Problem 4 - Advanced Engineering Mathematics

Question

Express \(e^{z}\) in the form \(a+i b\). \(z=2-\frac{\pi}{2} i\)

Step-by-Step Solution

Answer

The form is \( 0 - 7.389i \).

Step 1: Identify the Components of z

The given complex number is expressed as \( z = 2 - \frac{\pi}{2}i \). Here, the real part \( a = 2 \) and the imaginary part \( b = -\frac{\pi}{2} \).

Step 2: Use Euler's Formula

Recall Euler's formula: \[ e^{z} = e^{x + iy} = e^{x} (\cos y + i \sin y) \]In our problem, \( x = 2 \) and \( y = -\frac{\pi}{2} \).

Step 3: Calculate \( e^{x} \)

Calculate \( e^{2} \), which is straightforward since it is the real part exponentiation: \[ e^{2} \approx 7.389 \].

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

Evaluate \( \cos(-\frac{\pi}{2}) \) and \( \sin(-\frac{\pi}{2}) \):\[ \cos\left(-\frac{\pi}{2}\right) = 0, \quad \sin\left(-\frac{\pi}{2}\right) = -1 \].

Step 5: Combine the Results Using Euler's Formula

Substitute the values into Euler’s formula:\[ e^{z} = e^{2} (\cos(-\frac{\pi}{2}) + i \sin(-\frac{\pi}{2})) = e^{2} (0 - i \cdot 1) = -i \cdot e^{2} \].

Step 6: Express in the Form a + ib

Now, express \( e^{z} \) in the form \( a + ib \):\[ e^{z} = 0 + (-e^{2})i \].Therefore, \( a = 0 \) and \( b = -e^{2} \approx -7.389 \).

Key Concepts

Euler's Formula

Euler's Formula is a foundational equation in mathematics, especially when dealing with complex numbers. It says that for any real number \( x \) and imaginary number \( y \), the complex exponential \( e^{x + iy} \) can be expressed using trigonometric functions as follows: \[ e^{x + iy} = e^x (\cos y + i \sin y) \] This formula beautifully relates the exponential function to trigonometric functions, which is why it is so widely admired.

• It simplifies the computation of complex exponentials by breaking them down into manageable parts: a real exponential part and trigonometric parts.
• Euler's Formula is instrumental in converting polar forms of complex numbers to their rectangular forms.

For example, in the problem we have, we recognize the formula is already structured using a complex number \( z = 2 - \frac{\pi}{2}i \), where \( x = 2 \) and \( y = -\frac{\pi}{2} \). This breakdown allows us to apply Euler’s Formula directly to solve for \( e^z \).

Complex Numbers

Complex numbers extend the concept of traditional real numbers to include the square root of negative one, denoted as \( i \). A complex number is typically written in the form \( a + bi \) where \( a \) is the real part and \( b \) is the imaginary part.

• The imaginary unit \( i \) is defined such that \( i^2 = -1 \).
• Complex numbers have both a real and an imaginary component, allowing them to be plotted on a two-dimensional plane called the complex plane.

For example, the complex number given in the problem is \( z = 2 - \frac{\pi}{2}i \).- Here, \( 2 \) is the real part.- \(- \frac{\pi}{2} \) is the coefficient of the imaginary unit \( i \), representing the imaginary part.Complex numbers are crucial in fields like engineering and physics as they can effectively represent phenomena with two varying dimensions, such as wave functions or alternating current circuits.

Real and Imaginary Parts

When dealing with complex numbers, it is essential to differentiate between their real and imaginary components.

• The **real part** is the section without the imaginary unit \( i \). For \( z = a + bi \), the real part is \( a \).
• The **imaginary part** is the portion multiplied by \( i \). In \( z = a + bi \), the imaginary part is \( b \).

Why is separating these parts so important? It allows for operations such as addition, subtraction, and studying the properties of complex functions. In our problem, we were keen to identify these parts to plug into Euler's Formula. For \( z = 2 - \frac{\pi}{2}i \):- The **real part** \( a = 2 \), contributes to exponential computations but doesn't interact with \( i \).- The **imaginary part** \( b = -\frac{\pi}{2} \), contributes to the sine and cosine computations in Euler's Formula. This distinction is vital when translating complex exponentials into tangible outputs, as it clarifies each element's role in the complex number's overall value.