Problem 38 - Advanced Engineering Mathematics

Question

By the quadratic formula, \(e^{z}=-\frac{1}{2}+\frac{\sqrt{3}}{2} i\) or \(e^{z}=-\frac{1}{2}-\frac{\sqrt{3}}{2} i .\) Hence $$z=\ln \left(-\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)=\left(\frac{2 \pi}{3}+2 n \pi\right) i \quad \text { or } \quad z=\ln \left(-\frac{1}{2}-\frac{\sqrt{3}}{2} i\right)=\left(\frac{4 \pi}{3}+2 n \pi\right) i$$

Step-by-Step Solution

Answer

\(z=\frac{2\pi i}{3}+2n\pi i\) or \(z=\frac{4\pi i}{3}+2n\pi i\).

Step 1: Analyze Given Exponential Equation

Recognize that we have two separate solutions given by the exponential equation, either \(e^{z} = -\frac{1}{2} + \frac{\sqrt{3}}{2} i\) or \(e^{z} = -\frac{1}{2} - \frac{\sqrt{3}}{2} i\). These expressions involve complex numbers in polar form.

Step 2: Convert Exponential Form to Logarithmic Form

We need to find \(z\) by converting the exponential expressions into logarithmic form: \(z = \ln(-\frac{1}{2} + \frac{\sqrt{3}}{2} i)\) and \(z = \ln(-\frac{1}{2} - \frac{\sqrt{3}}{2} i)\).

Step 3: Express the Complex Number in Polar Form

Note that \(-\frac{1}{2} + \frac{\sqrt{3}}{2} i\) represents a point on the unit circle at an angle of \(\frac{2\pi}{3}\) radians (120 degrees). Similarly, \(-\frac{1}{2} - \frac{\sqrt{3}}{2} i\) is at \(\frac{4\pi}{3}\) radians (240 degrees).

Step 4: Use the Logarithm of a Complex Number Formula

The logarithm of a complex number given by \(re^{i\theta}\) is \(\ln|r| + i\theta\). Here \(|r| = 1\), so the logarithm reduces to simply \(z = i\theta\). So, \(z = i\left(\frac{2\pi}{3} + 2n\pi\right)\) for the first expression, and \(z = i\left(\frac{4\pi}{3} + 2n\pi\right)\) for the second, where \(n\) is an integer.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are commonly used in various fields, such as finance, physics, and biology, to describe growth and decay processes. In the realm of complex numbers, exponential functions become particularly interesting. When dealing with complex exponents, we use Euler's formula:

• The formula states that for any real number \( \theta \), \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \).
• This converts complex exponential equations into trigonometric form, which can be graphically represented as a point on the unit circle.

In the original exercise, the given exponential expressions are converted to logarithmic form, enabling the extraction of angles denoted by \( \theta \). These angles are critical in identifying the complex logarithm's imaginary component, represented in the solution as the principal argument, with additional terms added to account for periodicity (\( 2n\pi \)).
This interplay between exponential functions and trigonometry is foundational to understanding complex analysis.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. They play an essential role in calculating the continuous growth rates or comprehending the natural scales involved in phenomena. In complex analysis, logarithms stretch beyond real numbers and help in determining the angle or phase of a complex number. For a complex number in polar form, \( re^{i\theta} \), its logarithm is expressed as:

• \( \ln(z) = \ln|r| + i\theta \)
• where \(|r|\) is the magnitude (or modulus) of the complex number, and \(\theta\) is the angle (or argument).

The original problem takes advantage of this formula. The expressions \(-\frac{1}{2} + \frac{\sqrt{3}}{2}i\) and \(-\frac{1}{2} - \frac{\sqrt{3}}{2}i\) are converted to their logarithmic form, simplifying to the imaginary parts \(i\left(\frac{2\pi}{3} + 2n\pi\right)\) and \(i\left(\frac{4\pi}{3} + 2n\pi\right)\). Here, the "\(2n\pi\)" component accounts for the infinite nature of the logarithmic spiral as it winds around the origin. Understanding logarithmic transformation is crucial for navigating problems in complex numbers.

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system where each point is determined by a distance from a reference point and an angle from a reference direction. This system is especially useful in representing complex numbers because each complex number corresponds naturally to a radius and angle.

• The polar form of a complex number is \( z = re^{i\theta} \), where \(r\) is the modulus and \(\theta\) is the argument.
• This form simplifies operations like multiplication or division, reducing them to addition or subtraction of angles.
• Moreover, polar coordinates shine in visualizing complex numbers as points or vectors originating from the origin.

In the given exercise, complex numbers like \(-\frac{1}{2} + \frac{\sqrt{3}}{2}i\) are instantly linked to polar coordinates by calculating angles \(\frac{2\pi}{3}\) radians and \(\frac{4\pi}{3}\) radians. These angles help portray the location and direction of the number on the complex plane. Interpreting complex logarithms as angles in polar coordinates offers an intuitive appreciation for the geometric meaning behind algebraic manipulations.