1. Definition and basic concepts
1.1 Analyticity near a punctured neighborhood
Let \(f\) be a complex-valued function defined on a domain \(D\subset\mathbb{C}\). A point \(a\in D\) is said to be a *singularity* of \(f\) if \(f\) is not analytic at \(a\). A singularity is *isolated* if there exists some radius \(r>0\) such that \(f\) is analytic on the punctured disk \[
| 0< | z-a | <r. |
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\] In other words, the failure of analyticity is confined to the single point \(a\), while the function behaves analytically everywhere sufficiently close to \(a\) except at the point itself.
1.2 Singular points and isolated singularities
An *isolated singularity* is a singular point \(a\) for which the punctured neighborhood condition above holds. This concept is useful because complex analytic structure on punctured neighborhoods is highly constrained: powerful tools such as Laurent series, residue calculus, and uniqueness principles apply locally around \(a\).
1.3 Domain of definition
The notion depends on where \(f\) is initially defined. If \(a\notin D\), one can still discuss isolated singularities relative to the analytic continuation of \(f\) to a larger set. More commonly, \(f\) is considered analytic on \(D\setminus\{a\}\) for some domain \(D\) containing \(a\), and the singularity is characterized by how the behavior near \(a\) prevents analyticity on all of \(D\).
1.4 Examples of isolated singularities
- Removable: \(f(z)=\frac{\sin z}{z}\) has a hole at \(z=0\) if defined by the right-hand side without specifying \(f(0)\). Since \(\sin z / z\) extends analytically, the singularity at \(0\) is removable.
- Pole: \(f(z)=\frac{1}{z^3}\) is analytic on \(\mathbb{C}\setminus\{0\}\) but not at \(0\); the singularity is a pole of order \(3\).
- Essential: \(f(z)=e^{1/z}\) is analytic on \(\mathbb{C}\setminus\{0\}\) and has an essential singularity at \(0\).
2. Classification of isolated singularities
2.1 Removable singularities
A singularity at \(a\) is *removable* if \(f\) can be redefined at \(a\) (without changing its values elsewhere) so that the redefined function becomes analytic at \(a\).
2.1.1 Characterizations
| One characterization uses boundedness: if \(f\) is analytic on \(0< | z-a | <r\) and remains bounded as \(z\to a\), then the singularity at \(a\) is removable. Another characterization is via Laurent series: in the expansion around \(a\), all negative-power terms vanish. |
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2.1.2 Relation to boundedness
Boundedness near the singularity prevents the function from exhibiting the blow-up patterns associated with poles, while also ruling out the highly oscillatory behavior associated with essential singularities. The classical theorem that links boundedness with removability is sometimes presented as a corollary of more general results such as Riemann’s removable singularity theorem.
2.2 Poles
A singularity at \(a\) is a *pole* if \(f(z)\to\infty\) as \(z\to a\) in a controlled manner. Formally, \(a\) is a pole of order \(m\ge1\) if there exists an analytic function \(g\) near \(a\) such that \(g(a)\neq 0\) and \[ f(z)=\frac{g(z)}{(z-a)^m}. \]
2.2.1 Order of a pole
The order \(m\) can be read from the Laurent series of \(f\) at \(a\): if the principal part begins with \((z-a)^{-m}\) and not with any smaller negative power, then \(m\) is the order of the pole.
2.2.2 Behavior near the singularity
| Near a pole, the magnitude of \(f\) typically grows like \( | z-a | ^{-m}\). The direction-dependent oscillations are limited compared with essential singularities. Many qualitative features—such as asymptotic growth and the presence of finite principal parts—follow directly from the pole’s finite order. |
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2.3 Essential singularities
A singularity at \(a\) is *essential* if it is neither removable nor a pole. In Laurent-series terms, this means that the principal part contains infinitely many nonzero negative-power coefficients.
2.3.1 Casorati–Weierstrass theorem
The Casorati–Weierstrass theorem states that near an essential singularity, the set of values \(f(z)\) takes is dense in \(\mathbb{C}\). Thus, values of \(f\) come arbitrarily close to every complex number as \(z\) approaches \(a\).
2.3.2 Picard’s theorem
Picard’s theorem strengthens this statement: in any neighborhood of an essential singularity, \(f\) attains every complex value with at most one exception (the “omitted value”). The theorem reflects the maximal form of local complexity permitted by holomorphicity on the punctured neighborhood.
3. Laurent series representation
3.1 Laurent expansion in a punctured disk
| If \(a\) is an isolated singularity of \(f\) and \(f\) is analytic on \(0< | z-a | <r\), then \(f\) admits a Laurent expansion around \(a\): |
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\[ f(z)=\sum_{n=-\infty}^{\infty} c_n (z-a)^n, \]
| valid for all \(z\) in some punctured disk \(0< | z-a | <R\) (with \(R\le r\) depending on the domain of analyticity). |
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3.2 Principal part
The *principal part* is the portion of the Laurent series with negative powers: \[ \sum_{n=1}^{\infty} c_{-n}(z-a)^{-n}. \] It captures the singular behavior. For example, removable singularities correspond to the absence of negative powers, while poles correspond to a finite principal part.
3.3 Coefficients and uniqueness
Laurent coefficients \(c_n\) are uniquely determined by \(f\) on the punctured neighborhood. In particular, even though the singular point itself may be excluded from the domain, the surrounding analytic structure fixes the full expansion. Coefficients can be computed via contour integrals around \(a\).
3.4 Connection with classification
The classification is reflected directly in the Laurent expansion:
- Removable: \(c_{-n}=0\) for all \(n\ge1\).
- Pole of order \(m\): \(c_{-m}\neq 0\), \(c_{-n}=0\) for all \(n>m\).
- Essential: infinitely many coefficients \(c_{-n}\) are nonzero.
4. Criteria for identifying singularities
4.1 Limits near the singular point
| When the limit \(\lim_{z\to a} f(z)\) exists and is finite, the singularity is removable. If the function diverges to infinity in a manner consistent with a pole, one can often infer an integer order by comparing growth rates with \( | z-a | ^{-m}\). Absence of a finite limit and lack of pole-like algebraic behavior suggests an essential singularity. |
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4.2 Boundedness and continuity conditions
| If \(f\) is bounded on \(0< | z-a | <r\), then the singularity is removable, and \(f\) extends continuously (and indeed analytically) to \(a\). Conversely, poles typically cause unbounded growth, while essential singularities can produce unbounded behavior as well, though with far more irregularity. |
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4.3 Growth conditions
| Poles exhibit polynomial blow-up: for some \(m\), \( | f(z) | \) behaves like \(O( | z-a | ^{-m})\) near \(a\), and sharper estimates correspond to the exact order. Essential singularities do not admit such a finite-order growth restriction in general; they exhibit values that oscillate and fluctuate too strongly. |
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4.4 Behavior of derivatives
Analyticity on the punctured neighborhood implies that derivatives also have Laurent expansions. For a pole of order \(m\), the growth of derivatives is limited in a predictable way, roughly corresponding to increasing powers of \((z-a)^{-1}\). For removable singularities, derivatives extend holomorphically across \(a\). Essential singularities force derivatives to inherit similarly complicated Laurent structures.
5. Local behavior near isolated singularities
5.1 Removable extension
For a removable singularity at \(a\), one can define \[ \tilde{f}(a)=\lim_{z\to a} f(z) \] when the limit exists (and boundedness ensures it). The resulting function \(\tilde{f}\) is analytic at \(a\) and agrees with \(f\) on the punctured neighborhood. The singularity disappears under this extension, often simplifying further analysis.
5.2 Asymptotic behavior at poles
If \(a\) is a pole of order \(m\), then \[ f(z)=\frac{c_{-m}}{(z-a)^m}+\frac{c_{-(m-1)}}{(z-a)^{m-1}}+\cdots \] in a Laurent expansion. The leading term governs the asymptotic magnitude, and the coefficients of the principal part determine refined behavior, including how residues and principal parts influence integrals.
5.3 Dense image behavior near essential singularities
Essential singularities produce extreme local complexity. The Casorati–Weierstrass theorem indicates that values of \(f\) cluster densely across the complex plane. In practice, this means that small punctured neighborhoods around \(a\) contain points where \(f(z)\) is arbitrarily large, arbitrarily small, and approaches essentially any prescribed complex value.
6. Residues and contour integration
6.1 Residue at an isolated singularity
Given a Laurent expansion around \(a\), \[ f(z)=\sum_{n=-\infty}^{\infty} c_n (z-a)^n, \] the *residue* of \(f\) at \(a\) is the coefficient \(c_{-1}\) of \((z-a)^{-1}\). It measures the “local coefficient of the logarithmic term” behavior that governs contour integrals.
6.2 Computation from Laurent coefficients
The residue can be computed by integrating around a positively oriented simple closed contour \(\gamma\) contained in the domain of holomorphy of \(f\) except at \(a\): \[ \operatorname{Res}(f,a)=\frac{1}{2\pi i}\int_\gamma f(z)\,dz. \] Equivalently, \(c_{-1}\) equals the above integral divided by \(2\pi i\), tying the residue directly to the principal part.
6.3 Residue theorem
If \(f\) is meromorphic on a region containing finitely many isolated singularities \(a_k\) and \(\gamma\) encloses them, then \[ \int_\gamma f(z)\,dz = 2\pi i \sum_k \operatorname{Res}(f,a_k), \] where the sum runs over singularities inside \(\gamma\). This theorem generalizes the evaluation of many real-variable integrals by converting them into residue computations.
6.4 Applications in complex integration
Residue theory is especially effective for integrals of rational functions, and for integrals involving products of exponentials and trig functions when suitable contour choices are available. The method’s power comes from reducing global contour integrals to local information contained in residues and principal parts at isolated singularities.
7. Examples and standard functions
7.1 Rational functions
For a rational function \(f(z)=\frac{P(z)}{Q(z)}\), isolated singularities occur at zeros of \(Q\). If a zero of \(Q\) at \(a\) has multiplicity \(m\), then \(f\) has a pole of order \(m\) at \(a\) provided cancellation does not remove it. If the factor cancels completely, the singularity is removable after simplification.
7.2 Exponential and trigonometric functions
Functions like \(\exp(1/z)\) have essential singularities at \(z=0\). Trigonometric ratios can yield removable or pole-type behavior depending on whether the denominator introduces genuine negative powers after expansion; for instance, \(\sin z / z\) is removable at \(0\) because the numerator has a matching zero that cancels.
7.3 Logarithmic and branch-related examples
The logarithm \(\log z\) is multi-valued on \(\mathbb{C}\setminus\{0\}\). After choosing a branch cut and restricting to a branch of \(\log z\), the function becomes analytic on the slit domain, and the behavior at \(0\) is typically described as an isolated singularity of the branch chosen. The key point is that singular behavior can interact with branch structure; residues may or may not be defined depending on whether the function is single-valued and meromorphic near the point.
7.4 Meromorphic functions
A function is *meromorphic* on a domain if it is analytic except for isolated poles. For meromorphic functions, isolated singularities are always poles or removable (after cancellation), never essential. This makes meromorphic functions particularly amenable to residue calculus because their local behavior is governed by finite principal parts.
8. Generalizations and related notions
8.1 Isolated singularities in meromorphic functions
For meromorphic functions, the singular set consists of isolated poles. Consequently, every isolated singularity is classified as a pole (or becomes removable if the apparent pole cancels). This simplifies both local structure and integral computations, since essential behavior cannot occur in a meromorphic setting.
8.2 Singularities of multivalued functions
Multivalued analytic functions (such as algebraic functions or logarithms) can be studied by passing to a suitable branch or by working on a covering space where the function becomes single-valued. Once a branch is fixed, one can analyze the local behavior near the excluded points similarly to single-valued holomorphic functions, keeping in mind that analytic continuation around singularities may permute values.
8.3 Non-isolated singularities
If analyticity fails at more than one point accumulating near \(a\), the singularity is *not isolated*. In such cases, Laurent expansions about \(a\) may fail to exist in the same way, and residue theory still applies only under appropriate meromorphic hypotheses. The “isolated” condition is crucial for guaranteeing the local Laurent-series framework.
8.4 Singular points in several complex variables
In several complex variables, singularities are more intricate because holomorphicity depends on behavior along complex subspaces, not just along one complex variable. Nonetheless, analogous local classifications exist, and notions related to isolated singularities (such as singularities on analytic sets) guide the study of extension, integrability, and residue-type formulas in higher dimensions.