Problem 13 - Advanced Engineering Mathematics

Question

In Problems 13-24, 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

Each pole is of order 1.

Step 1: Identify Denominator Zeroes

Find the zeroes of the denominator by setting it to zero: \[ z^2 + 2z + 5 = 0 \]Use the quadratic formula \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1 \), \( b = 2 \), and \( c = 5 \).

Step 2: Apply Quadratic Formula

Calculate the discriminant: \[ b^2 - 4ac = 2^2 - 4 \times 1 \times 5 = 4 - 20 = -16 \]Since the discriminant is negative, the solutions are complex and given by:\[ z = \frac{-2 \pm \sqrt{-16}}{2} = -1 \pm 2i \] Thus, the zeroes are complex numbers \(-1+2i\) and \(-1-2i\).

Step 3: Determine Order of Poles

Observe that the denominator factors completely over the complex numbers and no factors to higher powers exist. Each zero, \(-1 + 2i\) and \(-1 - 2i\), is a simple root of the quadratic equation.

Step 4: Conclusion - Order of Poles

Since both zeroes of the denominator are simple roots, they correspond to simple poles of order 1 for the function \( f(z) \).

Key Concepts

Complex Analysis

Complex analysis is a fascinating subject that explores functions of complex numbers, often revealing insights and techniques applicable in various fields of mathematics and science. One central idea in complex analysis is the study of poles of functions, which are particularly important when evaluating integrals and exploring analytic properties. A pole is a point where a function becomes unbounded or tends to infinity. These singular points significantly affect the behavior of functions.

• When functions have poles, they can often be expressed as a ratio of two polynomials.
• Understanding where these poles occur, and their order, is crucial in analyzing complex functions.
• In the realm of complex analysis, each pole has associated with it a specific order, which determines how the function behaves near that pole.
• The order of a pole is essentially the power of the factor in the denominator that goes to zero at that point.

By pinpointing poles and assessing their order, complex analysts can undertake residue calculations, contour integrals, and much more, offering a deeper understanding of complex functions in diverse settings.

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of a quadratic equation of the form \[ ax^2 + bx + c = 0. \] It is given by \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \] This formula can solve any quadratic equation, whether it has real or complex roots.

• The discriminant, \( b^2 - 4ac \), determines the nature of the roots. If it is positive, the roots are real and distinct.
• If it is zero, the roots are real and repeated.
• However, if the discriminant is negative, as in our original exercise, the roots are complex.

For example, in solving \( z^2 + 2z + 5 = 0 \), we find \( b^2 - 4ac = -16 \), leading to complex roots \( -1 \pm 2i. \) This insight is vital when exploring poles of functions in complex analysis. The quadratic formula not only provides the solutions but also guides us in understanding how those solutions shape the behavior of functions around their poles.

Simple Poles

In complex analysis, poles are classified based on their order, with simple poles being the most straightforward. A simple pole is characterized by the order of one. This means the denominator of the function has a linear zero at this point, contributing to a fairly predictable behavior as the function approaches the pole.

• Functions with simple poles have Laurent Series expansions, where the part responsible for the pole is just \( \frac{1}{z-z_0}, \) where \( z_0 \) represents the location of the pole.
• These poles are crucial as they allow for residue calculations, which are integral in evaluating complex integrals.
• Knowing the function \( f(z) = \frac{3z-1}{z^2+2z+5} \), we determined the poles at \( z = -1 + 2i \) and \( z = -1 - 2i, \) which are simple since their order is 1.

Simple poles simplify many problems in complex analysis since their residues can be directly computed and used in various applications. Understanding them is foundational for anyone looking to delve deeper into the behavior of complex functions.