Problem 25
Question
Determine whether \(z=0\) is an isolated or nonisolated singularity of \(f(z)=\tan (1 / z)\)
Step-by-Step Solution
Verified Answer
The point \( z=0 \) is a nonisolated singularity of \( f(z)=\tan(1/z) \).
1Step 1: Understand the Problem
We need to determine if the point at \( z = 0 \) is an isolated or nonisolated singularity for the function \( f(z) = \tan(1/z) \). This involves examining the behavior of the function near \( z = 0 \). Remember that a singularity is isolated if there exists a neighborhood around the point where \( f(z) \) is analytic except possibly at the singularity itself.
2Step 2: Identify Tangent Singularities
The function \( an(z) \) is known to have singularities at \( z = (2k+1)\pi/2 \), where \( k \) is an integer. Transferring this idea to \( f(z) = \tan(1/z) \), singularities will occur where \( 1/z = (2k+1)\pi/2 \). Thus, \( z = 2/((2k+1)\pi) \) are the singular points of \( f(z) \).
3Step 3: Examine Singularity Density
As \( k \) increases or decreases without bound, the points \( z_k = 2/((2k+1)\pi) \) become arbitrarily close to \( z = 0 \). Therefore, \( z = 0 \) is not an isolated singularity, as there exists an accumulation of other singularities arbitrarily close to \( z = 0 \).
4Step 4: Conclusion
Based on the density of singularities around \( z = 0 \), \( f(z) = \tan(1/z) \) does not satisfy the criteria for an isolated singularity at \( z = 0 \). Instead, \( z = 0 \) is a nonisolated singularity.
Key Concepts
SingularitiesIsolated SingularityNonisolated SingularityAnalytic Functions
Singularities
In complex analysis, singularities are special points where an analytic function ceases to be well-defined or "nice". A function might explode to infinity, become undefined, or exhibit some irregular behavior at these points. Singularities are particularly interesting since they reveal much about the function's overall structure and behavior.
When analyzing a function, look beyond just the point of singularity. You should consider how the function behaves in the neighborhood of the singularity. This insight helps determine the type of singularity. It could be removable, essential, or a pole, each representing different types of irregularities:
When analyzing a function, look beyond just the point of singularity. You should consider how the function behaves in the neighborhood of the singularity. This insight helps determine the type of singularity. It could be removable, essential, or a pole, each representing different types of irregularities:
- Removable singularities: Points where a function could be redefined to become analytic.
- Poles: Points where a function goes to infinity in a particular manner.
- Essential singularities: Points exhibiting chaotic behavior, such that no redefining could rectify it into analyticity.
Isolated Singularity
An isolated singularity is a specifically intriguing type of singularity in complex functions. It is a point where a function is not analytic, but remains analytic in some neighborhood around the point, except at the singularity itself.
Isolated singularities are important because they allow for certain kinds of analysis and manipulation, such as calculating residues in the residue theorem. To determine if a singularity is isolated, check to see if there is a disk centered around the singularity, within which the function is analytic everywhere except the center. If such a neighborhood can be defined, you have an isolated singularity.
Isolated singularities are important because they allow for certain kinds of analysis and manipulation, such as calculating residues in the residue theorem. To determine if a singularity is isolated, check to see if there is a disk centered around the singularity, within which the function is analytic everywhere except the center. If such a neighborhood can be defined, you have an isolated singularity.
Nonisolated Singularity
Nonisolated singularities occur in scenarios where singularities accumulate densely around a point, making it impossible to isolate a single one. This means, unlike isolated singularities, you can't enclose this kind of singularity within a neighborhood where analytic behavior is restored.
A perfect example is the study of the function \( f(z) = \tan(1/z) \), where the singularities cluster increasingly close to \( z = 0 \). As we saw in the problem solution, these points, represented as \( z_k = 2/((2k+1)\pi) \), become densely packed near \( z = 0 \). This behavior prevents \( z = 0 \) from having a clear neighborhood of analytic points.
Such nonisolated cases often imply a more complicated structure of the function near that point. They indicate a degree of analytical complexity that can grow as you 'zoom' into the point.
A perfect example is the study of the function \( f(z) = \tan(1/z) \), where the singularities cluster increasingly close to \( z = 0 \). As we saw in the problem solution, these points, represented as \( z_k = 2/((2k+1)\pi) \), become densely packed near \( z = 0 \). This behavior prevents \( z = 0 \) from having a clear neighborhood of analytic points.
Such nonisolated cases often imply a more complicated structure of the function near that point. They indicate a degree of analytical complexity that can grow as you 'zoom' into the point.
Analytic Functions
In the realm of complex analysis, analytic functions, also known as holomorphic functions, are those that have derivatives at every point in their domain. This is akin to them being "smooth" in behavior, continuously differentiable, and free from abrupt changes.
Analyticity is a powerful property because of rules like Cauchy’s integral theorem, which simplifies integrations and calculus manipulations greatly. For a function \( f(z) \) to be considered analytic at a specific point \( z_0 \), it must have a well-defined and finite derivative at that point and in its vicinity.
Consideration of analytic functions leads to the investigation of singularities. Singularities disrupt analyticity, but within an isolated range, you might find restorative solutions. Thus, studying these functions provides insights into broader mathematical phenomena in complex spaces.
Analyticity is a powerful property because of rules like Cauchy’s integral theorem, which simplifies integrations and calculus manipulations greatly. For a function \( f(z) \) to be considered analytic at a specific point \( z_0 \), it must have a well-defined and finite derivative at that point and in its vicinity.
Consideration of analytic functions leads to the investigation of singularities. Singularities disrupt analyticity, but within an isolated range, you might find restorative solutions. Thus, studying these functions provides insights into broader mathematical phenomena in complex spaces.
Other exercises in this chapter
Problem 25
Evaluate the Cauchy principal value of the given improper integral. \(\int_{0}^{\infty} \frac{\cos 3 x}{\left(x^{2}+1\right)^{2}} d x\)
View solution Problem 25
Use Cauchy's residue theorem to evaluate the given integral along the indicated contour. \(\oint_{C} \frac{z e^{z}}{z^{2}-1} d z, C:|z|=2\)
View solution Problem 25
Expand \(f(z)=\frac{7 z-3}{z(z-1)}\) in a Laurent series valid for the indicated annular domain. \(0
View solution Problem 25
In Problems 11-30, evaluate the Cauchy principal value of the given improper integral. $$ \int_{0}^{\infty} \frac{\cos 3 x}{\left(x^{2}+1\right)^{2}} d x $$
View solution