Problem 6 - Advanced Engineering Mathematics

Question

In Problems 3-8, determine the zeros and their orders for the given function. $$ f(z)=z+\frac{9}{z} $$

Step-by-Step Solution

Answer

The zeros are \( z = 3i \) and \( z = -3i \), each of first order.

Step 1: Understand the Function

We have the function \( f(z) = z + \frac{9}{z} \). This function is defined for \( z eq 0 \). We need to find the values of \( z \) where \( f(z) = 0 \).

Step 2: Set the Function to Zero

To find the zeros of the function, set \( f(z) = 0 \). This leads to the equation: \[ z + \frac{9}{z} = 0 \]

Step 3: Solve the Equation

To solve the equation \( z + \frac{9}{z} = 0 \), multiply through by \( z \) to eliminate the fraction: \[ z^2 + 9 = 0 \]

Step 4: Solve the Quadratic Equation

The equation \( z^2 + 9 = 0 \) is a quadratic equation. Rearrange it to \( z^2 = -9 \). Taking the square root of both sides, we find \( z = \pm 3i \), where \( i \) is the imaginary unit.

Step 5: Determine the Order of the Zeros

The zeros of the function are \( z = 3i \) and \( z = -3i \). Since each zero appears once, they are of first order.

Key Concepts

Zeros of Functions

In complex analysis, zeros of functions are the points where the function's value becomes zero. If a function is given as \( f(z) = z + \frac{9}{z} \), we want to find all the points \( z \) such that \( f(z) = 0 \).
To achieve this, we set the function equal to zero:

• \( f(z) = 0 \)
• \( z + \frac{9}{z} = 0 \)

By solving this equation, we arrive at the potential zeros. These zeros are the potential solutions that make the entire function vanish at specific points.
Recognizing these zeros is crucial because they help determine the behavior of the function around those points and in complex analysis, such phenomenon often introduces interesting characteristics such as poles or singularities.

Quadratic Equations

A quadratic equation is a polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. In our problem, by setting \( f(z) = 0 \), we derive a quadratic equation:

• \( z + \frac{9}{z} = 0 \) simplifies to \( z^2 + 9 = 0 \) by multiplying through by \( z \).

This simplifies down to identifying a solution of the quadratic form \( z^2 + 9 = 0 \). Solving for \( z \) involves manipulating the equation into a standard quadratic form, which could include completing the square or using the quadratic formula.
These methods reveal solutions where \( z \) satisfies the equation and hence determines the zeros of our `f(z)` function.

Imaginary Unit

The imaginary unit \( i \) is a fundamental concept in complex analysis. It is defined by the property that \( i^2 = -1 \).
In the context of our quadratic equation \( z^2 + 9 = 0 \), solving gives us:

• \( z^2 = -9 \), leading to \( z = \pm 3i \).

These results indicate that the zeros of the function involve imaginary numbers. Imaginary numbers are essential because they extend the real number system, allowing more diverse solutions to polynomial equations. They often appear in solutions when dealing with non-real roots, expanding the scope of mathematical analysis.

Order of Zeros

The order of a zero is essentially the number of times that zero occurs as a solution for the equation. For the function \( f(z) = z + \frac{9}{z} \), the zeros are \( z = 3i \) and \( z = -3i \).
These zeros are said to have an order of one, meaning each appears once as a solution to the function equal to zero condition.
Understanding the order is important because zeros of higher order, known as multiple zeros, have more pronounced effects on the function's behavior.

• They may cause the function to flatten or approach a certain direction near that zero.
• They play a significant role in the function's Taylor series expansion.

In more advanced topics, such as residue theory or Laurent series, the order can influence the evaluation of integrals and the analysis of complex functions.