Problem 3 - Advanced Engineering Mathematics

Question

Determine the zeros and their orders for the given function. \(f(z)=(z+2-i)^{2}\)

Step-by-Step Solution

Answer

The zero is at \( z = -2 + i \) with an order of 2.

Step 1: Identify the Function

First, we need to identify the form of the function. The given function is \( f(z) = (z+2-i)^2 \). This is a polynomial function in the form \( (z-c)^n \), where \( c = -2+i \) and \( n = 2 \).

Step 2: Determine the Zero

The zero of a polynomial function \((z-c)^n\) is at \(z = c\). Thus, for the function \( f(z) = (z+2-i)^2 \), the zero is at \( z = -2 + i \).

Step 3: Determine the Order of the Zero

The order of the zero is given by the exponent \( n \) in the polynomial \( (z-c)^n \). In our case, since the function is \( (z+2-i)^2 \), the exponent is 2. Therefore, the order of the zero \( z = -2 + i \) is 2.

Key Concepts

Zero of a function

In complex analysis, a zero of a function refers to a point where the function evaluates to zero. For a function \( f(z) \), a zero is any complex number \( z_0 \) such that \( f(z_0) = 0 \). In the example given, where \( f(z) = (z+2-i)^2 \), the zero occurs when \( f(z) = 0 \). Here, you'll spot that if \( z = -2 + i \), then \( f(z) = ((-2+i)+(2-i))^2 = 0 \). Thus, \( z = -2+i \) is the zero of this function.

Identifying zeros is crucial for analyzing a function's behavior. It indicates where the function graph intersects the horizontal axis when plotted. This concept is not only fundamental in polynomial equations but extends to more complex functions as well, providing insights into the function's roots and structure.

• A zero of a function is a point where the function equals zero.
• Zeros are easy to spot in polynomial forms like \((z-c)^n\).
• Understanding zeros helps in graph interpretations and solving equations.

Order of a zero

The order or multiplicity of a zero offers insight into how many times the zero is repeated in the function. In polynomial terms like \((z-c)^n\), the order of the zero is defined by the exponent \( n \). For instance, in \( f(z) = (z+2-i)^2 \), we identify \( z = -2+i \) as the zero. Since it is raised to the power of 2, it is considered a zero of order 2.

This means that \( z = -2+i \) makes the function zero twice, highlighting its repeated root nature. A simple or distinct zero would have an order of 1. Knowing the order is key to understanding the behavior of polynomials, especially at their zeros.

• The order indicates how many times a function is zero at a point.
• A zero of order 2 implies the point is a repeated root.
• Functions may have zeros of different orders impacting their graph differently.

Polynomial functions

Polynomial functions are expressions involving variables raised to whole number powers and coefficients. They take the general form \( P(z) = a_nz^n + a_{n-1}z^{n-1} + \ldots + a_1z + a_0 \), where each \( a_i \) is a constant. A specific type of polynomial function is \((z-c)^n\), which highlights a single zero \( c \) of order \( n \).

For the function \( f(z) = (z+2-i)^2 \), this format is clearly seen, indicating it is a polynomial with a zero at \( z = -2+i \) of order 2. Polynomial functions are prevalent in mathematics due to their simplicity and the ease of analyzing their zeros. They allow us to systematically understand complex behavior through basic algebraic structures. They serve numerous roles in mathematics due to their predictable nature when it comes to differentiation, integration, and graphing.

• Polynomial functions include terms with variables raised to integer powers.
• The form \((z-c)^n\) is a simple yet insightful structure for polynomial analysis.
• These functions help in understanding and predicting the roots and behavior of mathematical relations.