Problem 8 - Advanced Engineering Mathematics

Question

$$e^{-0.3+0.5 i}=e^{-0.3} \cos 0.5+i e^{-0.3} \sin 0.5=0.6501+0.3552 i$$

Step-by-Step Solution

Answer

The expression equals \( 0.6501 + 0.3552i \).

Step 1: Recognize the Expression Structure

The given expression involves an exponent in the form of a complex number, which is generally expressed as \[ e^{a + bi} = e^a (\cos b + i \sin b) \]Here, \( a = -0.3 \) and \( b = 0.5 \). This is known as Euler's formula for complex exponentials.

Step 2: Calculate the Real Part

The real part of the expression is given by \[ e^{-0.3} \cos(0.5) \]. First, calculate \( e^{-0.3} \). Using a calculator, we find that \( e^{-0.3} \approx 0.7408 \).Next, calculate \( \cos(0.5) \). Again, using a calculator, \( \cos(0.5) \approx 0.8776 \).So the real part is approximately \[ 0.7408 \times 0.8776 \approx 0.6501 \].

Step 3: Calculate the Imaginary Part

The imaginary part of the expression is given by \[ e^{-0.3} \sin(0.5) \]. Using the calculated value of \( e^{-0.3} \approx 0.7408 \).Now, calculate \( \sin(0.5) \). Using a calculator, \( \sin(0.5) \approx 0.4794 \).So the imaginary part is approximately \[ 0.7408 \times 0.4794 \approx 0.3552 \].

Step 4: Combine Real and Imaginary Parts

Combine the results from Steps 2 and 3 to form the complex number: \[ 0.6501 + 0.3552i \].This matches the initial given expression, confirming that our calculations are correct.

Key Concepts

Euler's Formula

Euler's Formula is a fundamental concept in mathematics, particularly in the field of complex analysis. It shows a deep relationship between trigonometric functions and exponential functions. This formula is expressed as:\[ e^{i\theta} = \cos(\theta) + i \sin(\theta) \]Where \( \theta \) is a real number, \( e \) is the base of natural logarithms, and \( i \) is the imaginary unit, satisfying \( i^2 = -1 \). Using this formula, we can easily convert complex exponential expressions into their equivalent trigonometric form.
For example, given a complex number in the form \( a + bi \), Euler's formula helps us express it as \( e^{a + bi} = e^a (\cos b + i \sin b) \).
This is the basis of turning complex exponentials back into a form that we can graph or interpret as vectors in the complex plane. Understanding Euler's Formula is crucial as it provides a powerful tool for working with complex numbers in both theoretical and practical applications.

Complex Numbers

Complex numbers are numbers that have both a real and an imaginary part. They are expressed in the form \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part. The imaginary unit \( i \) is defined as \( \sqrt{-1} \), where \( i^2 = -1 \). This might initially seem abstract, but complex numbers have very practical uses in many fields like physics, engineering, and computer science.

• They allow us to solve equations that have no real solution.
• They are essential in signal processing and control systems.

When dealing with complex numbers, the real part is the component that lies on the horizontal axis of a complex plane, while the imaginary part is on the vertical axis. This visualization helps in understanding operations like addition and multiplication of complex numbers.

Real Part and Imaginary Part Calculations

Obtaining the real and imaginary parts from a complex exponential expression involves some simple yet crucial steps, illustrated by the original exercise.
First, identify and compute each part separately:**Real Part Calculation:**The real part of a complex exponential like \( e^{-0.3 + 0.5i} \) is calculated as \( e^{-0.3} \cos(0.5) \). Calculating this involves:

• Compute \( e^{-0.3} \). Using computational tools, we find it is approximately 0.7408.
• Next, find \( \cos(0.5) \), which is approximately 0.8776.
• The real part is then \( 0.7408 \times 0.8776 \approx 0.6501 \).

**Imaginary Part Calculation:**For the imaginary part, we find \( e^{-0.3} \sin(0.5) \):

• Firstly, use our previously calculated value \( e^{-0.3} \approx 0.7408 \).
• Then compute \( \sin(0.5) \), which is approximately 0.4794.
• So, the imaginary part becomes \( 0.7408 \times 0.4794 \approx 0.3552 \).

By combining both parts, we achieve the complex number \( 0.6501 + 0.3552i \), demonstrating the correctness and practicality of using Euler's formula in computations.