Problem 17 - Advanced Engineering Mathematics

Question

Evaluate the given function at the indicated points. \(f(z)=4 z+i \bar{z}+\operatorname{Re}(z)\) (a) \(4-6 i\) (b) \(-5+12 i\) (c) \(2-7 i\)

Step-by-Step Solution

Answer

(a) 14-20i, (b) -37+43i, (c) 3-26i.

Step 1: Review the Function Components

The function is given as \( f(z) = 4z + i\bar{z} + \operatorname{Re}(z) \). We need to evaluate this at different values of \( z \). Recall that \( \bar{z} \) is the complex conjugate, and \( \operatorname{Re}(z) \) is the real part of \( z \).

Step 2: Calculate f(4-6i)

For \( z = 4 - 6i \), we have \( \bar{z} = 4 + 6i \) and \( \operatorname{Re}(z) = 4 \). The function becomes:\[ f(4-6i) = 4(4 - 6i) + i(4 + 6i) + 4 \]Calculating each part, we have:- \( 4(4 - 6i) = 16 - 24i \)- \( i(4 + 6i) = 4i - 6 \)Substitute back:\[ f(4-6i) = 16 - 24i + 4i - 6 + 4 \]Simplify:\[ f(4-6i) = 14 - 20i \]

Step 3: Calculate f(-5+12i)

For \( z = -5 + 12i \), we have \( \bar{z} = -5 - 12i \) and \( \operatorname{Re}(z) = -5 \). Substitute into the function:\[ f(-5+12i) = 4(-5 + 12i) + i(-5 - 12i) - 5 \]Calculating each part, we get:- \( 4(-5 + 12i) = -20 + 48i \)- \( i(-5 - 12i) = -5i - 12 \)Substitute back:\[ f(-5+12i) = -20 + 48i - 5i - 12 - 5 \]Simplify:\[ f(-5+12i) = -37 + 43i \]

Step 4: Calculate f(2-7i)

For \( z = 2 - 7i \), we have \( \bar{z} = 2 + 7i \) and \( \operatorname{Re}(z) = 2 \). Substitute into the function:\[ f(2-7i) = 4(2 - 7i) + i(2 + 7i) + 2 \]Calculating each part, we get:- \( 4(2 - 7i) = 8 - 28i \)- \( i(2 + 7i) = 2i - 7 \)Substitute back:\[ f(2-7i) = 8 - 28i + 2i - 7 + 2 \]Simplify:\[ f(2-7i) = 3 - 26i \]

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real and an imaginary part. They are usually written in the form\( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part multiplied by the imaginary unit \( i \). This imaginary unit, \( i \), is defined by the property \( i^2 = -1 \).
While the real numbers can be graphed on a single number line, complex numbers require a two-dimensional plane.
This plane, known as the complex plane, lets the horizontal axis represent the real part, and the vertical axis represent the imaginary part.
The magnitude, or modulus, of a complex number is the distance from the origin to the point \( (a, b) \) on the complex plane, calculated as \( \sqrt{a^2 + b^2} \). Understanding complex numbers and their components allows us to delve into more advanced math concepts with clarity.

Function Evaluation

Function evaluation refers to the process of determining the result of a function for specific input values. In our context, we have a function defined as \( f(z) = 4z + i\bar{z} + \operatorname{Re}(z) \) and we need to evaluate it for specific points \( z \).
Evaluating a function involves substituting the given values into the function's formula and applying arithmetic operations to get the final result.
For complex functions, this often entails breaking down the input into its real and imaginary components. Then, we use these parts to determine the output.

• For example, let's consider evaluating \( f(z) \) at \( z = 4 - 6i \):
• First, substitute each relevant component such as \( \bar{z} \) and \( \operatorname{Re}(z) \) into the equation.
• Second, perform the arithmetic operations, remembering to treat the imaginary unit \( i \) in accordance with \( i^2 = -1 \).

In short, function evaluation allows us to see how a complex function behaves under different conditions.

Complex Conjugate

The complex conjugate of a complex number \( z = a + bi \) is \( \bar{z} = a - bi \). Basically, it involves changing the sign of the imaginary part.
Understanding complex conjugates is very important. As they help in various mathematical operations, such as division of complex numbers.
Furthermore, the product of a complex number with its conjugate results in a real number, given by
\( z \cdot \bar{z} = (a + bi)(a - bi) = a^2 + b^2 \). This means the multiplication removes the imaginary component.

• When dealing with the function expression \( i\bar{z} \), find the complex conjugate \( \bar{z} \) first.
• Then, multiply by \( i \) and add it into the equation.

Knowing how to determine and utilize the complex conjugate streamlines calculations and makes understanding complex functions simpler.

Real Part of a Complex Number

The real part of a complex number \( z = a + bi \) is simply \( a \). It's the component on the real axis of the complex plane.
In mathematical notation, this is represented as \( \operatorname{Re}(z) \). Grasping this basic notion helps in dividing the complex number into its constituents.
The real part is significantly important when performing operations on complex numbers, such as addition or forming real-number equations from complex ones.

• While evaluating functions, extract this real part directly.
• Use it as a normal real number in any further operations, always considering it separately from the imaginary portion.

This understanding aids in clarifying how different parts of a complex number work together in function evaluations and other calculations.