Problem 30 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

\( \operatorname{Ln}(-e^3) = 3 + i\pi \)

Step 1: Understanding the Problem

We are given a complex number in the form \( z = -e^3 \) and we need to express its natural logarithm \( \operatorname{Ln} z \) in the form \( a + i b \). This involves finding the magnitude and argument of the complex number, as the natural logarithm formula for complex numbers is \( \operatorname{Ln} z = \ln |z| + i(\text{arg}(z)) \).

Step 2: Determine the Magnitude

The magnitude of a complex number \( z = -e^3 \) is simply the absolute value, which is \( |z| = e^3 \). This is because the magnitude of a real number \( -e^3 \) is \( e^3 \).

Step 3: Determine the Argument

The argument of a complex number \( z \) is the angle \( \theta \) it makes with the positive real axis. Since \( z = -e^3 \) is on the negative real axis, the principal value of the argument \( \text{arg}(z) \) is \( \pi \).

Step 4: Apply the Complex Logarithm Formula

Using the formula \( \operatorname{Ln} z = \ln |z| + i(\text{arg}(z)) \), we substitute \(|z| = e^3\) and \(\text{arg}(z) = \pi\) to get \( \operatorname{Ln} z = \ln(e^3) + i\pi \).

Step 5: Simplify the Expression

The expression \( \ln(e^3) = 3 \ln e = 3 \), since \( \ln e = 1 \). Therefore, the logarithm simplifies to \( \operatorname{Ln} z = 3 + i\pi \).

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real and an imaginary part. They are written in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part with \( i \) being the imaginary unit. The imaginary unit \( i \) is defined as \( i^2 = -1 \). Complex numbers are useful in various fields including engineering, physics, and applied mathematics.

• The real part \( a \) represents the component on the horizontal axis in the complex plane.
• The imaginary part \( b \), with \( i \), represents the component on the vertical axis.

In our given exercise, the complex number \( z = -e^3 \) is situated entirely on the real axis as it has no imaginary component. However, we consider its position on the complex plane to understand its magnitude and argument.

Magnitude and Argument of Complex Numbers

The magnitude and argument are essential characteristics of complex numbers that help us express them in different forms. The magnitude, also known as the absolute value, measures the distance from the origin to the point representing the complex number on the complex plane. It is computed as \( |z| = \sqrt{a^2 + b^2} \). For purely real numbers like \( -e^3 \), the magnitude is simply the absolute value, \( e^3 \), since there's no imaginary part.
The argument refers to the angle made by the complex number with the positive real axis. It gives the direction of the vector in the complex plane.

• In the case of our example \( z = -e^3 \), it lies on the negative real axis so the argument is \( \pi \), which is 180 degrees or \( \pi \) radians.

Understanding these helps us express complex numbers accurately in their exponential form.

Exponential Form of Complex Numbers

The exponential form of a complex number elegantly incorporates both its magnitude and argument. It is expressed as \( z = re^{i\theta} \) where \( r \) is the magnitude and \( \theta \) is the argument. This form is quite useful when multiplying and raising complex numbers to powers.
In our exercise, the given complex number is already in exponential form: \( -e^3 = e^3e^{i\pi} \). Here:

• The magnitude \( r \) is \( e^3 \).
• The argument \( \theta \) is \( \pi \).

This representation allows for straightforward computation of the complex logarithm, as it uses the properties of exponentials and logarithms.

Principal Argument

The principal argument of a complex number is its argument confined within a particular range, typically \(-\pi < \theta \leq \pi\). It ensures a unique representation for angles that repeat every full circle, or 2\(\pi\) radians.
For a complex number \( z \) lying on the negative real axis, such as \( z = -e^3 \), the principal argument is \( \pi \), since it is exactly 180 degrees from the positive real axis. This ensures that we talk about this complex number's direction consistently without ambiguity.

• In essence, the principal argument acts like the angle wrapping function for complex numbers to provide a standardized angle measurement.

This consistency is particularly helpful when performing arithmetic operations involving complex numbers and when applying functions like the complex logarithm.