Problem 22 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

The order of the pole at \( z = 0 \) is 3.

Step 1: Identify the Potential Poles

To find the poles of the function \( f(z) = \frac{e^z - 1}{z^4} \), first note that poles occur where the denominator is zero and the numerator is nonzero. Here, the denominator \( z^4 \) becomes zero at \( z = 0 \). Thus, a potential pole is at \( z = 0 \).

Step 2: Analyze the Numerator Near the Potential Pole

Next, we need to determine whether \( e^z - 1 \) is nonzero at \( z = 0 \). We know that the Taylor series expansion of \( e^z \) is \( 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots \). Therefore, \( e^z - 1 = z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots \).

Step 3: Determine the Order of the Zero in the Numerator

From the expansion, \( e^z - 1 \) has a zero of order 1 at \( z = 0 \) since the smallest power of \( z \) in the expansion is 1.

Step 4: Calculate the Order of the Pole

Since the denominator \( z^4 \) has a zero of order 4 at \( z = 0 \) and the numerator has a zero of order 1 at \( z = 0 \), the order of the pole of \( f(z) \) at \( z = 0 \) is \( 4 - 1 = 3 \). The order of a pole is calculated by subtracting the order of the zero in the numerator from the order of zero in the denominator.

Key Concepts

Understanding Poles in Complex Analysis

In complex analysis, poles are special points in the domain of a complex function where the function extends to infinity in some manner.
These are singular points where the function doesn't possess a finite value. Understanding poles helps in analyzing the behavior of complex functions and their limits.

• A pole occurs when the denominator of the function is zero but the numerator is not zero.
• To determine potential poles, locate the zeros of the denominator.

In the given exercise, the function is \( f(z) = \frac{e^z - 1}{z^4} \).
Here, the denominator is \( z^4 \), which becomes zero at \( z = 0 \). At this point, we need to further inspect whether it's truly a pole by examining the numerator.

Exploring the Taylor Series

The Taylor series is a mathematical tool used to represent functions as infinite sums of their derivatives at a single point.
It offers a way to approximate complex functions using polynomials, which are often easier to handle in calculations.

• The Taylor series expansion of \( e^z \) around \( z = 0 \) is given by: \[ e^z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots \]
• For \( e^z - 1 \), this simplifies to: \[ e^z - 1 = z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots \]

This representation is crucial in understanding the behavior of \( e^z - 1 \) near \( z = 0 \). The smallest power of \( z \) shows the order of the zero at this point.

Diving into the Order of Zero

The order of a zero refers to the smallest power of \( z \) in a function's series expansion where the coefficient is non-zero.
It's important because it signifies how many times a function can be divided by \( z \) before it loses its zero value at a particular point.

• From the series expansion \( e^z - 1 = z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots \), the leading term is \( z \).
• This means \( e^z - 1 \) has a zero of order 1 at \( z = 0 \), indicating that the function dips to zero just once as \( z \to 0 \).

Understanding the order of zero helps determine the order of the pole when the function's numerator and denominator are considered together.

Calculating the Order of Pole

Determining the order of a pole involves comparing the orders of zeros in both the numerator and the denominator of a function.
In essence, it's about finding out how deeper a zero is buried in the denominator compared to the numerator.

• For \( f(z) = \frac{e^z - 1}{z^4} \), the denominator \( z^4 \) has a zero of order 4 at \( z = 0 \).
• The numerator \( e^z - 1 \), as seen, has a zero of order 1.
• The order of the pole is calculated by subtracting the order of the zero in the numerator from that in the denominator.

In this case, the pole's order is \( 4 - 1 = 3 \).
This means, at \( z = 0 \), the function \( f(z) \) has a pole of order 3, capturing the behavior of the function's infinity at this point.