Problem 1 - Advanced Engineering Mathematics

Question

In Problems 1-10, express \(e^{z}\) in the form \(a+i b\). \(z=\frac{\pi}{6} i\)

Step-by-Step Solution

Answer

The expression is \( e^{z} = \frac{\sqrt{3}}{2} + \frac{1}{2} i \).

Step 1: Identify the Formula for Complex Exponential

The exponential function for a complex number is given by Euler's formula: \( e^{z} = e^{x + yi} = e^x (\cos(y) + i \sin(y)) \). In this case, \( z = \frac{\pi}{6} i \), which means \( x = 0 \) and \( y = \frac{\pi}{6} \).

Step 2: Evaluate Exponential Component

Since \( x = 0 \), the exponential component is \( e^x = e^0 = 1 \). The expression simplifies to \( e^z = 1 \cdot (\cos\left(\frac{\pi}{6}\right) + i \sin\left(\frac{\pi}{6}\right)) \).

Step 3: Calculate Trigonometric Functions

Evaluate the trigonometric components: \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \) and \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \).

Step 4: Express in the Form a + ib

Substitute the values from Step 3 into the equation from Step 2: \( e^z = \frac{\sqrt{3}}{2} + i \cdot \frac{1}{2} \). Thus, \( e^{z} = \frac{\sqrt{3}}{2} + \frac{1}{2} i \).

Key Concepts

Euler's formula

Euler's formula is a powerful tool that connects complex numbers with trigonometry and exponentials. It is given by the expression: \[e^{z} = e^{x + yi} = e^x (\cos(y) + i \sin(y))\]This equation suggests that for any complex number, we can split it into two parts: a real part \(x\) and an imaginary part \(y\).

• \(e^x\) is the exponential part, responsible for the magnitude or growth behavior of the function.
• \(\cos(y) + i \sin(y)\) represents the rotation on the complex plane, where \(\cos(y)\) is the real horizontal component, and \(\sin(y)\) is the imaginary vertical component.

When applying Euler's formula to a given \(z = \frac{\pi}{6}i\), \(x\) becomes 0, simplifying the formula greatly because \(e^0 = 1\), leaving just the trigonometric rotation components. Euler’s formula makes it easier to work with complex exponentials by converting them into a more familiar trigonometric form.

Trigonometric functions

Trigonometric functions are critical to understanding Euler's formula and complex numbers. In complex exponentials, trigonometric functions \(\cos(y)\) and \(\sin(y)\) determine how the complex number rotates or is positioned on the complex plane.

For instance, taking \(z = \frac{\pi}{6}i\) and using Euler's formula, we need:

• \(\cos\left(\frac{\pi}{6}\right)\), which evaluates to \(\frac{\sqrt{3}}{2}\), providing the real component.
• \(\sin\left(\frac{\pi}{6}\right)\), which is \(\frac{1}{2}\), supplying the imaginary component.

These functions help break down a complex number into understandable parts, where \(\cos\) specifies how much the real part is elongated, while \(\sin\) stretches the imaginary part. Recognizing trigonometric values for common angles like \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), and \(\frac{\pi}{3}\) is very beneficial when dealing with such questions.

Complex numbers

Complex numbers expand our understanding of numbers beyond the real line, incorporating an imaginary dimension. They are usually expressed in the form \(a + bi\) where:

• \(a\) is the real part.
• \(b\) is the imaginary component, and \(i\) is the imaginary unit defined by \(i^2 = -1\).

Euler's formula provides a bridge between exponential functions and these complex numbers, allowing for transformations that are easier to manage when analyzing oscillations or waves.

In the exercise, you met \(z = \frac{\pi}{6}i\), where by using the trigonometric identities and considering \(x = 0\), the exponential \(e^z\) is beautifully expressed as a blend of \(\cos\) and \(\sin\). This makes working with complex numbers clearer, as their rotational and magnitude properties are unveiled through decomposition into trigonometric forms. Understanding complex numbers through Euler’s formula and trigonometry simplifies what would typically be a daunting task.