Problem 28 - Advanced Engineering Mathematics

Question

In Problems 23-28, express \(\ln z\) in the form \(a+i b\). $$ z=-\sqrt{3}+i $$

Step-by-Step Solution

Answer

\( \ln(z) = \ln(2) + i\frac{5\pi}{6} \)

Step 1: Express z in polar form

We begin by finding the magnitude of \( z \). The formula for magnitude is \( |z| = \sqrt{(-\sqrt{3})^2 + (1)^2} \). That gives us:\[|z| = \sqrt{3 + 1} = \sqrt{4} = 2\]Next, we find the argument \( \theta \) using the formula \( \tan \theta = \frac{\text{Imaginary part}}{\text{Real part}} = \frac{1}{-\sqrt{3}} \). Since \( z \) is in the second quadrant (negative real, positive imaginary), \( \theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \).

Step 2: Write ln(z) using Euler's formula

Now that we have that \( z = 2 \cdot e^{i\frac{5\pi}{6}} \), we use the Euler's formula to express ln(z). Thus, \[\ln(z) = \ln(|z|) + i \cdot \text{Arg}(z) = \ln(2) + i \frac{5\pi}{6}\]

Step 3: Express ln(z) in the form a + ib

From the previous step, we have:\[\ln(z) = \ln(2) + i\frac{5\pi}{6}\]This is already in the form \( a + i b \), where \( a = \ln(2) \) and \( b = \frac{5\pi}{6} \).

Key Concepts

Polar Form

The polar form of a complex number offers a different perspective compared to the standard rectangular form. While the rectangular form represents a complex number via real and imaginary components, namely as \( z = a + bi \), polar form provides a unique approach by using magnitude and angle.
In polar form, a complex number \( z \) is expressed as \( z = r \cdot ( \cos \theta + i \sin \theta ) \) or equivalently, \( z = r \cdot e^{i \theta} \) using Euler's formula. Here, \( r \) is the magnitude (or modulus) of the complex number, and \( \theta \) is the argument (or angle) which indicates the direction in the complex plane.
For example, the complex number \( -\sqrt{3} + i \) in polar form requires finding both its magnitude and argument. This conversion is particularly helpful in simplifying complex number operations like multiplication and division.

Magnitude of Complex Numbers

The process of finding the magnitude of a complex number is akin to finding the hypotenuse of a right triangle using the Pythagorean theorem. If you have a complex number \( z = a + bi \), the magnitude is denoted by \( |z| \) and calculated as:

• \(|z| = \sqrt{a^2 + b^2}\)

This formula manifests due to the rectangular representation of complex numbers and ensures that the magnitude captures the distance from the origin to the point \( (a, b) \) in the complex plane.
In the case of \( z = -\sqrt{3} + i \), the components are \( a = -\sqrt{3} \) and \( b = 1 \). The magnitude becomes \(|z| = \sqrt{(-\sqrt{3})^2 + 1^2} = \sqrt{4} = 2\).
A clear understanding of magnitude is crucial when converting complex numbers into their polar forms or when performing operations that involve distances in the complex plane.

Euler's Formula

One of the key tools in complex analysis is Euler's Formula, which elegantly connects trigonometry to exponential functions. Euler's formula states that for any real number \( x \):

• \( e^{ix} = \cos x + i \sin x \)

This connection is foundational in expressing complex numbers in polar form as \( z = r \cdot e^{i \theta} \), where \( r \) is the magnitude and \( \theta \) is the argument of the complex number.
Using Euler's formula simplifies many calculations, such as finding powers and roots of complex numbers, as well as transformations back and forth between polar and rectangular forms. In our example, by expressing \( z = 2 \cdot e^{i \theta} \) using Euler's formula, we efficiently determine \( \ln(z) = \ln(2) + i \frac{5\pi}{6} \).
Mastering Euler's formula enables deeper insights into complex numbers and fosters more intuitively solving problems within this mathematical realm.