Removable singularity

In complex analysis, a removable singularity is an isolated singularity of a holomorphic function at a point z_0 where the limit \lim_{z \to z_0} f(z) exists and is finite, allowing the function to be redefined at z_0 by setting f(z_0) equal to this limit, thereby extending f to a holomorphic function on a neighborhood including z_0.^[1] This contrasts with other isolated singularities, such as poles or essential singularities, where no such finite extension is possible, making removable singularities the "mildest" type that do not fundamentally disrupt holomorphy.^[2] The concept is formalized through the Laurent series expansion of f around z_0, where a singularity is removable if and only if the series contains no negative powers of (z - z_0), reducing to a power series that converges at z_0.^[3] Equivalently, if f is bounded in a punctured disk around z_0, or if \lim_{z \to z_0} (z - z_0) f(z) = 0, the singularity is removable.^[4] Riemann's removable singularity theorem provides a key characterization: if f is holomorphic on the punctured disk B(a, R) \setminus \{a\} and bounded there, then f extends holomorphically to the full disk B(a, R).^[2] This theorem, proved using properties of harmonic functions or Cauchy's integral formula, underscores the theorem's role in ensuring continuity and differentiability across the point.^[5] A classic example is the function f(z) = \frac{\sin z}{z}, which has a removable singularity at z = 0 because \lim_{z \to 0} \frac{\sin z}{z} = 1, so defining f(0) = 1 yields the entire holomorphic function \frac{\sin z}{z} on \mathbb{C}.^[1] Another instance occurs with rational functions where a factor cancels a denominator, such as f(z) = \frac{z^2 - 1}{z - 1} = z + 1 for z \neq 1, removable by setting f(1) = 2.^[6] Removable singularities arise in applications like contour integration, where they can be "removed" to simplify residue calculations, and in the study of entire functions, such as polynomials or exponentials, which have none.^[7]

Basic Concepts

Definition

In complex analysis, the complex plane \mathbb{C} is the set of all complex numbers z = x + iy, where x, y \in \mathbb{R} and i = \sqrt{-1}. A neighborhood of a point z_0 \in \mathbb{C} is an open disk centered at z_0, such as \{z \in \mathbb{C} : |z - z_0| < r\} for some r > 0. A function f: U \to \mathbb{C}, where U \subset \mathbb{C} is open, is holomorphic on U if it is complex differentiable at every point z \in U, meaning the limit \lim_{h \to 0} \frac{f(z + h) - f(z)}{h} exists and is the same regardless of how h approaches 0 in \mathbb{C}.^[8] An isolated singularity of a function f at a point z_0 \in \mathbb{C} occurs when f is holomorphic in some punctured neighborhood of z_0 (i.e., a neighborhood excluding z_0 itself) but is either undefined at z_0 or not holomorphic there.^[6] A singularity at z_0 is removable if \lim_{z \to z_0} f(z) exists and is finite; in this case, f can be extended to a holomorphic function on the full neighborhood by defining f(z_0) to be this limit value. On the Riemann sphere (the extended complex plane \mathbb{C} \cup \{\infty\}), such a finite limit corresponds to removability at z_0, distinguishing it from singularities where the limit is infinite. Boundedness of f near z_0 implies the existence of this finite limit, hence removability.^[6]^[1]

Examples

A classic example of a removable singularity is the function f(z) = \frac{\sin z}{z}, which is undefined at z = 0. To verify removability, compute the limit \lim_{z \to 0} \frac{\sin z}{z}. This indeterminate form \frac{0}{0} can be resolved using L'Hôpital's rule, yielding \lim_{z \to 0} \frac{\cos z}{1} = 1, or equivalently via the Taylor series expansion \sin z = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \cdots, which gives \frac{\sin z}{z} = 1 - \frac{z^2}{3!} + \frac{z^4}{5!} - \cdots, approaching 1 as z \to 0. Defining f(0) = 1 removes the singularity, extending f holomorphically to the entire complex plane.^[9] Another standard example is f(z) = \frac{e^z - 1}{z}, undefined at z = 0. The limit \lim_{z \to 0} \frac{e^z - 1}{z} is again \frac{0}{0}, and applying L'Hôpital's rule gives \lim_{z \to 0} \frac{e^z}{1} = 1; alternatively, the Taylor series e^z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots implies \frac{e^z - 1}{z} = 1 + \frac{z}{2!} + \frac{z^2}{3!} + \cdots, converging to 1 at z = 0. Setting f(0) = 1 eliminates the singularity, extending f holomorphically to the entire complex plane.^[9] For rational functions, consider f(z) = \frac{z^2 - 1}{z - 1}, which is undefined at z = [1](/page/1). Direct simplification by factoring the numerator as (z - 1)(z + 1) shows f(z) = z + 1 for z \neq 1, so \lim_{z \to 1} f(z) = 2. Defining f(1) = 2 removes the singularity, resulting in the entire holomorphic function f(z) = z + 1. These cases demonstrate removability by confirming the existence of the limit, as per the definition of isolated singularities where the function approaches a finite value.^[9]

Riemann's Theorem

Statement

Riemann's removable singularity theorem provides a key characterization of removable singularities for holomorphic functions. Specifically, suppose f is holomorphic on the punctured disk $0 < |z - z_0| < r for some r > 0, and suppose f is bounded near z_0, meaning there exists a constant M > 0 such that |f(z)| \leq M for all z satisfying $0 < |z - z_0| < r. Then, the limit \lim_{z \to z_0} f(z) exists and is finite, and f extends to a holomorphic function on the full disk |z - z_0| < r by defining f(z_0) = \lim_{z \to z_0} f(z).^[10] This theorem, named after the German mathematician Bernhard Riemann who formulated it in the 1850s, emphasizes that boundedness in the punctured neighborhood is sufficient to ensure the singularity is removable, without requiring prior knowledge of the limit's existence.^[11] A direct corollary follows from the definition of a removable singularity: if f is holomorphic on $0 < |z - z_0| < r and \lim_{z \to z_0} f(z) exists and is finite, then the singularity at z_0 is removable, allowing holomorphic extension to the full disk.^[10]

Proof Outline

To prove Riemann's theorem, which states that a bounded holomorphic function on a punctured disk has a removable singularity at the isolated point, consider the Laurent series expansion of f around z_0: f(z) = \sum_{k=-\infty}^{\infty} c_k (z - z_0)^k, valid in the punctured disk. The coefficients are given by c_k = \frac{1}{2\pi i} \int_{|\zeta - z_0| = \rho} \frac{f(\zeta)}{(\zeta - z_0)^{k+1}} d\zeta for $0 < \rho < r.^[12] Since |f(z)| \leq M, the magnitude of the integral satisfies |c_k| \leq \frac{1}{2\pi} \cdot 2\pi \rho \cdot \frac{M}{\rho^{k+1}} = \frac{M}{\rho^k} for k < 0. Letting \rho \to 0^+, the right side tends to 0 because k < 0 implies \rho^k \to +\infty, so c_k = 0 for all k < 0. Thus, the Laurent series has no principal part and reduces to a power series f(z) = \sum_{k=0}^{\infty} c_k (z - z_0)^k, which converges in the full disk |z - z_0| < r and defines a holomorphic extension of f there, with f(z_0) = c_0. This confirms the singularity is removable.^[12]

Classification and Detection

Isolated Singularities Overview

An isolated singularity of a holomorphic function f at a point z_0 in the complex plane is a point where f fails to be holomorphic, but f is holomorphic in some punctured disk $0 < |z - z_0| < r surrounding z_0 for a positive r./08%3A_Taylor_and_Laurent_Series/8.05%3A_Singularities) This isolation allows for a local analysis of the function's behavior near z_0 using series expansions, distinguishing it from non-isolated singularities where other singular points accumulate nearby.^[13] Isolated singularities are classified into three main types based on the nature of the function's behavior as z approaches z_0: removable singularities, poles, and essential singularities. A removable singularity occurs when the limit \lim_{z \to z_0} f(z) exists and is finite, allowing f to be extended holomorphically to z_0 by defining f(z_0) as this limit value./08%3A_Taylor_and_Laurent_Series/8.05%3A_Singularities) In this case, the singularity is removable, meaning it is not truly singular after redefinition, in contrast to the other types where no such holomorphic extension is possible. A pole arises when |f(z)| \to \infty as z \to z_0, characterized by a Laurent series expansion around z_0 with a finite number of negative powers in the principal part.^[14] Essential singularities, on the other hand, exhibit more erratic behavior, with the Laurent series having infinitely many negative powers, as exemplified by the function e^{1/z} at z_0 = 0. For essential singularities, the Casorati-Weierstrass theorem describes their wild nature: in any punctured neighborhood of z_0, the image of f comes arbitrarily close to every complex number, making it dense in \mathbb{C}.^[15] A stronger result, the Great Picard theorem, states that near an essential singularity, f assumes every complex value, with at most one exception, infinitely often.^[16] This classification provides a foundational taxonomy for understanding singularity types, with removable singularities standing out as the mildest form that can be eliminated through analytic continuation.^[3]

Methods to Identify Removable Singularities

One primary method to identify a removable singularity at an isolated point z_0 involves expanding the function in its Laurent series around z_0. The Laurent series is given by

f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n,

valid in some punctured disk $0 < |z - z_0| < R. If the principal part vanishes, meaning a_n = 0 for all n < 0, then the series reduces to a Taylor series, and the singularity is removable by defining f(z_0) = a_0.^[17]^[10] Equivalently, a singularity at z_0 is removable if and only if f admits a power series expansion (Taylor series) around z_0 with no negative powers, allowing holomorphic extension to the full disk |z - z_0| < R. This extension is unique by the identity theorem for holomorphic functions.^[17] A practical approach uses limit tests to classify the singularity without computing the full series. Specifically, compute \lim_{z \to z_0} (z - z_0)^k f(z) for successive integers k = 0, 1, 2, \dots. The singularity is removable if the limit for k=0 exists and is finite (i.e., \lim_{z \to z_0} f(z) = L < \infty), in which case define f(z_0) = L to remove it. For higher k > 0, the limits will be zero under this condition, confirming no pole.^[17]^[18] Riemann's theorem provides another criterion: if |f(z)| \leq M for some constant M < \infty in a punctured neighborhood of z_0, then the singularity is removable. To apply this, verify boundedness near z_0, such as by showing |f(z)| \leq M on $0 < |z - z_0| < \delta for some \delta > 0. The proof constructs a holomorphic extension using Cauchy's integral formula on auxiliary functions.^[17]^[10] For rational functions f(z) = P(z)/Q(z), where P and Q are polynomials with no common factors initially, a potential singularity at z_0 (where Q(z_0) = 0) is removable if P(z_0) = 0 with multiplicity at least that of Q at z_0. Factor and cancel common powers of (z - z_0), reducing the apparent singularity to a holomorphic point.^[19]