Question

Determine the types of singularities (if any) possessed by the following functions at \(z=0\) and \(z=\infty\) (a) \((z-2)^{-1}\) (b) \(\left(1+z^{3}\right) / z^{2}\) (c) \(\sinh (1 / z)\), (d) \(e^{z} / z^{3}\) (e) \(z^{1 / 2} /\left(1+z^{2}\right)^{1 / 2}\)

Short answer

At \(z=0\) and \(z=\infty\), respectively: (a) no singularity, no singularity; (b) pole (order 2), essential; (c) essential, no singularity; (d) pole (order 3), essential; (e) branch point, no singularity.

Step-by-step solution

- Determine singularities for \((z-2)^{-1}\)

Identify where the function \((z-2)^{-1}\) becomes undefined. The function is undefined at \(z=2\). At \(z=0\) and \(z=\infty\), the function is well-defined and regular.

- Determine singularities for \((1+z^{3}) / z^{2}\)

Identify where \((1+z^{3}) / z^{2}\) becomes undefined. The function is undefined at \(z=0\) with a pole of order 2 (since \(z^{2}\) in the denominator). At \(z=\infty\), since \z^{3} / z^{2} = z\, there's an essential singularity.

- Determine singularities for \(\sinh (1 / z)\)

The function \(\sinh (1 / z)\) is undefined at \(z=0\), resulting in an essential singularity. At \(z=\infty\), the behavior is regular as \(1 / z\) tends towards zero.

- Determine singularities for \(e^{z} / z^{3}\)

The function \(e^{z} / z^{3}\) is undefined at \(z=0\), having a pole of order 3 due to the \z^{3}\ in the denominator. At \(z=\infty\), \(e^{z}\) grows faster than any polynomial, indicating an essential singularity.

- Determine singularities for \(z^{1 / 2} / (1+z^{2})^{1 / 2}\)

The function \(z^{1 / 2} / (1+z^{2})^{1 / 2}\) is undefined at \(z=0\), as \(z^{1/2}\) is not analytic at 0, indicating a branch point. At \(z=\infty\), the function behaves regularly.

Key concepts

Types of Singularities

In complex analysis, a singularity is a point where a function is not defined or not analytic. There are several types of singularities:

• Poles: Points where the function goes to infinity in a specific manner.
• Essential Singularities: Points where the function exhibits chaotic behavior.
• Branch Points: Points where the function takes on multiple values.

Singularities can greatly affect the behavior of complex functions and are crucial in the study and applications of complex analysis.

Poles in Complex Functions

A pole is a type of singularity where a complex function takes an infinite value at a certain point.
Poles are characterized by the order of the pole, which describes how fast the function goes to infinity.
For example, the function \((z-2)^{-1}\) has a pole of order 1 at \(z=2\). This is because the denominator becomes zero, causing the function to blow up.
In contrast, the function \((1+z^{3}) / z^{2}\) has a pole of order 2 at \(z=0\). The denominator \(z^{2}\) causes the function to go to infinity at a rate proportional to \(1 / z^{2}\).
Poles can be isolated and are important in residue calculus and in evaluating complex integrals.

Essential Singularities

An essential singularity is a point at which the function behaves erratically.
Unlike poles, essential singularities do not have a well-defined order. Instead, the function can exhibit wild and unpredictable behavior near the singularity.
For instance, the function \(\text{sinh}(1 / z)\) has an essential singularity at \(z=0\). As \(z\) approaches 0, \(1 / z\) can take any value, leading to erratic behavior.
Similarly, \(e^{z} / z^{3}\) has an essential singularity at \(z=\bold{\frac {1 \text{or}} \text{ is.very .big}}\). \(e^{z}\) grows faster than any polynomial, causing unpredictable behavior at large \(z\).
Understanding essential singularities is crucial in complex analysis as they often indicate complex dynamic systems.

Branch Points

Branch points are singularities where a function takes on multiple values.
These points are closely related to multi-valued functions, which are functions that can give more than one value for the same input.
For instance, the function \(\finsqrt (){z^{1 / 2} }}})( 1+\textstyle{}\) has branch points at \({0,{+z2}}\). The square root function is multi-valued and not analytic at 0, thus creating a branch point.
Branch points are important in understanding complex planes and Riemann surfaces, which provide a way to visualize multi-valued functions by spreading them out over several layers or sheets.