Problem 15 - Advanced Engineering Mathematics

Question

Determine the order of the poles for the given function. \(f(z)=\frac{1+4 i}{(z+2)(z+i)^{4}}\)

Step-by-Step Solution

Answer

Pole at \(z = -2\) is order 1; pole at \(z = -i\) is order 4.

Step 1: Identify the Function's Denominator

The function given is \( f(z) = \frac{1+4i}{(z+2)(z+i)^4} \). The poles of the function correspond to the values of \( z \) where the denominator becomes zero. Hence, the poles are determined from the denominator \((z+2)(z+i)^4\).

Step 2: Determine the Points of Singularities

Set the denominator equal to zero: \[(z+2)(z+i)^4 = 0\]Thus, the values that make the expression zero are the singularities, \(z = -2\) and \(z = -i\).

Step 3: Identify the Order of Each Pole

The term \((z+2)\) contributes a simple pole at \(z = -2\) because its power is 1. The term \((z+i)^4\) contributes a pole of order 4 at \(z = -i\) because its power is 4.

Step 4: Compilation of Results

Therefore, the function has a simple pole of order 1 at \(z = -2\) and a pole of order 4 at \(z = -i\).

Key Concepts

Understanding Poles in Complex Analysis

In complex analysis, poles are special points in the complex plane where a complex function becomes undefined. A function has a pole at a particular point if it can be expressed in such a way that the function goes to infinity as it approaches that point. Poles are an essential part of understanding how a function behaves.

• A pole can be visualized as a kind of "spike" or "hole" in the graph of a complex function, where the function's value shoots off to infinity.
• Understanding where these points are helps us better analyze the character and boundary behavior of complex functions.

For example, in the given function, poles occur where the denominator is zero, such as at points \(z = -2\) and \(z = -i\). Knowing the location of the poles helps us identify potential issues when evaluating the function and predict where it might be undefined.

The Role of Singularities

Singularities in a complex function are the points where the function fails to be analytic, i.e., it cannot be represented as a convergent power series. There are different types of singularities, including poles and essential singularities.

• Singularities are crucial for understanding the domain of a function, defining how and where a function can operate.
• Poles are a type of singularity where the function behaves predictably, yet non-analytically, typically involving a division by zero.

In the context of the provided function, finding the singularities involved identifying the values of \(z\) that make the denominator zero, such as \(z=-2\) and \(z=-i\). Recognizing these values allows us to grasp the essential properties of the complex function and understand its limits.

Determining the Order of Poles

The order of a pole of a complex function is the number that indicates how powerful the singularity is at that point. It measures how fast the function grows to infinity as the variable approaches the pole.

• The concept of the order of a pole is related to the power of the term in the denominator that causes the pole.
• A simple pole has an order of 1, meaning the function goes to infinity in a linear manner with respect to its variable.
• Higher-order poles, such as order 4, indicate a steeper, more intense spike as the function approaches the point of singularity.

In the solution, we identified that \(z=-2\) is a simple pole with an order of 1. In contrast, \(z=-i\) corresponds to a pole of order 4, due to the term \((z+i)^4\). This tells us how the function uniquely grows around these singularities.

Denominator of a Complex Function

In complex functions, the denominator holds a significant role—it determines the location and nature of the poles. Where the denominator equals zero, the function will have a pole, assuming the numerator is non-zero at those points.

• The expression of the denominator provides insights into singularities and can guide how the function is simplified or factored.
• To find poles, we focus on the critical part of the denominator where the terms equal zero.

For instance, in the function \( f(z) = \frac{1+4i}{(z+2)(z+i)^4} \), we examined \((z+2)(z+i)^4\) to discover values that zero the denominator. This examination shows the critical role of the denominator in identifying pole locations and their corresponding orders. Understanding the structure and factorization of the denominator helps us predict and control the function's behavior in the complex plane.