Problem 32 - Advanced Engineering Mathematics

Question

In Problems \(29-34\), express \(\operatorname{Ln} z\) in the form \(a+i b\). $$ z=3-4 i $$

Step-by-Step Solution

Answer

\(\operatorname{Ln}(3-4i) = \ln(5) - 0.93i\).

Step 1: Find the magnitude of z

The magnitude of the complex number \( z \), denoted as \( |z| \), is given by the formula: \[ |z| = \sqrt{x^2 + y^2} \] where \( x \) is the real part and \( y \) is the imaginary part of \( z \). For \( z = 3 - 4i \), we have: \( x = 3 \) and \( y = -4 \). So, \[ |z| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] Thus, the magnitude \( |z| = 5 \).

Step 2: Compute the argument of z

The argument \( \theta \) of a complex number \( z = x + yi \) is given by: \[ \theta = \arctan\left( \frac{y}{x} \right) \] Since the complex number \( z = 3 - 4i \) is in the fourth quadrant (since \( x > 0 \) and \( y < 0 \)), the correct angle is computed by: \[ \theta = \arctan\left( \frac{-4}{3} \right) \approx -0.93 \]

Step 3: Express \( \operatorname{Ln} z \) in terms of natural logarithm

The natural logarithm of a complex number \( z \), \( \operatorname{Ln} z \), is defined as: \[ \operatorname{Ln} z = \ln |z| + i \theta \] We found previously that \( |z| = 5 \) and \( \theta \approx -0.93 \). Therefore, \[ \operatorname{Ln} z = \ln(5) + i (-0.93) \]

Key Concepts

Magnitude of Complex Numbers

In the realm of complex numbers, the concept of magnitude is pivotal. The magnitude (also called the modulus) of a complex number gives us a measure of its size, or distance, from the origin in the complex plane. Think of it like finding the length of the hypotenuse in a right-angled triangle having the real and imaginary parts as the other two sides.

To compute the magnitude of a complex number, denoted as \( z = x + yi \), use the formula:

• \( |z| = \sqrt{x^2 + y^2} \)

Easily plug in the values for \( z \), such as in the example \( z = 3 - 4i \), where \( x = 3 \) and \( y = -4 \). The calculation becomes:

• \( |z| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)

This result indicates that our complex number has a magnitude of 5, suggesting its position is 5 units away from the origin within the complex plane.

Argument of Complex Numbers

The argument of a complex number is another essential concept, which represents the angle between the positive real axis and the line representing the complex number in the complex plane. Understanding this angle helps to know the direction or orientation of the complex number.

To determine the argument (denoted as \( \theta \)) for a complex number \( z = x + yi \), you use this relationship:

• \( \theta = \arctan\left( \frac{y}{x} \right) \)

However, special attention is needed when the complex number is not in the first quadrant. For example, with a complex number \( z = 3 - 4i \), we need to consider its position in the fourth quadrant (since \( x > 0 \) and \( y < 0 \)) when calculating:

• \( \theta = \arctan\left( \frac{-4}{3} \right) \approx -0.93 \)

The negative angle here reflects a clockwise rotation in the complex plane, consistent with the fourth quadrant location of the number.

Natural Logarithm of Complex Numbers

Using the natural logarithm on complex numbers translates them into a form that's often easier to work with, particularly in exponential growth or decay scenarios.

When computing the natural log of a complex number \( z = x + yi \), represented as \( \operatorname{Ln} z \), we employ this formula:

• \( \operatorname{Ln} z = \ln |z| + i \theta \)

This combines the magnitude \( |z| \) and the argument \( \theta \) into a single, complex natural logarithm. From our earlier example where \( z = 3 - 4i \), with a magnitude \( |z| = 5 \) and argument \( \theta \approx -0.93 \), it is calculated as:

• \( \operatorname{Ln} z = \ln(5) + i (-0.93) \)

The result, \( \ln(5) + i (-0.93) \), fully describes the logarithmic state of the complex number, encapsulating both size and direction within the complex plane.