Question

Say whether the indicated point is regular, an essential singularity, or a pole, and if a pole of what order it is. (a) \(\frac{\sin z}{z}, \quad z=0\) (b) \(\frac{\cos z}{z^{3}}, \quad z=0\) (c) \(\frac{z^{3}-1}{(z-1)^{3}}, \quad z=1\) (d) \(\frac{e^{z}}{z-1}, \quad z=1\)

Short answer

(a) Regular Point. (b) Pole of order 3. (c) Pole of order 2,\quad (d) Simple Pole (order 1).

Step-by-step solution

Identify Singularities

Determine if the given point is a singularity by checking if the denominator equals zero at that point for the given functions.

Analyze Each Function Separately

Evaluate each function at the indicated point and determine whether it is a pole, an essential singularity, or a regular point.

Case (a): \frac{\sin z}{z}, z=0

Since \(\sin z\) is analytic at \(z=0\), and \(\sin 0 = 0\), this is a removable singularity. By defining \(f(z) = \sin z / z, \quad f(0) = 1\), the point \(z=0\) becomes a regular point.

Case (b): \frac{\cos z}{z^{3}}, z=0

The function \(\cos z\) is analytic and \(\cos 0 = 1\). Thus, the function becomes \(\frac{1}{z^{3}}\) at \(z=0\), which indicates a pole of order 3.

Case (c): \frac{z^{3}-1}{(z-1)^{3}}, z=1

Evaluate the numerator and denominator at \(z=1\). The numerator \((z^3 - 1)\) has a zero at \(z=1\) making \(\lim_{z \to 1} \frac{(z-1)(something)}{(z-1)^3} = \frac{constant}{(z-1)^2}\). So there is a pole of order 2.

Case (d): \frac{e^{z}}{z-1}, z=1

The function \(e^z\) is analytic and non-zero at \(z=1\). Thus, the function becomes \(\frac{e}{z-1}\) at \(z=1\), which indicates a simple pole (order 1).

Key concepts

Singularity

A singularity in a complex function is a point where the function does not behave normally, meaning it is not analytic or cannot be defined in a standard way.
These are critical points where the behavior of a function changes dramatically.
There are different types of singularities, including removable singularities, poles, and essential singularities.

To identify singularities, we often look for points where the denominator of a function's expression equals zero.
This step helps us understand how the function behaves near that point.

Pole

A pole is a type of singularity where the function approaches infinity as the variable approaches the singular point.
Poles can be of different orders (e.g., first order, second order), depending on how the function behaves near the singularity.
A function has a pole of order n at point a if it can be written as \( f(z) = \frac{g(z)}{(z-a)^n} \), where \( g(z) \) is analytic and non-zero at \( z=a \).

For example, consider the function \( \frac{1}{z^3} \).
At \( z=0 \), this function has a pole of order 3 because it becomes unbounded as z gets closer to 0.
In the exercise above, the function \( \frac{\cos z}{z^3}\) at \( z=0 \) has a third-order pole.

Essential Singularity

An essential singularity is a type of singularity where the function behaves in a wild and unpredictable way near the singular point.
Unlike poles, the function neither goes to infinity nor acts in a removable manner.
Instead, it oscillates between values without settling.According to Picard's theorem, in the neighborhood of an essential singularity, the function takes on almost all possible complex values, infinitely often.

A classic example is the function \( e^{\frac{1}{z}} \) at \( z=0 \).
This type of singularity results in values that cover the entire complex plane, except possibly one value.

Mathematical Functions

In complex analysis, functions are mappings from one set of complex numbers to another.
These functions need to be analytic (differentiable) for them to be well-behaved.
Any irregular behavior in a complex function is often due to singularities.

Common functions in complex analysis include trigonometric functions like \( \sin z \) and \( \cos z \), exponential functions like \( e^z \), and rational functions that are ratios of polynomials.
Different functions are used for various purposes, but singularities dictate how we understand and manipulate these functions in practice.

For instance, in the given exercise, we deal with rational functions such as \( \frac{e^z}{z-1} \) and trigonometric functions like \( \frac{\sin z}{z} \), each of which behaves differently near their respective points of interest.