Problem 26 - Advanced Engineering Mathematics

Question

Show that \(z=0\) is an essental singularity of \(f(z)=z^{3} \sin (1 / z)\).

Step-by-Step Solution

Answer

The point \( z = 0 \) is an essential singularity of \( f(z) = z^3 \sin(1/z) \).

Step 1: Identifying the Singular Point

The function given is \( f(z) = z^3 \sin\left(\frac{1}{z}\right)\). We want to analyze the behavior of \( f(z) \) near \( z = 0 \). Since the term \( \sin\left(\frac{1}{z}\right) \) involves \( \frac{1}{z} \), \( z = 0 \) is a potential point of singularity for the function.

Step 2: Checking for Removable Singularity

To check if \( z = 0 \) is a removable singularity, we need to see if \( \lim_{z \to 0} f(z) \) exists and is finite. Calculating:\[\lim_{z \to 0} z^3 \sin\left(\frac{1}{z}\right) = 0 \cdot \lim_{z \to 0} \sin\left(\frac{1}{z}\right) = 0\]Since the term \( \sin\left(\frac{1}{z}\right) \) oscillates infinitely between -1 and 1 as \( z \to 0 \), the product goes to 0. However, since \( \sin\left(\frac{1}{z}\right) \) is oscillating, the limit does not exist definitively.

Step 3: Determining if it's a Pole

Next, we determine if \( z = 0 \) is a pole. A point is a pole if \( f(z) \) behaves like \( \frac{A}{(z - z_0)^n} \) as \( z \to z_0 \). Since \( \sin\left(\frac{1}{z}\right) \) oscillates widely and does not become asymptotically \(1/z^n\), \( z = 0 \) is not a pole.

Step 4: Establishing Essential Singularity

A point is an essential singularity if neither removable nor pole conditions are satisfied, and if the behavior near the point is chaotic or highly oscillatory. The term \( \sin\left(\frac{1}{z}\right) \) causes \( f(z) \) to exhibit chaotic behavior as \( z \to 0 \). Hence, by definition, the function \( f(z) \) has an essential singularity at \( z = 0 \).

Key Concepts

Essential Singularity

In complex analysis, an essential singularity is a fascinating and complex type of singular point. Unlike removable singularities or poles, essential singularities create a dramatically unpredictable behavior in functions. For a function to have an essential singularity at a point, neither the limit as you approach the point should exist, nor should it exhibit the predictable blowing-up behavior typical of a pole. Instead, the function behaves in a chaotic or highly oscillatory manner as it approaches the point.
After analyzing the function given in the exercise, we can conclude that the function \( f(z) = z^3 \sin\left(\frac{1}{z}\right) \) has an essential singularity at \( z=0 \). This conclusion arises because the function does not satisfy the conditions for a removable singularity or a pole. Instead, it exhibits chaotic behavior due to the oscillations of \( \sin\left(\frac{1}{z}\right) \) as \( z \to 0 \).
The famous Casorati-Weierstrass theorem helps us understand essential singularities further. It states that near an essential singularity \( z_0 \), the values of the function \( f(z) \) are dense in the complex plane. This means for any complex number and any small neighborhood around \( z_0 \), \( f(z) \) will come arbitrarily close to any point in the complex plane, making the function's behavior extremely unpredictable.

Singular Points

Singular points in complex analysis are key to understanding the behavior of functions under various conditions. These points are where a function ceases to be analytic. In other words, the function faces some disruption at these points. Singular points are categorized mainly into three types: removable singularities, poles, and essential singularities.

• **Removable singularity**: A point where the function is not initially defined but can be redefined so that the function continues to be analytic at that point.
• **Pole**: A point where the function behaves like \( \frac{A}{(z-z_0)^n} \) as \( z \to z_0 \), causing the function to "blow up" to infinity.
• **Essential singularity**: A point where neither of the above conditions is satisfied. Instead, the function shows unpredictable and chaotic behavior.

In our case, \( z=0 \) quickly draws attention as a singular point for the given function. The involvement of \( \sin\left(\frac{1}{z}\right) \) indicates potentially complex behavior as \( z \rightarrow 0 \).
Understanding these categories and identifying which type of singularity a function possesses is crucial in deeply grasping complex functions' nature and how they will act in specific regions of the complex plane.

Oscillatory Behavior

Oscillatory behavior refers to the rapid fluctuations a function may exhibit as it approaches a particular point, especially in the context of complex functions. In complex analysis, such behavior often appears around essential singularities. This is because functions can swing widely between values without settling towards a stable limit.
Consider the function from the exercise, \( f(z) = z^3 \sin\left(\frac{1}{z}\right) \), to appreciate this behavior. As \( z \to 0 \), the term \( \sin\left(\frac{1}{z}\right) \) oscillates infinitely between -1 and 1. This means as close as \( z \) may get to zero, \( \sin\left(\frac{1}{z}\right) \)'s rapid oscillations induce a wide range of values, leading the function to behave erratically.
Such oscillatory behavior makes essential singularities unique. The value of \( f(z) \) does not converge to a single point or diverge to infinity; instead, it covers many values repetitively. This chaotic pattern showcases why \( z = 0 \) is an essential singularity for the given function. These oscillations prevent the function's limit from existing and explain why the function is not defined as analytic at that point.