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Isolated singularity

In , an isolated singularity of a f is a point z_0 in the where f fails to be holomorphic, but is holomorphic throughout some deleted disk $0 < |z - z_0| < r for a positive radius r. Such singularities are distinguished from non-isolated ones, like branch points, by the existence of a punctured neighborhood free of other singularities. Isolated singularities are classified into three principal types based on the Laurent series expansion of f around z_0, which includes a principal part consisting of negative powers of (z - z_0). A removable singularity arises when the principal part is zero, meaning \lim_{z \to z_0} f(z) exists and is finite; by Riemann's removable singularity theorem, f can be redefined at z_0 to make it holomorphic there. A pole of order m (a positive integer) occurs when the principal part has finitely many terms up to (z - z_0)^{-m}, with the coefficient of that term nonzero, leading to |f(z)| \to \infty as z \to z_0; in this case, f(z) = \frac{h(z)}{(z - z_0)^m} where h is holomorphic and h(z_0) \neq 0. An essential singularity, by contrast, features infinitely many negative powers in the Laurent series, resulting in highly erratic behavior near z_0, such as unbounded oscillation. The study of isolated singularities is fundamental to residue theory and contour integration, as residues are computed from the coefficient of the (z - z_0)^{-1} term in the Laurent series. For essential singularities, the Casorati–Weierstrass theorem states that the image of any punctured neighborhood under f is dense in the complex plane, underscoring their wild nature. Classic examples include the pole at z = 0 for f(z) = 1/z, the removable singularity at z = 0 for \sin z / z, and the essential singularity at z = 0 for e^{1/z}.

Definition and Basic Concepts

Formal Definition

In complex analysis, a point z_0 in the complex plane is called a singularity of a function f if f fails to be analytic at z_0. Specifically, z_0 is an isolated singularity of f if there exists some r > 0 such that f is analytic in the punctured disk $0 < |z - z_0| < r, but f is not analytic at z_0 itself. Here, analyticity refers to the property of being holomorphic, meaning that f is complex differentiable at every point in some open domain containing that point. In contrast, at a regular point z_0, f is analytic in some full neighborhood including z_0, so f is holomorphic there without exception.

Punctured Neighborhood

In complex analysis, the punctured neighborhood of an isolated singularity at a point z_0 \in \mathbb{C} is defined as the punctured disk \{ z \in \mathbb{C} : 0 < |z - z_0| < r \} for some radius r > 0, where the function in question is holomorphic throughout this region but not necessarily at z_0 itself. This setup excludes the singular point z_0 while capturing the immediate surroundings, allowing the function to be analytic in a disk centered at z_0 minus the origin point. The isolation condition requires that no other singularities of the function lie within this punctured disk, meaning z_0 is the sole singular point in some full disk |z - z_0| < r. This topological separation ensures the singularity behaves independently, without interference from nearby singularities, and facilitates local analysis around z_0. While the punctured disk is central to the geometric setup for isolated singularities, a more general annular region r_1 < |z - z_0| < r_2 with $0 < r_1 < r_2 arises in contexts like Laurent series expansions, where the inner radius r_1 can be positive to avoid the singularity, though the isolation focuses on the case starting from the origin. This annular structure extends the domain for convergence but preserves the key property that the function remains holomorphic in the region excluding z_0.

Classification of Isolated Singularities

Removable Singularities

A removable singularity at an isolated singular point z_0 of a holomorphic function f defined on a punctured neighborhood of z_0 is characterized by the existence of a finite limit \lim_{z \to z_0} f(z) = L \in \mathbb{C}. In this case, the function fails to be defined at z_0, but redefining f(z_0) = L extends f to a holomorphic function on the entire neighborhood, including z_0. This property distinguishes removable singularities as the mildest type among isolated singularities, allowing the function to behave analytically after a simple redefinition. Riemann's removable singularity theorem provides a key criterion: if f is holomorphic and bounded on a punctured disk centered at z_0, then the singularity at z_0 is removable, and f extends holomorphically to the full disk. Equivalently, the singularity is removable if and only if the Laurent series of f around z_0 has no principal part, meaning all coefficients of negative powers of (z - z_0) vanish, reducing the expansion to a Taylor series. A representative example is the function f(z) = \frac{\sin z}{z}, which is undefined at z = 0 but has \lim_{z \to 0} f(z) = 1, confirming a removable singularity there. Defining f(0) = 1 yields the entire , which is holomorphic everywhere.

Poles

A pole is an isolated singularity at a point z_0 where the function f(z) exhibits unbounded growth in a manner controlled by a finite power of (z - z_0). Specifically, z_0 is a pole of order m (with m \geq 1) if m is the smallest positive integer such that \lim_{z \to z_0} (z - z_0)^m f(z) exists and is a nonzero finite complex number, while for all smaller positive integers k < m, the limit \lim_{z \to z_0} (z - z_0)^k f(z) either does not exist or is infinite. Equivalently, in the Laurent series expansion of f(z) around z_0, given by f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n valid in a punctured neighborhood of z_0, the principal part consists of finitely many negative powers, specifically up to -m, with a_{-m} \neq 0 and a_n = 0 for all n < -m. The principal part is thus \sum_{k=1}^{m} a_{-k} (z - z_0)^{-k}, which dominates the behavior near z_0. This finite principal part distinguishes poles from other isolated singularities. The order m can also be determined as the smallest integer such that g(z) = (z - z_0)^m f(z) is analytic at z_0 and g(z_0) \neq 0. Near z_0, f(z) behaves asymptotically like a_{-m} (z - z_0)^{-m}, reflecting polynomial-like growth in the reciprocal sense. As z approaches z_0, |f(z)| \to \infty, confirming the singularity is non-removable and unbounded. This divergence occurs along all paths to z_0, with the rate governed by the order m.

Essential Singularities

In complex analysis, an essential singularity at a point z_0 is an isolated singularity that cannot be classified as either removable or a pole. It arises when the Laurent series expansion of the function around z_0 contains infinitely many terms with negative powers of (z - z_0), indicating unbounded irregularity without the finite principal part seen in poles. This infinite descent in the series reflects a profound lack of analytic continuation, distinguishing essential singularities from milder types where the function either extends holomorphically or grows like a rational power. The behavior of a holomorphic function near an essential singularity is markedly chaotic. In any punctured neighborhood of the singularity, the image under f is dense in the complex plane, as stated by the . This wild oscillation prevents any consistent limiting behavior, such as approaching a finite value or infinity in a controlled manner, and underscores the "essential" nature of the disruption to analyticity. A prototypical example is the function f(z) = e^{1/z}, which has an essential singularity at z = 0. As z approaches 0 along the positive real axis, f(z) tends to infinity, but along the negative real axis, it approaches 0; in any small punctured disk around 0, the image of f is dense in the entire complex plane, exemplifying the theorem's density property. This exponential form highlights how transcendental functions can produce such singularities, contrasting with the algebraic simplicity of poles in rational functions.

Laurent Series and Residues

Laurent Series Expansion

The Laurent series expansion serves as the fundamental analytic tool for representing holomorphic functions in a punctured neighborhood of an isolated singularity at z_0, allowing decomposition into parts that reveal the nature of the singularity. For a function f holomorphic in the annulus $0 < |z - z_0| < R for some R > 0, the series takes the general form f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n, which converges uniformly on compact subsets of this punctured disk. This representation extends the by incorporating negative powers, enabling analysis of behavior near z_0. The series divides into two distinct components: the principal part, consisting of the terms with negative exponents, \sum_{n=-\infty}^{-1} a_n (z - z_0)^n, which captures the singular behavior at z_0, and the regular (or holomorphic) part, \sum_{n=0}^{\infty} a_n (z - z_0)^n, which is analytic at z_0 and resembles a standard . The principal part determines the type of isolated singularity: if it vanishes (all a_n = 0 for n < 0), the singularity is removable; if it contains finitely many nonzero terms, the singularity is a pole of order equal to the highest negative power with nonzero coefficient; and if it has infinitely many nonzero terms, the singularity is essential. The coefficients a_n of the Laurent series are uniquely determined by the Cauchy integral formula adapted to the annulus. Specifically, for any simple closed contour C enclosing z_0 and lying within the region of holomorphy, a_n = \frac{1}{2\pi i} \oint_C \frac{f(\zeta)}{(\zeta - z_0)^{n+1}} \, d\zeta, valid for all integers n. This uniqueness theorem ensures that the expansion is independent of the choice of contour or annulus, provided the function is holomorphic there, and it underpins the classification by allowing direct computation of the principal part from the function's values.

Computation of Residues

The residue of a function f at an isolated singularity z_0, denoted \operatorname{Res}(f, z_0), is defined as the coefficient a_{-1} of the term (z - z_0)^{-1} in the Laurent series expansion of f around z_0: f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n, \quad \operatorname{Res}(f, z_0) = a_{-1}. This coefficient captures the contribution of the singularity to contour integrals enclosing z_0. For a removable singularity at z_0, the Laurent series has no negative powers, so the principal part vanishes and \operatorname{Res}(f, z_0) = 0. At a pole of order m (where m \geq 1) located at z_0, the residue is given by the formula \operatorname{Res}(f, z_0) = \frac{1}{(m-1)!} \lim_{z \to z_0} \frac{d^{m-1}}{dz^{m-1}} \left[ (z - z_0)^m f(z) \right]. This expression arises from differentiating the analytic part after multiplying by (z - z_0)^m to remove the pole. For a simple pole (m=1), it simplifies to \operatorname{Res}(f, z_0) = \lim_{z \to z_0} (z - z_0) f(z). For an essential singularity at z_0, the residue must generally be extracted from the full Laurent series, as there is no finite-order pole formula. For example, the function f(z) = e^{1/z} has an essential singularity at z=0, with Laurent series \sum_{n=0}^{\infty} \frac{1}{n!} z^{-n}, yielding \operatorname{Res}(f, 0) = a_{-1} = \frac{1}{1!} = 1. Residues at isolated singularities enable the evaluation of contour integrals via the residue theorem, which states that for a closed contour C enclosing singularities at z_k, \oint_C f(z) \, dz = 2\pi i \sum \operatorname{Res}(f, z_k).

Properties and Theorems

Casorati-Weierstrass Theorem

The Casorati–Weierstrass theorem characterizes the behavior of holomorphic functions near an essential singularity. Specifically, if f is holomorphic in the punctured disk $0 < |z - z_0| < r for some r > 0 and has an essential singularity at z_0, then the image f(\{ z : 0 < |z - z_0| < r \}) is dense in the complex plane \mathbb{C}. This means that for any complex number w \in \mathbb{C} and any \varepsilon > 0, there exists a sequence \{ z_n \} in the punctured disk with z_n \to z_0 such that |f(z_n) - w| < \varepsilon. The result highlights the "wild" oscillation of functions at essential singularities, distinguishing them from other types. A proof proceeds by contradiction, leveraging the Laurent series expansion of f around z_0. Suppose the image is not dense; then there exist w \in \mathbb{C} and \delta > 0 such that |f(z) - w| \geq \delta for all z in the punctured disk. Consider g(z) = 1/(f(z) - w), which is holomorphic and bounded by $1/\delta in the punctured disk, implying a at z_0 by Riemann's theorem. Extending g holomorphically to z_0, if g(z_0) \neq 0, then f(z) - w extends holomorphically, contradicting the . If g(z_0) = 0, then f(z) - w has a at z_0, again contradicting the essential nature, as the would have a finite principal part. The infinite principal part of the thus ensures the density. This density contrasts sharply with the behavior at poles and removable singularities. At a pole of order m, |f(z)| \to \infty as z \to z_0, so the image avoids any neighborhood of 0, lying instead in \{ w : |w| > R \} for some R > 0 sufficiently close to z_0. For a removable singularity, f extends to a holomorphic function at z_0, remaining bounded in a full neighborhood and thus not dense in the entire plane unless the extension is constant. Essential singularities, by the theorem, fill the plane densely due to their unbounded and oscillatory nature. The theorem is named after the Italian mathematician Felice Casorati, who first published it in his 1868 work Teorica delle funzioni di variabili complesse, and , who independently proved it in his 1876 paper "Zur Theorie der eindeutigen analytischen Funktionen." It was also discovered independently by Yulian Sokhotski in 1868, leading to its occasional designation as the Casorati–Sokhotski–Weierstrass theorem in some literature.

Great Picard Theorem

The Great Picard Theorem asserts that if a f has an at a point z_0 \in \mathbb{C}, then in every punctured neighborhood of z_0, the function f(z) assumes every complex value infinitely often, with at most one possible exception. This result highlights the extreme behavior of functions near essential singularities, where the image is not merely dense but recurrent for nearly all values. This theorem refines the Casorati-Weierstrass theorem by elevating the mere density of the image in punctured neighborhoods to infinite attainments for all complex numbers except possibly one. The proof typically employs the Montel theorem on normal families, noting that holomorphic functions on the plane omitting three fixed values form a normal family. If f omitted two values near the singularity, rescaling or composition arguments would imply normality, contradicting the essential singularity unless the function is constant or has a milder singularity. Alternative approaches use the modular function \lambda(\tau) from uniformization theory or Harnack-type inequalities for subharmonic functions to establish the infinite repetitions. A global counterpart is the Little Picard Theorem, which states that any non-constant omits at most one complex value from its range. This follows similar ideas, applying to suitably transformed functions that omit multiple values.

Examples

Rational Functions

Rational functions provide a fundamental class of examples for isolated singularities in , as their singularities are algebraic and occur precisely at the zeros of the denominator . A is expressed in the general form f(z) = \frac{p(z)}{q(z)}, where p(z) and q(z) are polynomials in the complex variable z, and the function is holomorphic everywhere except possibly at points where q(z) = 0. These points are isolated singularities, provided they are finite and distinct from the zeros of p(z). If q(z_0) = 0 but p(z_0) \neq 0, then z = z_0 is a pole of f(z), and the order of the pole equals the multiplicity of the root z_0 of q(z). For instance, if q(z) has a zero of order m at z_0, the Laurent series of f(z) around z_0 has principal part consisting of terms up to (z - z_0)^{-m}, with the coefficient of (z - z_0)^{-m} being nonzero. This behavior reflects the finite-order blow-up of |f(z)| as z \to z_0, scaling as |z - z_0|^{-m}. In contrast, if p(z) and q(z) share a common zero at z_0 with the multiplicity of the zero in p(z) at least as large as that in q(z), the apparent singularity at z_0 is removable. Canceling the common factor (z - z_0)^k (where k is the minimum multiplicity) yields a holomorphic function at z_0, which can be extended continuously by defining f(z_0) as the limit value. For example, f(z) = \frac{z^2 - 1}{z - 1} = z + 1 for z \neq 1, removing the singularity at z = 1 by setting f(1) = 2. The of a explicitly reveals its at the poles, facilitating the classification of singularities. Assuming p(z) and q(z) have no common factors and \deg p < \deg q, f(z) decomposes as a sum of a polynomial (possibly constant) plus terms of the form \sum_{j=1}^k \frac{R_j(z)}{(z - \beta_j)^{d_j}}, where each R_j(z) is a polynomial of degree less than d_j, the \beta_j are the distinct poles, and d_j is the order of the pole at \beta_j. The principal part at each pole is the negative powers in the expansion of these terms, confirming the pole order and enabling residue computation. A concrete illustration is f(z) = \frac{1}{z^2 - 1} = \frac{1}{(z-1)(z+1)}, which has simple poles (order 1) at z = 1 and z = -1, since the denominator factors have simple zeros and the numerator is nonzero there. The partial fraction decomposition is f(z) = \frac{1/2}{z-1} - \frac{1/2}{z+1}, so the residues—the coefficients of (z - z_k)^{-1}—are +1/2 at z = 1 and -1/2 at z = -1. This decomposition highlights the isolated nature of these poles and their principal parts.

Transcendental Functions

Transcendental functions often exhibit isolated singularities with more intricate behaviors than those of rational functions, particularly essential singularities arising from their non-algebraic nature. A quintessential example is the function f(z) = e^{1/z}, which possesses an essential singularity at z = 0. The Laurent series expansion of e^{1/z} about this point is \sum_{n=0}^{\infty} \frac{1}{n!} z^{-n}, featuring infinitely many negative powers in the principal part, confirming the essential nature of the singularity. Near z = 0, the behavior of e^{1/z} is highly erratic; along the positive real axis as x \to 0^+, e^{1/x} tends to infinity, while along the negative real axis as x \to 0^-, it approaches 0, illustrating wild oscillations. Moreover, in any punctured neighborhood of 0, e^{1/z} assumes every complex value except 0 infinitely often, a consequence of the applied to essential singularities. In contrast, some transcendental functions display isolated poles. The cotangent function \cot(\pi z) has simple poles at every integer z = n, where n \in \mathbb{Z}, with each residue equal to $1/\pi. This periodic arrangement of poles underscores the function's meromorphic character across the complex plane. Another illustrative essential singularity occurs in g(z) = \sin(1/z) at z = 0, where the Laurent series includes infinitely many negative odd powers, such as \sum_{k=0}^{\infty} (-1)^k \frac{1}{(2k+1)!} z^{-(2k+1)}. The zeros of \sin(1/z) at z = 1/(k\pi) for integers k \neq 0 accumulate at 0, forming dense clusters in the domain that contribute to the singularity's complexity, while the function's image near 0 exhibits dense coverage of the complex plane with exceptional density around certain values.

Non-Isolated Singularities

Natural Boundaries

In complex analysis, a natural boundary of a holomorphic function is a curve or line segment, such as the unit circle, along which the singularities of the function are dense, thereby preventing analytic continuation across that boundary. This density implies that every point on the boundary is a singularity or a limit point of singularities, rendering it impossible to extend the function holomorphically into any neighborhood that crosses the boundary. Unlike isolated singularities, where the function remains analytic in a punctured disk around the point, a natural boundary lacks such a disk free of singularities on one side. A classic example is the lacunary power series f(z) = \sum_{n=1}^{\infty} z^{n!}, which converges to a holomorphic function inside the unit disk |z| < 1 but has the unit circle |z| = 1 as its natural boundary. For points z = e^{2\pi i p/q} on the unit circle, where p/q is rational, the partial sums of the series grow unbounded in every neighborhood of such points, and since these points are dense on the circle, singularities accumulate everywhere along it. This prevents any analytic continuation beyond the disk, as the function becomes unbounded near every boundary point. The Ostrowski-Hadamard gap theorem provides a general condition under which a power series exhibits a natural boundary. Specifically, if the exponents n_k in the series \sum a_k z^{n_k} satisfy n_{k+1}/n_k \geq 1 + \delta for some \delta > 0 and all k, with \limsup |a_k|^{1/n_k} = 1, then the circle of convergence serves as a natural boundary for the function. This , building on earlier work by Hadamard, highlights how large gaps in the exponents lead to dense singularities on the boundary, contrasting sharply with series lacking such gaps, which may allow continuation.

Cluster Singularities

A cluster singularity, also known as an or limit point of singularities, arises when a of isolated singularities converges to a point in the . For instance, if a has simple poles at the points z = 1/n for each positive n, then z = 0 serves as the cluster singularity, as these poles accumulate at the origin. Unlike an isolated singularity, where there exists a punctured disk around the point in which the function is analytic, a cluster singularity is non-isolated by nature. In every punctured neighborhood of the cluster point, infinitely many singularities are present, preventing the function from being analytic in any such deleted disk. This accumulation violates the condition for isolation, rendering standard Laurent series expansions inapplicable at the cluster point itself. A classic example is the function f(z) = \tan(1/z), which exhibits poles where \cos(1/z) = 0, specifically at z_k = 1/(\pi/2 + k\pi) for nonzero integers k. These points z_k approach 0 as |k| \to \infty, forming a cluster of poles at the origin and establishing z = 0 as a non-isolated singularity. Near this cluster point, the function displays highly irregular behavior, densely approaching all complex values in accordance with generalizations of the Casorati-Weierstrass theorem. The presence of a cluster singularity has significant implications for the and meromorphicity of the function. It precludes classification of the point using the categories of removable singularities, poles, or isolated essential singularities, as no finite principal part in a Laurent expansion exists. Consequently, the function cannot be meromorphic in any domain encompassing the cluster point, limiting extensions to regions excluding the accumulation.