Problem 20 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

The function has a pole of order 2 at \( z = 0 \).

Step 1: Understand the function

The function given is \( f(z) = \frac{e^z}{z^2} \). It is a fraction where the numerator is \( e^z \) and the denominator is \( z^2 \). This implies a potential pole at \( z = 0 \).

Step 2: Identify the poles

A pole occurs when the function's denominator equals zero, as it's not defined at that point. Here, the denominator \( z^2 \) implies a pole at \( z = 0 \).

Step 3: Determine the order of the pole

The order of a pole is determined by the highest power of \( z \) that divides the function. Since the denominator is \( z^2 \), \( z = 0 \) is a pole of order 2. There are no additional zeros in the numerator of \( e^z \) at \( z=0 \).

Step 4: Confirm that no other factors affect the pole

Since \( e^z \) is an entire function (analytic everywhere on the complex plane with no zeros at \( z = 0 \)), it does not cancel or introduce a higher power of \( z \) in the numerator. Thus, the order remains influenced solely by \( z^2 \) in the denominator.

Key Concepts

Poles of a function

In complex analysis, poles of a function are specific points in the complex plane where the function becomes unbounded or undefined. A pole occurs when the denominator of a rational function is zero, making the function value approach infinity. For the function given in the exercise, \( f(z) = \frac{e^z}{z^2} \), the pole is identified at \( z = 0 \), since at this point, the denominator \( z^2 \) equals zero.

Poles are fundamental elements in understanding the behavior of complex functions. They help us understand how a function behaves near certain critical points, and they can be used to evaluate complex integrals using techniques like residue theory. Identifying poles is essential in analyzing and solving complex function problems.

Order of a pole

The order of a pole refers to the number of times the denominator of a function needs to be differentiated to eliminate the singularity at that pole. This is a crucial aspect as it gives insight into how the function grows in the vicinity of the pole.

For our function \( f(z) = \frac{e^z}{z^2} \), the order of the pole at \( z = 0 \) is 2, because the denominator is \( z^2 \), indicating the pole repeats twice. The numerator, \( e^z \), is analytic and does not contribute to increasing or reducing the order of the pole at \( z = 0 \), as it is non-zero and has no zero at that point. Understanding the order of poles helps in applying techniques such as Laurent series expansions around these points.

Complex functions

Complex functions are functions that take complex numbers as inputs and produce complex numbers as outputs. These functions are often expressed in terms of complex variables, such as \( z \), where \( z = x + iy \) (with \( i \) being the imaginary unit, and \( x \) and \( y \) are real numbers). In the case of the function \( f(z) = \frac{e^z}{z^2} \), it involves both exponential functions and polynomials, which are common components in complex analysis.

Complex functions exhibit unique properties, such as continuity, differentiability, and analyticity, that open up a variety of powerful analysis techniques. They are applied across various fields, from physics to engineering, due to their ability to model phenomena with both real and imaginary components. Comprehending these functions' behaviors, including at their poles, is essential in mastering complex analysis.

Analytic functions

Analytic functions, also known as holomorphic functions, are functions that are differentiable at every point in their domain of complexity. A function is analytic if it can be represented by a power series expansion in the neighborhood of each point in its domain.

For example, the function \( e^z \) in the numerator of \( f(z) = \frac{e^z}{z^2} \) is an entire function, meaning it is analytic over the whole complex plane. This characteristic of \( e^z \) ensures that it does not interfere with the pole at \( z = 0 \) created by the \( z^2 \) denominator. Analytic functions have significant implications, such as obeying the Cauchy-Riemann equations, and they play a critical role in the foundational theories within complex analysis, including the development of Taylor and Laurent series.