Problem 16 - Advanced Engineering Mathematics

Question

In Problems 15-18, evaluate the given function at the indicated points. $$ f(z)=(x+1+1 / x)+i\left(4 x^{2}-2 y^{2}-4\right) $$ (a) \(1+i\) (b) \(2-i\) (c) \(1+4 i\)

Step-by-Step Solution

Answer

(a) 3 - 2i, (b) 3.5 + 10i, (c) 3 - 32i.

Step 1: Understand the Function and Points

The given function is \( f(z) = (x+1+1/x) + i(4x^2 - 2y^2 - 4) \). Here, \( z = x + yi \) where \( x \) and \( y \) are real numbers representing the real and imaginary parts of \( z \). We need to evaluate this function for specific values of \( z \).

Step 2: Evaluate at Point (a) 1+i

For \( z = 1+i \), \( x = 1 \) and \( y = 1 \). Substitute these into the function: \( f(1+i) = (1 + 1 + \frac{1}{1}) + i(4 \times 1^2 - 2 \times 1^2 - 4) \). Calculate the components: Real part: \( 1 + 1 + 1 = 3 \); Imaginary part: \( 4 - 2 - 4 = -2 \). Therefore, \( f(1+i) = 3 - 2i \).

Step 3: Evaluate at Point (b) 2-i

For \( z = 2-i \), \( x = 2 \) and \( y = -1 \). Substitute these values: \( f(2-i) = (2 + 1 + \frac{1}{2}) + i(4 \times 2^2 - 2 \times (-1)^2 - 4) \). Calculate: Real part: \( 2 + 1 + 0.5 = 3.5 \); Imaginary part: \( 16 - 2 - 4 = 10 \). Thus, \( f(2-i) = 3.5 + 10i \).

Step 4: Evaluate at Point (c) 1+4i

For \( z = 1 + 4i \), \( x = 1 \) and \( y = 4 \). Substitute these into the function: \( f(1+4i) = (1 + 1 + \frac{1}{1}) + i(4 \times 1^2 - 2 \times 4^2 - 4) \). Calculate: Real part: \( 1 + 1 + 1 = 3 \); Imaginary part: \( 4 - 32 - 4 = -32 \). Therefore, \( f(1+4i) = 3 - 32i \).

Key Concepts

Complex Numbers

Complex numbers form an essential part of mathematics and are quite fascinating. A complex number is expressed as \( z = x + yi \), where \( x \) and \( y \) are real numbers. The variable \( i \) is known as the imaginary unit and is defined by the property \( i^2 = -1 \). This unique characteristic allows us to explore numbers beyond the traditional real number line.

• The real part of the complex number \( z \) is \( x \),
• The imaginary part is \( yi \).

Any number that can be written in this form is considered a complex number. These numbers are crucial for solving many problems in fields such as engineering, physics, and applied mathematics. They allow us to handle equations that do not have real solutions, thus broadening our mathematical toolkit.

Function Evaluation

Evaluating functions involving complex numbers is an important skill in complex analysis. For the given function \( f(z) = (x+1+1/x) + i(4x^2 - 2y^2 - 4) \), function evaluation involves substituting the given complex numbers into the function to find the resulting value.
First, identify the real \( x \) and imaginary \( y \) components from the complex number \( z = x + yi \). Substitute these values into both the real and imaginary parts of the function separately.

• Calculate the real part: \( x+1+1/x \)
• Evaluate the imaginary component: \( 4x^2 - 2y^2 - 4 \)

Finally, combine these results to form the evaluated complex function \( f(z) \). This approach simplifies the process and ensures accurate results, making it easy to analyze how functions behave at various complex points.

Real and Imaginary Components

The expression of a function involving complex numbers often results in two components: a real part and an imaginary part. For instance, in our function \( f(z) = (x+1+1/x) + i(4x^2 - 2y^2 - 4) \), each part provides valuable insights into the behavior of the function.
The real component \((x+1+1/x)\) is processed by substituting \( x \) from the complex input \( z \). Meanwhile, the imaginary portion \(i(4x^2 - 2y^2 - 4)\) requires us to carefully substitute both \( x \) and \( y \), as both contribute to the result.

• The real part affects the horizontal shift in the complex plane.
• The imaginary part influences the vertical displacement.

Understanding each component's impact helps in visualizing how the function interacts with the complex inputs and provides clarity when graphically representing these functions.