8.5 Removable singularities

Removable singularities are points where a function is undefined but can be redefined to make it continuous and differentiable. They're crucial in potential theory, often arising in complex analysis and harmonic functions.

Understanding different types of removable singularities helps simplify and analyze complex functions. Techniques like examining limits, derivatives, and comparing with known examples are used to identify and remove these singularities, extending a function's domain and ensuring continuity.

Types of removable singularities

• Removable singularities are points where a function is undefined but can be redefined to make the function continuous and differentiable at that point
• Understanding the different types of removable singularities is crucial in potential theory as they often arise in complex analysis and the study of harmonic functions

Isolated singularities

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• Isolated singularities are points where a function is undefined but the limit of the function exists as the variable approaches the singularity
• These singularities can be removed by redefining the function's value at the singular point to be equal to the limit
• Example: $f(z) = \frac{z^2 - 1}{z - 1}$ has an isolated singularity at $z = 1$

Poles

• Poles are isolated singularities where the function approaches infinity as the variable approaches the singular point
• The order of a pole determines how quickly the function grows near the singularity
• Example: $f(z) = \frac{1}{(z - 1)^2}$ has a pole of order 2 at $z = 1$

Essential singularities

• Essential singularities are isolated singularities that are neither removable nor poles
• The function's behavior near an essential singularity is more complex and cannot be characterized by a finite limit or infinite limit with a specific order
• Example: $f(z) = e^{\frac{1}{z}}$ has an essential singularity at $z = 0$

Identifying removable singularities

• Recognizing removable singularities is an essential skill in potential theory as it allows for the simplification and analysis of complex functions
• Several methods can be used to identify removable singularities, including examining the function's limit, derivative, and comparing it with known removable singularities

Limit of function near singularity

• If the limit of a function exists as the variable approaches the singular point, the singularity is likely to be removable
• Calculate the limit using various techniques such as substitution, L'Hôpital's rule, or series expansions
• Example: For $f(z) = \frac{z^2 - 1}{z - 1}$, $\lim_{z \to 1} f(z) = 2$, indicating a removable singularity at $z = 1$

Behavior of derivative near singularity

• Analyze the function's derivative near the singular point to determine if the singularity is removable
• If the derivative exists and is bounded near the singularity, it is likely to be removable
• Example: For $f(z) = \frac{z^2 - 1}{z - 1}$, $f'(z) = \frac{z^2 - 2z + 1}{(z - 1)^2}$, which is defined and bounded near $z = 1$, confirming a removable singularity

Comparison with known removable singularities

• Compare the function with known examples of removable singularities to identify similarities in their structure and behavior
• Common examples include rational functions with factors canceling out, or functions with indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$
• Example: $f(z) = \frac{z^2 - 1}{z - 1}$ has a similar structure to $g(z) = \frac{z - 1}{z - 1}$, which has a known removable singularity at $z = 1$

Removing singularities

• Removing singularities is a crucial technique in potential theory as it allows for the extension of a function's domain and ensures continuity
• The process involves redefining the function at the singular point, extending its domain, and verifying continuity after the removal

Redefining function at singularity

• To remove a singularity, redefine the function's value at the singular point to be equal to the limit of the function as the variable approaches the singularity
• This new definition makes the function continuous at the previously singular point
• Example: For $f(z) = \frac{z^2 - 1}{z - 1}$, redefine $f(1) = 2$ to remove the singularity at $z = 1$

Extending domain of function

• After redefining the function at the singular point, extend its domain to include the previously excluded value
• This extension allows the function to be defined and continuous at the singular point
• Example: For $f(z) = \frac{z^2 - 1}{z - 1}$, extend the domain to include $z = 1$, making the function defined on the entire complex plane

Ensuring continuity after removal

• Verify that the function is continuous at the previously singular point after redefining its value and extending its domain
• Use the definition of continuity or the limit definition to confirm that the function is continuous at the point
• Example: For $f(z) = \frac{z^2 - 1}{z - 1}$ with $f(1) = 2$, $\lim_{z \to 1} f(z) = f(1) = 2$, ensuring continuity at $z = 1$

Applications of removable singularities

• Removable singularities have numerous applications in various branches of mathematics and physics, particularly in potential theory
• Understanding and handling removable singularities is essential for solving problems in complex analysis, harmonic functions, and physical systems

Complex analysis

• In complex analysis, removable singularities often arise when studying complex functions and their properties
• Removing singularities allows for the extension of analytic functions and the application of powerful theorems like the Cauchy integral formula
• Example: The function $f(z) = \frac{\sin(z)}{z}$ has a removable singularity at $z = 0$, and removing it makes the function entire

Harmonic functions

• Harmonic functions, which satisfy Laplace's equation, are fundamental in potential theory and often encounter removable singularities
• Removing singularities in harmonic functions enables the application of the maximum principle and the study of their behavior at infinity
• Example: The harmonic function $u(x, y) = \log(x^2 + y^2)$ has a removable singularity at $(0, 0)$, and removing it allows for the analysis of its properties

Electrostatics and fluid dynamics

• In electrostatics, removable singularities can occur in the electric potential due to point charges or discontinuities in the charge distribution
• Fluid dynamics problems involving irrotational flow may also encounter removable singularities in the velocity potential
• Example: The electric potential $V(r) = \frac{1}{r}$ has a removable singularity at $r = 0$, representing the potential due to a point charge at the origin

Theorems related to removable singularities

• Several important theorems in complex analysis and potential theory deal with removable singularities and their properties
• These theorems provide powerful tools for analyzing and understanding the behavior of functions near singularities

Riemann's theorem on removable singularities

• Riemann's theorem states that if a function is analytic in a punctured disk around a point and bounded near that point, then the singularity is removable
• The theorem provides a sufficient condition for a singularity to be removable and allows for the extension of the function to the entire disk
• Example: If $f(z)$ is analytic in $0 < |z - z_0| < r$ and bounded near $z_0$, then $z_0$ is a removable singularity of $f(z)$

Painlevé's theorem

• Painlevé's theorem deals with the removability of singularities for bounded harmonic functions
• It states that if a harmonic function is bounded near an isolated singularity, then the singularity is removable
• Example: If $u(x, y)$ is a bounded harmonic function near $(0, 0)$, then $(0, 0)$ is a removable singularity of $u(x, y)$

Picard's great theorem

• Picard's great theorem is a stronger result that relates the behavior of a function near an essential singularity to its range
• It states that in any neighborhood of an essential singularity, a non-constant analytic function takes on all possible complex values, with at most one exception
• Example: The function $f(z) = e^{\frac{1}{z}}$ has an essential singularity at $z = 0$ and takes on all complex values, except possibly one, in any neighborhood of 0

Examples of removable singularities

• Studying examples of removable singularities helps develop intuition and problem-solving skills in potential theory
• Common examples include rational functions, exponential and logarithmic functions, and trigonometric functions

Rational functions

• Rational functions, which are quotients of polynomials, often have removable singularities when factors cancel out
• These singularities can be removed by simplifying the function and redefining its value at the singular point
• Example: $f(z) = \frac{z^2 - 4}{z - 2}$ has a removable singularity at $z = 2$, which can be removed by simplifying to $f(z) = z + 2$ and defining $f(2) = 4$

Exponential and logarithmic functions

• Exponential and logarithmic functions may have removable singularities when combined with other functions or when their arguments have singularities
• Removing these singularities often involves using properties of exponents and logarithms or applying L'Hôpital's rule
• Example: $f(z) = z\log(z)$ has a removable singularity at $z = 0$, which can be removed by defining $f(0) = 0$

Trigonometric functions

• Trigonometric functions, such as sine and cosine, can have removable singularities when divided by appropriate factors
• These singularities can be removed using trigonometric identities or by analyzing the function's limit at the singular point
• Example: $f(z) = \frac{\sin(z)}{z}$ has a removable singularity at $z = 0$, which can be removed by defining $f(0) = 1$

Techniques for handling removable singularities

• Several techniques are used to analyze and handle removable singularities in potential theory
• These techniques include Laurent series expansions, residue calculus, and conformal mappings

Laurent series expansions

• Laurent series are a generalization of Taylor series that allow for the representation of functions with singularities
• Expanding a function into its Laurent series near a singularity can help identify the type of singularity and provide a means for removing it
• Example: For $f(z) = \frac{z + 1}{z(z - 1)}$, the Laurent series expansion around $z = 0$ is $f(z) = \frac{1}{z} + 1 + z + z^2 + \cdots$, revealing a simple pole at $z = 0$

Residue calculus

• Residue calculus is a powerful tool in complex analysis that relates the integral of a function around a closed contour to the residues of its singularities inside the contour
• Computing residues at removable singularities can help evaluate integrals and analyze the function's behavior
• Example: For $f(z) = \frac{z^2 - 1}{z - 1}$, the residue at the removable singularity $z = 1$ is $\lim_{z \to 1} (z - 1)f(z) = 2$

Conformal mappings

• Conformal mappings are angle-preserving transformations that can be used to simplify the geometry of a problem or to remove singularities
• By mapping a region with a removable singularity to another region where the singularity is absent, the function can be analyzed more easily
• Example: The Joukowski transformation $f(z) = \frac{1}{2}(z + \frac{1}{z})$ maps the exterior of the unit disk to the exterior of an airfoil, removing the singularity at infinity