Problem 17 - Advanced Engineering Mathematics

Question

In Problems 15-18, 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) -13 + 43i, (c) 3 - 26i.

Step 1: Understand the function components

The function is given by \( f(z) = 4z + i\bar{z} + \text{Re}(z) \). Here, \( z \) is a complex number, \( \bar{z} \) is the complex conjugate of \( z \), and \( \text{Re}(z) \) denotes the real part of \( z \).

Step 2: Calculate for part (a) \( z = 4 - 6i \)

First, find \( \bar{z} = 4 + 6i \) and \( \text{Re}(z) = 4 \). Substitute these into the function: \[ f(4 - 6i) = 4(4 - 6i) + i(4 + 6i) + 4 \] Calculate \[ 4(4 - 6i) = 16 - 24i \] \[ i(4 + 6i) = 4i + 6i^2 = 4i - 6 \] Finally,\[ f(4 - 6i) = (16 - 24i) + (4i - 6) + 4 = 14 - 20i \].

Step 3: Calculate for part (b) \( z = -5 + 12i \)

Identify the conjugate and real part: \( \bar{z} = -5 - 12i \), \( \text{Re}(z) = -5 \). Substitute these in:\[ f(-5 + 12i) = 4(-5 + 12i) + i(-5 - 12i) - 5 \]Calculate:\[ 4(-5 + 12i) = -20 + 48i \]\[ i(-5 - 12i) = -5i - 12i^2 = -5i + 12 \]Then,\[ f(-5 + 12i) = (-20 + 48i) + (-5i + 12) - 5 = -13 + 43i \].

Step 4: Calculate for part (c) \( z = 2 - 7i \)

The conjugate and real parts are: \( \bar{z} = 2 + 7i \), \( \text{Re}(z) = 2 \). Substitute these in:\[ f(2 - 7i) = 4(2 - 7i) + i(2 + 7i) + 2 \]Calculate:\[ 4(2 - 7i) = 8 - 28i \]\[ i(2 + 7i) = 2i + 7i^2 = 2i - 7 \]Finally,\[ f(2 - 7i) = (8 - 28i) + (2i - 7) + 2 = 3 - 26i \].

Key Concepts

Complex Numbers

Complex numbers are a fundamental concept in mathematics, especially in fields like engineering and physics. A complex number is expressed in the form \( z = a + bi \) where:

• \( a \) is the real part of the complex number.
• \( b \) is the imaginary part, and \( i \) is the imaginary unit, satisfying \( i^2 = -1 \).

Every complex number can be represented on a plane, known as the complex plane, with the real part \( a \) on the horizontal axis and the imaginary part \( b \) on the vertical axis. This representation is crucial as it allows for easy addition and multiplication of complex numbers geometrically.
When dealing with complex functions, especially when calculating function values at given points, understanding both real and imaginary parts is essential. Working through the exercise, you notice how you must manipulate both parts separately before combining results. This is intuitive with the geometric representation, as operations affect both axes of the complex plane.

Complex Conjugate

The complex conjugate of a complex number is an important concept in simplifying calculations and solving equations involving complex numbers. For a complex number \( z = a + bi \), its complex conjugate is \( \bar{z} = a - bi \).
Key properties of complex conjugates include:

• The complex conjugate of the sum of two complex numbers is the sum of their complex conjugates: \( \overline{z_1 + z_2} = \bar{z_1} + \bar{z_2} \).
• Multiplying a complex number by its conjugate results in a real number: \( z \bar{z} = a^2 + b^2 \).

In the given exercise, the function you analyze utilizes the complex conjugate of \( z \). By substituting \( \bar{z} \) in the function components, and calculating step-by-step, you simplify otherwise complex expressions. This approach is common when ensuring results are in their simplest or most interpretable form.

Real Part

The real part of a complex number \( z = a + bi \) is \( a \). It's the non-imaginary part of the complex number. Identifying the real part is essential in processes involving complex numbers since it often contains the tangible result in practical applications.
In the formula used in the exercise, \( \text{Re}(z) \) indicates the real part of the complex number \( z \). This part is treated separately from the imaginary part in mathematical expressions, especially those involving functions like \( f(z) = 4z + i\bar{z} + \text{Re}(z) \).

• The property \( \text{Re}(z) \) equals \( \frac{z + \bar{z}}{2} \).
• In integration or differentiation involving complex functions, maintaining clarity about which part (real vs. imaginary) is useful for closing steps, specifically when interpreting results.

Making sure you identify and use the real part of the number efficiently in calculations, especially when dealing with complex conjugates and imaginary units, can prevent errors and aid in reaching the correct solution in mathematical problems.