6 Isolated singularities and residues

6.1 The extended complex plane

Definition 6.1 The Riemann sphere

The extended complex plane is the set

\begin{align*} \hat{\mathbb {C}} = \mathbb {C}\cup \left\{ \infty \right\} \, , \end{align*}

where \(\infty \) is a symbol added to \(\mathbb {C}\) to represent a single point at infinity. The set \(\hat{\mathbb {C}}\) is given a topology in such a way that open sets in \(\mathbb {C}\) remain open in \(\hat{\mathbb {C}}\), and sets of the form \(\big\{ z \in \mathbb {C}\; \big| \; |z|{\gt}M \big\} \) for \(M{\gt}0\) form a neighborhood basis at \(\infty \).

(This topology makes \(\hat{\mathbb {C}}\) homeomorphic to the 2-dimensional sphere in three-dimensional space, and \(\hat{\mathbb {C}}\) is also called the Riemann sphere.)

For example a function \(f \colon U \to \mathbb {C}\) has limit \(\lim _{z \to z_0} f(z) = \infty \) at \(z_0\) if for any \(M{\gt}0\) there exists a \(\delta {\gt} 0\) such that \(|f(z)|{\gt}M\) whenever \(0 {\lt} |z-z_0| {\lt} \delta \).

6.2 Isolated singularities of analytic functions

Definition 6.2 Isolated singularity [Palka1991, Sec. VIII.2.1]

Let \(f : U \to \mathbb {C}\) be an analytic function on an open set \(U \subset \mathbb {C}\). We say that \(f\) has an isolated singularity at \(z_0 \in \mathbb {C}\) if \(\mathcal{B}(z_0; r) \setminus \left\{ z_0 \right\} \subset U\) for some \(r{\gt}0\) but \(z_0 \notin U\).

Definition 6.3 Classification of isolated singularities [Palka1991, Sec. VIII.2.1]

Let \(z_0 \in \mathbb {C}\) be an isolated singularity of an analytic function \(f : U \to \mathbb {C}\). Let \(r {\gt} 0\) be such that \(\mathcal{B}(z_0; r) \setminus \left\{ z_0 \right\} \subset U\), so that by Theorem 5.37 \(f\) can be represented in \(\mathcal{B}(z_0; r) \setminus \left\{ z_0 \right\} \) uniquely as a Laurent series

\begin{align*} f(z) = \sum _{n=-\infty }^\infty a_n (z - z_0)^n . \end{align*}

Depending on the coefficients \(a_n\) of negative indices \(n {\lt} 0\), we distinguish three types of singularities:

\(f\) has a removable singularity at \(z_0\) if \(a_n = 0\) for all \(n {\lt} 0\);
\(f\) has a pole of order \(m \in \mathbb {N}\) at \(z_0\) if \(a_{-m} \ne 0\) and \(a_n = 0\) for all \(n {\lt} -m\);
\(f\) has an essential singularity at \(z_0\) if \(a_n \ne 0\) for infinitely many \(n {\lt} 0\).

Definition 6.4 Residue at an isolated singularity [Palka1991, Sec. VIII.2.1]

\begin{align*} f(z) = \sum _{n=-\infty }^\infty a_n (z - z_0)^n . \end{align*}

The coefficient \(a_{-1}\) is called the residue of \(f\) at \(z_0\), and is denoted \(\mathrm{Res}_{{z_0}}({f}) = a_{-1} \in \mathbb {C}\).

Theorem 6.5 Removable singularity characterization [Palka1991, Thm VIII.2.1 and Thm VIII.2.2]

Let \(z_0 \in \mathbb {C}\) be an isolated singularity of an analytic function \(f : U \to \mathbb {C}\). Then the following conditions are equivalent:

(R-1): The singularity of \(f\) at \(z_0\) is removable (i.e., all negative index Laurent series coefficients of \(f\) expanded near \(z_0\) vanish).
(R-2): There exists an analytic function \(\tilde{f} \colon U \cup \left\{ z_0 \right\} \to \mathbb {C}\) such that \(f(z) = \tilde{f}(z)\) for all \(z \in U\).
(R-3): The limit \(\lim _{z \to z_0} f(z)\) exists in \(\mathbb {C}\).
(R-4): The function \(f\) is bounded in some punctured disk \(\mathcal{B}(z_0; r) \setminus \left\{ z_0 \right\} \) with \(r{\gt}0\).

Proof ▶

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Theorem 6.6 Characterization of poles [Palka1991, Thm VIII.2.3 and Thm VIII.2.4]

Let \(z_0 \in \mathbb {C}\) be an isolated singularity of an analytic function \(f : U \to \mathbb {C}\). Then the following conditions are equivalent:

(P-1): The singularity of \(f\) at \(z_0\) is a pole (i.e., finitely many Laurent series coefficients of \(f\) near \(z_0\) are nonzero).
(P-2): There exists an \(m \in \mathbb {N}= \left\{ 1,2,\ldots \right\} \) such that \(z \mapsto (z-z_0)^m \, f(z)\) has a removable singularity and a nonzero limit as \(z \to z_0\).
(P-3): The function \(f\) has the limit \(\lim _{z \to z_0} f(z) = \infty \) at \(z_0\).

Proof ▶

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Theorem 6.7 Characterization of essential singularities [Palka1991, Thm VIII.2.6 and Thm VIII.2.7]

Let \(z_0 \in \mathbb {C}\) be an isolated singularity of an analytic function \(f : U \to \mathbb {C}\). Then the following conditions are equivalent:

(E-1): The singularity of \(f\) at \(z_0\) is essential (i.e., infinitely many Laurent series coefficients of \(f\) near \(z_0\) are nonzero).
(E-2): For any small \(\delta {\gt} 0\), the image \(f \big[ \mathcal{B}(z_0; \delta ) \setminus \left\{ z_0 \right\} \big]\) is dense in \(\mathbb {C}\).
(E-3): The limit \(\lim _{z \to z_0} f(z)\) does not exist in the extended complex plane \(\hat{\mathbb {C}} = \mathbb {C}\cup \left\{ \infty \right\} \).

Proof ▶

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6.3 The residue theorem

Theorem 6.8 Residue theorem [Palka1991, Thm VIII.3.1]

Let \(U \subset \mathbb {C}\) be an open set and \(\gamma \) a contractible closed contour in \(U\). Let \(f : U \setminus S \to \mathbb {C}\) be an analytic function with isolated singularities at a countable set \(S \subset U\) of points. Then

\begin{align*} \oint _\gamma f(z) \, dz = 2 \pi \mathfrak {i}\sum _{w \in S} \mathfrak {n}_{{w}}(\gamma ) \, \mathrm{Res}_{{w}}({f}) . \end{align*}

Proof ▶

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Corollary 6.9 Residue theorem for Jordan contours [Palka1991, Cor VIII.3.2]

Let \(U \subset \mathbb {C}\) be an open set and \(S \subset U\) a discrete subset of it. Let \(\mathcal{D}\) be a Jordan domain such that \(\overline{{\mathcal{D}}} \subset U\) and \(\partial \mathcal{D}\cap S = \emptyset \). Let \(\gamma \) be a closed contour traversing the boundary \(\partial \mathcal{D}\) of the Jordan domain in the positive orientation. Let \(f : U \setminus S \to \mathbb {C}\) be an analytic function with isolated singularities at the points of \(S\). Then

\begin{align*} \oint _\gamma f(z) \, dz = 2 \pi \mathfrak {i}\sum _{w \in S \cap \mathcal{D}} \mathrm{Res}_{{w}}({f}) . \end{align*}

Proof ▶

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