Problem 33 - Advanced Engineering Mathematics

Question

In Problems \(29-34\), express \(\operatorname{Ln} z\) in the form \(a+i b\). $$ z=(1+\sqrt{3} i)^{5} $$

Step-by-Step Solution

Answer

\( \operatorname{Ln} z = 5 \ln 2 + i \frac{2\pi}{3} \)

Step 1: Recognize the Problem

We are given a complex number \[ z=(1+\sqrt{3} i)^{5} \]We need to express its natural logarithm, \( \operatorname{Ln} z \), in the form \( a + ib \), where \( a \) and \( b \) are real numbers.

Step 2: Convert the Complex Number to Polar Form

To express the complex number \(1+\sqrt{3}i\) in polar form, we use the formula \( z = r (\cos \theta + i \sin \theta) \), where:- \( r = |1+\sqrt{3}i| \) is the magnitude- \( \theta = \arg(1+\sqrt{3}i) \) is the argumentCalculate the magnitude:\[ r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2 \]Calculate the argument:\[ \theta = \arctan\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3} \]Thus, \( 1+\sqrt{3}i \) in polar form is \( 2(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}) \).

Step 3: Apply the Exponent

Using De Moivre’s Theorem, which states \((r (\cos \theta + i \sin \theta))^n = r^n (\cos(n\theta) + i \sin(n\theta))\), we apply it to our complex number:\[ (2(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}))^5 = 2^5 (\cos(5 \cdot \frac{\pi}{3}) + i \sin(5 \cdot \frac{\pi}{3})) \]Calculate the new magnitude and angle:\[ 2^5 = 32 \]\[ 5 \cdot \frac{\pi}{3} = \frac{5\pi}{3} = \frac{2\pi}{3} + 2\pi \cdot 1 = \frac{2\pi}{3} + 2\pi\cdot 0 \] as angles repeat every \(2\pi\).Thus, the resulting expression is:\[ z = 32(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}) \]

Step 4: Find the Natural Logarithm

The natural logarithm of a complex number \( z = r (\cos \theta + i \sin \theta) \) is given by:\[ \operatorname{Ln}(z) = \ln r + i\theta \]Here:\[ \ln r = \ln(32) = \ln(2^5) = 5 \ln(2) \]\[ \theta = \frac{2\pi}{3} \]Thus, the natural logarithm \( \operatorname{Ln}(z) \) is:\[ \operatorname{Ln}(z) = 5 \ln 2 + i \frac{2\pi}{3} \]

Step 5: Final Expression

Substitute the values from Step 4 into the form \( a + ib \):\[ a = 5 \ln 2 \]\[ b = \frac{2\pi}{3} \]Therefore, the expression for \( \operatorname{Ln} z \) is:\[ \operatorname{Ln} z = 5 \ln 2 + i \frac{2\pi}{3} \]

Key Concepts

De Moivre’s Theorem

De Moivre’s Theorem is a powerful tool in the study of complex numbers. It provides a simple way to raise complex numbers expressed in polar form to any power. According to the theorem, if you have a complex number in the form \( z = r(\cos \theta + i \sin \theta) \), the expression for \( z^n \) becomes \( r^n (\cos(n\theta) + i \sin(n\theta)) \). This method simplifies the process of exponentiation for complex numbers, which might otherwise be very complicated if attempted using Cartesian form.
Let's break this down: when a complex number is raised to a power, both the magnitude and the angle undergo transformations. The magnitude, denoted by \( r \), is raised to the power \( n \), while the angle itself is multiplied by \( n \). This dual transformation allows the result to remain accurate without getting excessively complex in calculation.

• The new magnitude is simply \( r^n \).
• The new angle is \( n\theta \), which is the original angle multiplied by the power.

This theorem is foundational for working with complex numbers in various fields like engineering and physics.

Polar Form of Complex Numbers

To work effectively with complex numbers, it is often helpful to express them in their polar form. A complex number \( z = x + yi \) can be rewritten as \( z = r(\cos \theta + i \sin \theta) \). In this expression, \( r \) is the magnitude of the complex number, and \( \theta \) is the angle formed with the positive real axis, known as the argument.
The transition from rectangular (Cartesian) form to polar form is achieved using the relationships:

• Magnitude: \( r = \sqrt{x^2 + y^2} \)
• Argument: \( \theta = \arctan\left(\frac{y}{x}\right) \)

Polar form offers a more intuitive understanding when dealing with multiplication, division, and exponentiation of complex numbers as it leverages angular and distance measures.
In polar form, multiplication and division become more straightforward, with angles simply adding and subtracting, respectively. This simplification is why polar form is incredibly useful, especially when combined with De Moivre's Theorem.

Complex Number Magnitude

The magnitude of a complex number, also known as its modulus, is a measure of its size or distance from the origin in the complex plane. For a complex number \( z = x + yi \), the magnitude \( r \) is given by:\[ r = \sqrt{x^2 + y^2} \]
This formula arises from the Pythagorean theorem, where \( x \) and \( y \) are the real and imaginary parts of the complex number, acting as the sides of a right triangle. The hypotenuse, which is the line connecting the origin to the point \( (x, y) \) in the complex plane, represents the magnitude.
Understanding the magnitude is crucial because it translates to the 'length' or absolute size of the complex number, regardless of its direction. This concept plays a pivotal role when switching to polar coordinates, which provides a unique representation of the number's geometric properties.

Complex Number Argument

The argument of a complex number is the angle formed between the positive real axis and the line representing the complex number in the complex plane. It gives direction and is denoted by \( \theta \). For a complex number \( z = x + yi \), \( \theta \) can be calculated using:

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

Determining the argument is vital for converting a complex number into its polar form because it provides the necessary angle for the expression \( \cos \theta + i \sin \theta \).
It's important to remember that the argument has multiple equivalent values, differing by integer multiples of \( 2\pi \). Thus, the standard range for the principal value of \( \theta \) is \( -\pi < \theta \leq \pi \). This is especially useful when calculating powers or roots of complex numbers, as angles can be added or subtracted from multiples of \( 2\pi \) without changing the number itself.