Problem 6 - Advanced Engineering Mathematics

Question

$$ \text { In Problems } 1-10 \text {, express } e^{z} \text { in the form } a+i b \text {. } $$ $$ z=-0.3+0.5 i $$

Step-by-Step Solution

Answer

The expression for \( e^z \) is approximately \( 0.6507 + 0.3552i \).

Step 1: Understand the problem

We are given a complex number \( z = -0.3 + 0.5i \) and we need to convert \( e^z \) into the form \( a + ib \), where \( a \) and \( b \) are real numbers.

Step 2: Use Euler's formula

Recall that for any complex number \( z = x + yi \), Euler's formula allows us to express \( e^z \) as \( e^x (\cos(y) + i \sin(y)) \). Here, \( x = -0.3 \) and \( y = 0.5 \).

Step 3: Calculate \( e^x \)

Calculate \( e^{-0.3} \). Using a calculator, we find \( e^{-0.3} \approx 0.7408 \).

Step 4: Compute \( \cos(y) \) and \( \sin(y) \)

Calculate \( \cos(0.5) \) and \( \sin(0.5) \). Using a calculator:1. \( \cos(0.5) \approx 0.8776 \)2. \( \sin(0.5) \approx 0.4794 \)

Step 5: Substitute back into Euler's formula

Substitute both the exponential and trigonometric values back into the formula:\[ e^z = 0.7408 \times (0.8776 + 0.4794i) \].

Key Concepts

Complex Numbers

Complex numbers are an extension of the real numbers that include an imaginary part. They are expressed in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part, with \( i \) representing the imaginary unit. The imaginary unit \( i \) is defined by the property that \( i^2 = -1 \). This simple yet powerful concept allows us to solve equations that have no solutions in the real number system, such as \( x^2 + 1 = 0 \).

• The real part \( a \) represents the number on the horizontal axis of the complex plane.
• The imaginary part \( b \) is positioned along the vertical axis.

Any complex number can be visualized as a point or a vector in this plane. Complex numbers make it easier to perform rotations and reflections, and they are commonly used in engineering, physics, and advanced mathematics.
Additionally, complex numbers are crucial for expressing exponential functions with imaginary exponents, which will be explored in our next section.

Exponential Functions

Exponential functions involving complex numbers carry a unique and interesting behavior. When we talk about \( e^z \) where \( z \) is a complex number, we use Euler's formula: \( e^z = e^{x + yi} = e^x \times (\cos(y) + i\sin(y)) \). This formula connects exponential functions with trigonometric ones.

• The term \( e^x \) behaves like a standard exponential decay or growth function based on the real part \( x \).
• The expression \( \cos(y) + i\sin(y) \) is a circle in the complex plane for the imaginary part \( y \).

This representation helps in transforming complex exponentials into a form purely consisting of their real and imaginary parts. It is widely used in fields such as signal processing and electrical engineering as it simplifies the handling of waveforms and oscillations.
In our example, we calculated \( e^{-0.3} \approx 0.7408 \) to find the real exponential component in Euler's formula.

Trigonometric Identities

Trigonometric identities transform complex numbers into manageable terms. When we apply Euler's formula in the context of complex numbers, the trigonometric identities \( \cos(y) \) and \( \sin(y) \) come into play. For our exercise problem, the trigonometric identities help us decompose the imaginary component.

• \( \cos(y) \) provides the real part of the trigonometric circle.
• \( \sin(y) \) provides the imaginary part of the circle.

By substituting these values into Euler’s formula, we captured both real and imaginary components of \( e^z \). In our example, \( \cos(0.5) \approx 0.8776 \) and \( \sin(0.5) \approx 0.4794 \), which factor into the multiplication with \( e^{-0.3} \) to find the final expression \( a + bi \).
This ties everything together by bridging exponential and circular (trigonometric) functions to offer a simplified, elegant form for complex numbers under exponential functions.