Problem 30 - Advanced Engineering Mathematics

Question

Let \(z=x+i y .\) Find the indicated expression. $$ \operatorname{Im}\left(\bar{z}^{2}+z^{2}\right) $$

Step-by-Step Solution

Answer

The imaginary part is 0.

Step 1: Understand Conjugate and Squaring

First, recognize that \( \bar{z} \), the conjugate of \( z \), is given by \( \bar{z} = x - iy \). When we square \( z \) and \( \bar{z} \), we need to remember the squaring rules for complex numbers. The square of a complex number \( z = x + iy \) is \( z^2 = (x+iy)^2 = x^2 - y^2 + 2ixy \). Similarly, \( \bar{z}^2 = (x-iy)^2 = x^2 - y^2 - 2ixy \).

Step 2: Compute \( \bar{z}^2 + z^2 \)

Using the results from Step 1, \( z^2 = x^2 - y^2 + 2ixy \) and \( \bar{z}^2 = x^2 - y^2 - 2ixy \). Adding these, we get:\[ \bar{z}^2 + z^2 = (x^2 - y^2 - 2ixy) + (x^2 - y^2 + 2ixy) = 2(x^2 - y^2) \]Notice that the imaginary components cancel out.

Step 3: Determine the Imaginary Part

After simplifying the expression, we obtain \( 2(x^2 - y^2) \), which is a real number. Therefore, its imaginary part is \( 0 \).

Step 4: Conclude the Imaginary Part

Since the sum \( \bar{z}^2 + z^2 \) yields no imaginary component, the solution is clear. Hence, the imaginary part of the expression is indeed zero.

Key Concepts

Complex Conjugate

In the world of complex numbers, the complex conjugate plays an important role. Consider a complex number represented as \( z = x + iy \). The complex conjugate of this number is expressed as \( \bar{z} = x - iy \). Basically, you flip the sign of the imaginary component.
This operation is crucial when working with complex numbers because it helps simplify expressions and find other properties like magnitude or phase.

• The symmetry becomes evident: \( z + \bar{z} = 2x \), a result devoid of imaginary components.
• Used frequently in division: Multiplying by the conjugate helps eliminate the imaginary part when dividing complex numbers.

Understanding the complex conjugate is a foundational step when encountering problems involving complex number operations. It's a tool for simplifying parameters to a more manageable form.

Imaginary Part

The imaginary part of a complex number is equally as critical as the real part. In a complex number \( z = x + iy \), the imaginary part is \( y \), represented by the imaginary unit \( i \).
The imaginary part is what makes complex numbers unique and useful in numerous practical applications, like signal processing or waves.
Here’s how you typically work with the imaginary part:

• Assists in describing phenomena that have 90-degree phase shifts, such as alternating currents.
• In expressions, identify the imaginary part by what multiplies \( i \).

In the provided solution, the imaginary part of the sum \( \bar{z}^2 + z^2 \) turned out to be zero after the terms were simplified. This is often a goal in complex expressions, allowing them to be used for real-world interpretation.

Complex Number Operations

Operating with complex numbers includes strategies which, at first glance, extend regular algebra to this wider number field. Key operations include addition, subtraction, multiplication, and division in the complex plane.
When you perform these operations, you usually:

• Add or subtract real and imaginary parts independently.
• Multiply using distributive law, just as with polynomials.

For instance, squaring a complex number \( z = x + iy \) involves not only applying the formula \( (x+iy)^2 = x^2 - y^2 + 2ixy \), but also identifying how the terms transform.
These operations facilitate solving more intricate equations which can't easily be handled using only real numbers, thus showcasing the utility and flexibility of complex numbers.