Problem 16 - A First Course in Complex Analysis with Applications

Question

Determine the order of the poles for the given function. $$ f(z)=5-\frac{6}{z^{2}} $$

Step-by-Step Solution

Answer

The pole at \(z = 0\) is of order 2.

Step 1: Identify the Terms in the Function

The function given is \(f(z) = 5 - \frac{6}{z^2}\). There are two parts to this function: the constant \(5\) and the term \(-\frac{6}{z^2}\). The pole arises from the latter term \(-\frac{6}{z^2}\).

Step 2: Identify the Denominator and Potential Singularities

Examine the term \(-\frac{6}{z^2}\). The denominator is \(z^2\). A pole occurs where the denominator is zero, i.e., at \(z = 0\).

Step 3: Determine the Order of the Pole

A pole is of order \(n\) if \((z - a)^n\) divides the denominator and \((z - a)^{n+1}\) does not. Here, \(\frac{6}{z^2}\) indicates the denominator is \(z^2\), leading to a singularity when \(z = 0\). Thus, the pole at \(z = 0\) is of order \(2\), as the denominator is \(z^2 = (z-0)^2\).

Key Concepts

Poles of a Function

In complex analysis, poles of a function are special types of singularities that highlight fascinating aspects of a function's behavior. Poles occur specifically where the function "blows up" or becomes infinite. For a given function expressed as a fraction, a pole manifests where the denominator is zero while the numerator is not. This can lead to the function approaching infinity at that specific point.

For example, consider a function \( f(z) = \frac{6}{z^2} \). Here, the denominator \( z^2 \) equals zero when \( z = 0 \). Therefore, at \( z = 0 \),\( f(z) \) becomes undefined, creating what is called a pole. This behavior is crucial as it outlines the points where the function departs from normality and exhibits extraordinary behavior, such as heading towards infinity.

Poles allow for deeper insights:

• They identify critical points that define the function's boundary behavior.
• They are pivotal in performing complex function analysis and residue calculations.
• Poles are distinct from zeros, as zeros occur when the numerator is zero while the denominator is non-zero.

Order of a Pole

The order of a pole gives additional insight into the severity of the singularity. This concept refers to how many times the denominator's variable, raised to a power, contributes to the pole. If \((z - a)^n\) divides fully into the denominator but \((z - a)^{n+1}\) does not, then the function is said to have a pole of order \(n\) at \(z = a\).

Take, for example, the function \( f(z) = \frac{6}{z^2} \). The crucial observation here is that the denominator \(z^2\) can be rewritten as \((z - 0)^2\). Since \((z-0)^3\) does not divide into the denominator without remainder, the function has a pole of order 2 at \(z=0\).

The order of pole provides meaningful insights:

• Higher order poles indicate a more severe singularity.
• Understanding the order allows for precise residue calculations in complex integration.
• It aids in determining the behavior of the function near the singularity, thus essential for analyzing function limits.

Singularities in Complex Functions

Singularities in complex functions represent points where the function does not behave in its usual manner. These can be thought of points of "non-normal" behavior, such as becoming infinite, undefined, or oscillating wildly. Recognizing and classifying singularities form a key part of complex analysis.

Singularities can be categorized several ways, but the main types include:

• Removable Singularities: If a singularity can be "removed" by redefining the function at that point, it is removable.
• Poles: Points where the function approaches infinity; these are described in detail via their order.
• Essential Singularities: These create dramatic changes, where the behavior of the function is chaotic and not confined to just tending towards infinity or zero.

Understanding singularities enables mathematicians to map a function's properties more completely and accurately. They are instrumental in determining integration paths, calculating the residues of contour integrals and evaluating limits.