91Ó°ÊÓ

Euler's formula is a fundamental bridge between complex numbers and trigonometry. It states that for any real number \( x \) and \( y \), \[ e^{(x + jy)} = e^x \times (\text{cos}(y) + j\text{sin}(y)) \] This formula shows how exponential functions can map complex numbers, making it a powerful tool in many fields of science and engineering. Here's how it can be broken down:

• \( e^x \) represents the growth factor.
• The trigonometric functions, \( \text{cos}(y) \) and \( \text{sin}(y) \), describe rotation around the complex plane.

In our problem, we use Euler's formula to convert the complex number \( z = \text{ln}(3) + j \frac{\text{Ï€}}{2} \) into its exponential form. This enables us to separate and identify the real and imaginary components.

The exponential form of a complex number is a way to express complex numbers in terms of their magnitude and angle. Using Euler's formula, any complex number \( z = a + jb \) can be written as: \[ z = e^{(x + jy)} \] Applying this to our problem, we have: \[ e^z = e^{\text{ln}(3) + j \frac{\text{Ï€}}{2}} \] Expanding this using Euler's formula provides: \[ e^{\text{ln}(3)} \times (\text{cos}(\frac{\text{Ï€}}{2}) + j\text{sin}(\frac{\text{Ï€}}{2})) \] Breaking it down:

• \( e^{\text{ln}(3)} = 3 \)
• The trigonometric parts: \( \text{cos}(\frac{\text{Ï€}}{2}) = 0 \) and \( \text{sin}(\frac{\text{Ï€}}{2}) = 1 \)

This simplifies to: \[ 3 \times (0 + j \times 1) = 3j \] Here, we can clearly see that the real part \(a\) is 0, and the imaginary part \(b\) is 3.

The imaginary unit, represented by \( j \) (or sometimes \( i \) in mathematics), is a fundamental concept when working with complex numbers. It is defined by the property: \[ j^2 = -1 \] This allows complex numbers to be expressed in the form \( a + jb \), where \( a \) and \( b \) are real numbers. Here, \( a \) is the real part, and \( b \) is the imaginary part.When we say imaginary, it doesn't mean that it doesn't exist; instead, it represents values that can't be mapped directly onto the real line. In our problem, the presence of \( \text{ln}(3) + j \frac{\text{Ï€}}{2} \) means that the imaginary unit is being multiplied by \( \frac{\text{Ï€}}{2} \). By simplifying through Euler's formula and understanding the imaginary unit, we determine the complex number's real and imaginary parts effectively.

• The real part \( a = 0 \).
• The imaginary part \( b = 3 \).

This breakdown helps demystify the slightly abstract notion of the imaginary unit and demonstrates its utility in solving complex number problems.