§4.2 Isolated Singularities

4.2 Isolated Singularities

An isolated singularity of a complex function is a point $𝑎 \in ℂ$ where a function $𝑓$ is holomorphic on some open punctured neighborhood of $𝑎$ (namely, for some $𝑟 > 0$ , the punctured disk $𝐷^{*} (𝑎, 𝑟)$ ), but not necessarily defined or holomorphic at $𝑎$ itself. The nature of this isolated singularity is characterized by the principal part $𝜓 (𝑧)$ (let $𝜑 (𝑧)$ be the holomorphic part) of the Laurent series of $𝑓$ at the point $𝑎$ . Specifically, we can analyze the behavior of $𝑓 (𝑧)$ as $𝑧 \to 𝑎$ .

If ${lim}_{𝑧 \to 𝑎} 𝑓 (𝑧)$ exists and is finite, then $𝑧 = 𝑎$ is a removable singularity and can be analytically continued to $𝐷 (𝑎, 𝑟)$ by Theorem 3.2.9 $Theorem 3.2.9 Theorem 3.2.9$ . Consequently, $𝑓 (𝑧)$ has a convergent Taylor expansion and the principal part of its Laurent expansion vanishes, and $𝑓 (𝑧) = 𝜑 (𝑧)$ .
If ${lim}_{𝑧 \to 𝑎} 𝑓 (𝑧) = \infty$ , then $𝑧 = 𝑎$ is a pole of $𝑓$ (from the stereographic projection and the Riemann sphere, the $\infty$ is a single point in $\hat{ℂ}$ , and approaching $\infty$ does not distinguish between different directions, unlike the use of $+ \infty$ and $- \infty$ ).
Theorem 4.2.1.
The condition ${lim}_{𝑧 \to 𝑎} 𝑓 (𝑧) = \infty$ is equivalent to there being a finite number of nonzero $𝑐_{- 𝑛}$ ’s, where $𝑛 \in ℕ$ .
In other words the principal part of $𝑓$ is equal to
$𝜓 (𝑧) = \frac{𝑐_{- 1}}{𝑧 - 𝑎} + \dots + \frac{𝑐_{- 𝑚}}{{(𝑧 - 𝑎)}^{𝑚}}, 𝑐_{- 𝑚} \neq 0$
for some $𝑚 \in ℕ$ . Therefore,
$𝑓 (𝑧) = 𝜑 (𝑧) + 𝜓 (𝑧) = \sum_{𝑛 = - 𝑚}^{\infty} 𝑐_{𝑛} {(𝑧 - 𝑎)}^{𝑛} = \frac{𝑔 (𝑧)}{{(𝑧 - 𝑎)}^{𝑚}}$
on the punctured disk $𝐷^{*} (𝑎, 𝑟)$ , where
$𝑔 (𝑧) = \sum_{𝑛 = 0}^{\infty} 𝑐_{𝑛 - 𝑚} {(𝑧 - 𝑎)}^{𝑛}$
is holomorphic on $𝐷 (𝑎, 𝑟)$ and does not attain a zero at $𝑧 = 𝑎$ . Then $𝑓 (𝑧)$ has a pole at $𝑧 = 𝑎$ with order $𝑚$ . If $𝑚 = 1$ , 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, ${lim}_{𝑧 \to 𝑎} 𝑓 (𝑧) = \infty$ . Now we will prove the converse. Let
$𝑔 (𝑧) = \frac{1}{𝑓 (𝑧)} .$
Then ${lim}_{𝑧 \to 𝑎} 𝑔 (𝑧) = 0$ . There exists a $𝛿 > 0$ such that $𝑓$ is nonzero on $𝐷^{*} (𝑎, 𝛿)$ . Then $𝑔 (𝑧)$ is holomorphic on $𝐷^{*} (𝑎, 𝛿)$ and has a removable singularity at $𝑧 = 𝑎$ . By Theorem 3.2.9 $Theorem 3.2.9 Theorem 3.2.9$ , $𝑔$ can be analytically continued to $𝐷 (𝑎, 𝛿)$ . Let the multiplicity of the zero at $𝑧 = 𝑎$ be $𝑚$ . Then
$𝑔 (𝑧) = 𝜑 (𝑧) {(𝑧 - 𝑎)}^{𝑚},$
where $𝜑 (𝑧)$ is holomorphic and nonzero at $𝑧 = 𝑎$ . Then there exists a $𝛿^{'} > 0$ such that $𝜑$ is nonzero on $𝐷 (𝑎, 𝛿^{'})$ . It follows that $\frac{1}{𝜑}$ is holomorphic and nonzero on $𝐷 (𝑎, 𝛿^{'})$ . We can then write its Taylor expansion as
$\frac{1}{𝜑 (𝑧)} = 𝑐_{- 𝑚} + 𝑐_{1 - 𝑚} (𝑧 - 𝑎) + \dots,$
where $𝑐_{- 𝑚} \neq 0$ . It follows that
$𝑓 (𝑧) = \frac{1}{𝑔 (𝑧)} = \frac{{(𝑧 - 𝑎)}^{- 𝑚}}{𝜑 (𝑧)} = 𝑐_{- 𝑚} {(𝑧 - 𝑎)}^{- 𝑚} + \dots + 𝑐_{0} + \dots .$
By the uniqueness of the Laurent series, the conclusion follows. ⁠ $□$
If ${lim}_{𝑧 \to 𝑎} 𝑓 (𝑧)$ does not exist, then $𝑎$ is known as an essential singularity.
Example 4.2.2.
Classify the singularity of the function $e^{\frac{1}{𝑧}}$ on the punctured plane $ℂ^{*} = ℂ ∖ {0}$ at $𝑧 = 0$ .
Solution to Example 4.2.2 $Example 4.2.2 Example 4.2.2$ .
Observe that ${lim}_{𝑥 \to 0^{+}} e^{\frac{1}{𝑥}} = \infty$ . Similarly, ${lim}_{𝑥 \to 0^{-}} e^{\frac{1}{𝑥}} = 0$ , and for $𝑧 = i 𝑦$ with $𝑦 \to 0^{+}$ ,
$e^{\frac{1}{𝑧}} = e^{- \frac{i}{𝑦}},$
which is divergent. Hence, the limit does not exist, and thus $𝑧 = 0$ is an essential singularity. ⁠ $□$
The implication on its Laurent expansion at $𝑎$ is:
Theorem 4.2.3.
The necessary and sufficient condition for ${lim}_{𝑧 \to 𝑎} 𝑓 (𝑧)$ to not exist is that infinitely many of $𝑐_{- 𝑛}$ (where $𝑛 \in ℕ$ ) are nonzero.
This follows by elimination from the established trichotomy; if the limit as $𝑧 \to 𝑎$ does not exist, then the singularity is neither removable nor a pole (results from Part 1 $Part 1 Part 1$ and Part 2 $Part 2 Part 2$ ). Similar logic can be applied to the coefficients of the Laurent expansion.
Indeed, in Example 4.2.2 $Example 4.2.2 Example 4.2.2$ , the Laurent expansion is equal to:
$e^{\frac{1}{𝑧}} = \sum_{𝑛 = 0}^{\infty} \frac{𝑧^{- 𝑛}}{𝑛!},$
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.

Theorem 4.2.4 (Casorati–Sokhotski–Weierstrass).

Let

𝑎 \in ℂ

and

𝑈 \subseteq ℂ

be an open region. Suppose

𝑓 : 𝑈 ∖ {𝑎} \to ℂ

is holomorphic with an essential singularity at

𝑎

. Then the set of values that

𝑓

attains on any open punctured neighborhood of

𝑎

is dense. In other words,

\forall 𝜀, 𝛿 > 0

\forall 𝑤 \in ℂ

\exists 𝑧 \in 𝐷^{*} (𝑎, 𝛿)

such that

| 𝑓 (𝑧) - 𝑤 | < 𝜀

Proof.

Assume for the sake of contradiction that $\exists 𝜀, 𝛿 > 0$ , and $\exists 𝑤 \in ℂ$ such that $\forall 𝑧 \in 𝐷^{*} (𝑎, 𝛿)$ , $| 𝑓 (𝑧) - 𝑤 | > 𝜀$ . Define the auxiliary function

𝑔 (𝑧) = \frac{𝑓 (𝑧) - 𝑤}{𝑧 - 𝑎},

which is holomorphic and non-vanishing on the punctured neighborhood of $𝑎$ . Since as $𝑧 \to 𝑎$ , $𝑔 (𝑧) \to \infty$ , it follows that $𝑔 (𝑧)$ has a pole at $𝑎$ . Let the order of the pole be $𝑚 \in ℕ$ . By Theorem 4.2.1 $Theorem 4.2.1 Theorem 4.2.1$ , $𝑔 (𝑧)$ has the Laurent expansion

\frac{𝑐_{- 𝑚}}{{(𝑧 - 𝑎)}^{𝑚}} + \dots + 𝑐_{0} + 𝑐_{1} (𝑧 - 𝑎) + \dots

for some $𝑚 \in ℕ$ . It follows that

𝑓 (𝑧) = \frac{𝑐_{- 𝑚}}{{(𝑧 - 𝑎)}^{𝑚 - 1}} + \dots + 𝑐_{- 1} + 𝑤 + 𝑐_{0} (𝑧 - 𝑎) + \dots .

𝑚 = 1

, then

𝑓

has a removable singularity at

𝑎

. If

𝑚 \geq 2

, then

𝑓

has a pole at

𝑎

. Hence, we have a contradiction. ⁠

□

An analogous proof yields the following result for entire functions.

Theorem 4.2.5.

The set of values that a non-constant entire function

𝑓

assumes is dense in

ℂ

Proof.

For the sake of contradiction, assume there exists $𝑤 \in ℂ$ and $𝜀 > 0$ such that $𝐷 (𝑤, 𝜀) \cap 𝑓 (ℂ) = ⌀$ . Define

𝑔 (𝑧) = \frac{1}{𝑓 (𝑧) - 𝑤} .

It follows that

| 𝑔 | \leq \frac{1}{𝜀}

ℂ

. By Liouville’s Theorem (Theorem 3.2.5

Theorem 3.2.5 Theorem 3.2.5

𝑔

is a constant function, and hence,

𝑓

is also constant, which is a contradiction of the statement. ⁠

□

In Section 8 $Section 8 Section 8$ , we will prove a profound generalization of the two results (Theorem 8.5.1 $Theorem 8.5.1 Theorem 8.5.1$ and Theorem 8.3.6 $Theorem 8.3.6 Theorem 8.3.6$ ), which was first proved by Emile Picard in 1879:

Theorem 8.3.6 (Picard’s Little Theorem).

Let

𝑓 : ℂ \to 𝑈

be entire such that

ℂ ∖ 𝑈

contains two or more points. Then

𝑓

is constant.

Theorem 8.5.1 (Picard’s Great Theorem).

Suppose

𝑓

is holomorphic on a punctured neighborhood

𝐷^{*} (𝑧_{0}, 𝛿)

𝑧_{0}

. If

𝑧_{0}

is an essential singularity of

𝑓

, then

𝑓 (𝐷^{*} (𝑧_{0}, 𝛿))

omits at most one value of

ℂ