Problem 18 - Advanced Engineering Mathematics

Question

In Problems 13-24, determine the order of the poles for the given function. $$ f(z)=\frac{\cot \pi z}{z^{2}} $$

Step-by-Step Solution

Answer

Poles are of order 1 at \( z = n, n \neq 0 \) and order 3 at \( z = 0 \).

Step 1: Identify potential poles of the function

The function is given by \( f(z) = \frac{\cot \pi z}{z^2} \). The cotangent function, \( \cot \pi z = \frac{\cos \pi z}{\sin \pi z} \), has poles where \( \sin \pi z = 0 \). This occurs at \( z = n \), where \( n \) is an integer. The function \( \frac{1}{z^2} \) also has a pole of order 2 at \( z = 0 \). Thus, we must consider both types of poles.

Step 2: Determine the order of poles at non-zero integers

For \( z = n \) where \( n eq 0 \) and \( n \in \mathbb{Z} \), the pole of \( \cot \pi z \) is of order 1, because \( \cot \pi z \) behaves like \( \frac{1}{\pi(z-n)} \) near \( z = n \). At these points, \( \frac{1}{z^2} \) is analytic and non-zero, contributing no poles themselves. Thus, the order of the pole at these points is 1.

Step 3: Analyze the pole at \( z = 0 \)

At \( z = 0 \), we have overlapping poles from both functions in the composition: the pole of order 1 from \( \cot \pi z \) and of order 2 from \( \frac{1}{z^2} \). To find the order of the pole of \( f(z) \) at \( z = 0 \), add the orders (1 from \( \cot \pi z \) and 2 from \( \frac{1}{z^2} \)), yielding a total of 3.

Key Concepts

Poles of a function

Poles of a function are points in the complex plane where the function's value becomes unbounded. These are the points where a function tends to infinity. Understanding poles is crucial in complex analysis because they reveal much about the behavior and characteristics of functions.

There are a few key things to know about poles:

• Locations: Identifying where a function has poles involves solving for points where the function is undefined. For example, if a function has a denominator, the poles are where this denominator equals zero.
• Nature: Poles can be simple or involve higher order, which influences how the function behaves near these points.
• Role in Integration: Poles are critical when evaluating complex integrals as they determine the residues contributing to the final integral results via the residue theorem.

In the function provided, \(f(z) = \frac{\cot \pi z}{z^2}\), there are poles at any integer \(n\) where \(\sin \pi z = 0\), and an additional pole from the \(\frac{1}{z^2}\) term at \(z = 0\). These points are crucial for analyzing where the function ceases to be finite.

Order of poles

The order of a pole is an important concept that describes the severity of a pole. It tells you how rapidly the function tends toward infinity at that particular point. The order is defined as the smallest positive integer \(m\) such that the limit of \((z-z_0)^m f(z)\) exists and is finite as \(z\) approaches \(z_0\).

Here's how you can determine the order of a pole:

• Simple Pole (Order 1): A pole is considered simple if the function resembles \(\frac{1}{z-z_0}\) near the pole.
• Higher-Order Poles: These occur when the function can be locally expressed as \(\frac{C}{(z-z_0)^n}\) for \(n > 1\), indicating a stronger tendency towards infinity.

In the function \(f(z) = \frac{\cot \pi z}{z^2}\), at \(z = n\) (where \(n\) is a non-zero integer), each pole is of order 1, as \(\cot \pi z\) behaves like \(\frac{1}{\pi(z-n)}\). At \(z = 0\), overlapping poles exist, with a total order of 3, formed by combining the orders from \(\cot \pi z\) and \(\frac{1}{z^2}\).

Cotangent function

The cotangent function \(\cot z\) is a fundamental trigonometric and complex function, playing a significant role in mathematics. In terms of sine and cosine, it is expressed as \(\cot z = \frac{\cos z}{\sin z}\), highlighting its undefined nature wherever \(\sin z = 0\).

Some characteristics of \(\cot z\) include:

• Periodic Nature: \(\cot z\) is periodic with a period of \(\pi\), meaning it repeats its values over intervals of length \(\pi\).
• Pole Locations: In the complex plane, the poles of \(\cot \pi z\) are found at any integer \(z = n\) because \(\sin \pi n = 0\).
• Behavior Near Poles: Near its poles, \(\cot \pi z\) behaves like \(\frac{1}{\pi (z-n)}\), indicating it has simple poles of order 1.

In the given function \(f(z) = \frac{\cot \pi z}{z^2}\), the cotangent's poles at integer values contribute to understanding how the function behaves in those regions, allowing for complex integration and residue calculations.

Analytic functions

Analytic functions are an essential class of functions in complex analysis, characterized by their smoothness and differentiability. A function is analytic if it is differentiable at every point in its domain and its Taylor series converges to the function in some neighborhood around every point.

Key points about analytic functions include:

• Holomorphic: Analytic functions are often referred to as holomorphic when defined over a complex plane region.
• Series Representation: Any analytic function can be expressed as a power series, which aids in understanding its behavior.
• Convention: Analytic functions lack poles, but if a function can be split into a quotient where the denominator vanishes, it may still possess poles locally.

For \(f(z) = \frac{\cot \pi z}{z^2}\), outside of its poles, this function can be considered analytic. Understanding its analytic nature helps manage integration tasks and ensures that, away from poles, the function behaves smoothly across the complex plane.