Problem 13 - Advanced Engineering Mathematics

Question

Determine the order of the poles for the given function. \(f(z)=\frac{3 z-1}{z^{2}+2 z+5}\)

Step-by-Step Solution

Answer

The function has two simple poles at \(-1 + 2i\) and \(-1 - 2i\) with an order of 1 each.

Step 1: Identify potential poles

Poles of a function occur where the denominator is equal to zero. Thus, to find the potential poles of the function, we need to solve the equation: \[ z^2 + 2z + 5 = 0. \]

Step 2: Use quadratic formula

The quadratic formula \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is used to solve for \( z \) where \( a = 1, b = 2, \) and \( c = 5 \). Substituting these values gives:\[ z = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1} = \frac{-2 \pm \sqrt{4 - 20}}{2}. \]

Step 3: Calculate the discriminant

Calculate the discriminant \( 4 - 20 = -16 \). Since the discriminant is negative, there are no real roots, and the denominator has complex roots.

Step 4: Find complex roots

The roots are complex and can be found as follows:\[ z = \frac{-2 \pm \sqrt{-16}}{2} = \frac{-2 \pm 4i}{2}. \]Simplifying gives the roots: \[ z = -1 + 2i \quad \text{and} \quad z = -1 - 2i. \] These are the potential poles of the function.

Step 5: Determine the order of the poles

For each pole, determine the order by checking the multiplicity of the factor causing the zero in the denominator. Since each root has a multiplicity of 1 (the exponent of each factor is 1), each pole \(-1 + 2i\) and \(-1 - 2i\) has an order of 1.

Key Concepts

Poles of a Function

In complex analysis, a **pole** refers to a certain kind of singularity in a complex-valued function. Simply put, poles occur at points where a function grows to infinity. For instance, if you have a function like \( f(z) = \frac{1}{(z - z_0)^n} \), the function reaches infinity at \( z = z_0 \). This location is called a **pole**.
Poles are important because they help us understand the behavior of functions, especially when evaluating integrals along complex paths.

Key aspects to consider about poles:

• **Order:** The order of a pole is determined by the highest power of \((z - z_0)\) in the denominator.
• **Simple Pole:** If the order is 1, it is called a simple pole.
• **Location:** Calculating exactly where a function has poles usually involves setting the denominator equal to zero and solving for those exact values.

In the provided solution, the poles were found after solving the quadratic equation in the denominator, which helped locate these specific points where the function is undefined.

Quadratic Formula

The **quadratic formula** is a crucial tool in algebra, especially when dealing with complex functions like the one in the original exercise.
The formula used is:\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula helps you find the roots of a quadratic equation of the form \( az^2 + bz + c = 0 \).

Here’s how it helps:

• **Identifies Roots:** It directly provides the roots of the quadratic equation, which are essential in locating the poles of a function.
• **Works with Complex Numbers:** Through the discriminant \( b^2 - 4ac \), the formula tells us about the nature of the roots — whether they are real or complex.
• **Systematic Approach:** This structured method ensures you find all roots, essential for understanding a function's behavior.

In this exercise, substituting the values \( a = 1, b = 2, \) and \( c = 5 \), allowed us to identify that the roots are complex, revealing poles in the function.

Complex Roots

**Complex roots** emerge when the discriminant of a quadratic equation \( b^2 - 4ac \) is negative.
In such cases, no real solutions exist, hence the roots are expressed with imaginary terms involving \( i \), where \( i = \sqrt{-1} \).

Understanding complex roots involves:

• **Visualization:** In the complex plane, complex roots appear as conjugate pairs. For example, \( z = -1 + 2i \) and \( z = -1 - 2i \).
• **Application:** In processes like finding poles, these roots reveal positions where a function is not defined.
• **Reliance on Quadratic Formula:** The roots are calculated using the quadratic formula, which naturally incorporates imaginary components when needed.

From the exercise, the roots \(-1 + 2i\) and \(-1 - 2i\) indicate complex poles, pivotal in analyzing function behavior in complex analysis.

Multiplicity of Roots

The **multiplicity of roots** refers to how many times a particular root is repeated as a solution of a polynomial equation.
This concept aids in assessing the behavior and properties of polynomial functions, critical when examining poles.

Key points about root multiplicity:

• **Single Root:** If a root occurs once, it has a multiplicity of 1, often referred to as a simple root.
• **Repeated Roots:** If a root appears \( n \) times in the factorization of a polynomial, its multiplicity is \( n \).
• **Impact on Function:** Higher multiplicities can influence how a function approaches infinity or zero near the root.

In the initial problem, both roots, \(-1 + 2i\) and \(-1 - 2i\), have a multiplicity of 1. This indicates each is a simple pole, crucial in deducing the function’s behavior at these points.