the uniqueness of the Laurent series, the conclusion follows.

• If is nonexistent, then is known as an essential singularity.

  Example 4.2.1.

  The function has an essential singularity at .

  Proof.

  Observe that . Similarly, , and for with ,

  which is divergent. Therefore, the limit does not exist.

  The implication on its Laurent expansion at is:

  Theorem 4.2.2.

  The necessary and sufficient condition for to not exist is that infinitely many of (where ) are nonzero.

  This follows by elimination from the established trichotomy; if the limit as does not exist, then the singularity is neither removable nor a pole (results from Part 1 and Part 2). Similar logic can be applied to the coefficients of the Laurent expansion.

  Indeed, in Example 4.2.1, the Laurent expansion is equal to:

  which has infinitely many nonzero coefficients of negative powers.

A function with an essential singularity exhibits striking behavior. We will first introduce the following famous result.

An analogous proof yields the following result for entire functions.

In Section 8, we will prove a profound generalization of the two results (@ thm:greatpicard and Theorem 8.3.4), which was first proved by Emile Picard in 1879.

4.2.1 At the Point

Given the one-point compactification of , , we can now define and analyze the behavior of functions near the point at . Similar to the classification of isolated singularities in , we can classify as a removable singularity, a pole, or an essential singularity of a holomorphic function.

Let be holomorphic for some . Then is an isolated singularity of . To analyze the nature of the singularity, let . We define a new function , which is holomorphic on . Then at , has the Laurent expansion of

where and are the holomorphic and principal parts of , respectively. Let , . At , then has the Laurent expansion of

The classification of the singularity at is then reduced to the classification of the singularity of at :

• If is a removable singularity of , then has the form of

• If is a pole of with degree , then can be written as

  where .

• If is an essential singularity of , then can be expanded as

  where , such that (infinitely many coefficients of or are nonzero).