Problem 28 - Advanced Engineering Mathematics

Question

Express \(\ln z\) in the form \(a+i b\). \(z=-\sqrt{3}+i\)

Step-by-Step Solution

Answer

The form is \(\ln z = \ln 2 + i\frac{5\pi}{6}\).

Step 1: Express the Complex Number in Polar Form

Given the complex number \(z = -\sqrt{3} + i\), we start by converting it to polar form. The magnitude \(r\) is given by \(r = \sqrt{(-\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = 2\).

Step 2: Find the Argument of the Complex Number

Next, we determine the argument \(\theta\). Since the complex number is in the second quadrant, \(\theta = \pi - \tan^{-1}\left(\frac{1}{\sqrt{3}}\right)\). This simplifies to \(\theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6}\).

Step 3: Write the Complex Number in Exponential Form

We can now express \(z\) in exponential form as \(z = 2e^{i(\frac{5\pi}{6})}\). This uses the polar coordinates \((r, \theta)\).

Step 4: Apply the Logarithm to the Exponential Form

To find \(\ln z\), apply the natural logarithm: \(\ln z = \ln(2e^{i(\frac{5\pi}{6})}) = \ln 2 + i\frac{5\pi}{6}\).

Step 5: Identify Real and Imaginary Components

Finally, express \(\ln z\) in the form \(a + ib\). Here, \(a = \ln 2\) and \(b = \frac{5\pi}{6}\).

Key Concepts

Polar Form of Complex Numbers

To understand the polar form of complex numbers, think of them as coordinates on a plane. A complex number can be represented as the point \(z = x + yi\), where \(x\) is the real part and \(y\) is the imaginary part. Instead of setting them on a rectangular coordinate system, we use polar coordinates. In polar form, a complex number is written as:

• \(r \cdot (\cos \theta + i \sin \theta)\)

Here, \(r\) is the magnitude or absolute value of the complex number, and \(\theta\) (theta) is the argument of the complex number. To convert to polar form, we use:

• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\)

This way of writing makes it easy to visualize the number on the complex plane.

Understanding Complex Numbers

Complex numbers extend the idea of one-dimensional numbers like real numbers into the two-dimensional space. Each complex number has a real part and an imaginary part. The standard form of a complex number is \(z = x + yi\), where:

• \(x\) is the real component
• \(y\) is the imaginary component, associated with the imaginary unit \(i\), where \(i^2 = -1\)

These numbers encompass solutions to equations that do not have real solutions. For example, the equation \(x^2 + 1 = 0\) doesn't have a real solution, but it has two complex solutions, \(i\) and \(-i\).Understanding complex numbers unlocks a broader sense of mathematical problems and solutions.

Complex Numbers in Exponential Form

Writing complex numbers in exponential form is a succinct and powerful way of expressing them. Once we have a complex number in polar form, transforming it into exponential form uses Euler's formula:

• \(z = r e^{i\theta}\)

Here, \(r\) is the magnitude, and \(\theta\) is the argument of the complex number. Euler's formula states that:

• \(e^{i\theta} = \cos \theta + i \sin \theta\)

This means that a complex number \(z\) can also be expressed as \(r e^{i\theta}\).The exponential form is especially useful in simplifying the multiplication and division of complex numbers, and it also plays a crucial role in complex logarithms.

Argument of a Complex Number

The argument of a complex number is the angle formed with the positive real axis when the complex number is represented in the complex plane. It is an essential component of the polar form and can be denoted as \(\theta\).To find \(\theta\), you calculate the arctangent of the ratio of the imaginary part to the real part:

• \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\)

However, care must be taken depending on which quadrant the complex number is situated in:

• If the complex number is in the first quadrant, \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\)
• In the second quadrant, \(\theta = \pi - \tan^{-1}\left(\frac{y}{-x}\right)\)
• For the third quadrant, \(\theta = \pi + \tan^{-1}\left(\frac{y}{x}\right)\)
• And in the fourth quadrant, \(\theta = 2\pi - \tan^{-1}\left(\frac{-y}{x}\right)\)

By determining \(\theta\), you have an essential part of expressing a complex number in both polar and exponential forms.