Problem 28 - Advanced Engineering Mathematics

Question

In Problems 27-32, let \(z=x+i y\). Find the indicated expression. $$ \operatorname{Re}\left(z^{2}\right) $$

Step-by-Step Solution

Answer

The real part of \( z^2 \) is \( x^2 - y^2 \).

Step 1: Express the Problem in Terms of x and y

Given the complex number \( z = x + i y \), we want to find the real part of \( z^2 \). First, express \( z^2 \) in terms of \( x \) and \( y \). Start by expanding \( z^2 = (x + i y)^2 \).

Step 2: Expand the Squaring Expression

Now expand the expression: \[ z^2 = (x + i y)^2 = x^2 + 2 x(i y) + (i y)^2 = x^2 + 2 i xy - y^2 \].Note that \( (i y)^2 = -y^2 \) because \( i^2 = -1 \).

Step 3: Identify the Real and Imaginary Parts

From the expansion \( x^2 + 2i xy - y^2 \), identify the real part and the imaginary part. The real part is \( x^2 - y^2 \) and the imaginary part is \( 2xy \).

Step 4: Extract the Real Part

The real part of \( z^2 \) is \( x^2 - y^2 \). Therefore, the solution to the problem \( \operatorname{Re}(z^2) \) is \( x^2 - y^2 \).

Key Concepts

Real Part of Complex Numbers

Complex numbers are composed of a real part and an imaginary part. When we have a complex number like \( z = a + bi \), the real part is simply the \( a \) in that expression, which is the component that doesn't involve the imaginary unit \( i \). It's important because the real part gives us a view into the 'real world' component of the complex number, disregarding the imaginary component. In calculations involving complex numbers, we often need to isolate this real portion, particularly when the context we're dealing with is on the real number line.

Finding the real part of a complex expression can help in understanding various properties of complex numbers or solving problems when the real world behavior of the system is considered. For example, in our exercise with \( z^2 \), the goal is to identify the real part of \( z^2 \) after performing the algebraic operations, which was extracted as \( x^2 - y^2 \). Recognizing these differences between real and imaginary parts is fundamental in complex number arithmetic.

Complex Number Multiplication

When multiplying complex numbers, the distributive property is used, treating the imaginary unit \( i \) as any other algebraic term, but with the caveat that \( i^2 = -1 \). This property is crucial, as it transforms the expression back into its real and complex components.

Consider our example \( z = x + i y \), and we want to find \( z^2 \). The step involves multiplying \( (x + i y) \) by itself:

• First multiply: \( x(x) = x^2 \) and \( x(i y) = i xy \)
• Next: \( i y(x) = i xy \) and \( i y(i y) = i^2 y^2 = -y^2 \)

Putting this together gives us \( z^2 = x^2 + 2i xy - y^2 \). Note the use of \( i^2 = -1 \) to transition \( (i y)^2 \) to \( -y^2 \). Understanding complex number multiplication is fundamental, as it serves as a building block for more complicated expressions and operations.

Expanding Complex Expressions

Expanding a complex expression involves breaking down the expression into its simpler components based on algebraic rules. In the context of complex numbers, this often means applying the distributive property to achieve a full expansion. This process not only simplifies the expression but also distinguishes between real and imaginary parts.

When expanding \((x + i y)^2\), you apply the distributive property, treating \(x\) and \(i y\) as separate variables at first. Each term is multiplied by every other term:

• \((x)^2\) is straightforward, resulting in \(x^2\)
• \(2 (x)(i y)\) combines the real with imaginary, giving \(2 i xy\), which is the combined imaginary part
• \((i y)^2\) results in \(-y^2\) using \(i^2 = -1\)

Expanding helps to identify the terms that are real (like \(x^2 - y^2\)) and those that are imaginary (like \(2i xy\)). By working through the expansion, complex number operations become clearer, laying the foundation for more advanced mathematical manipulations involving complex expressions.