Problem 6 - Advanced Engineering Mathematics

Question

Express \(e^{z}\) in the form \(a+i b\). \(z=-\pi+\frac{3 \pi}{2} i\)

Step-by-Step Solution

Answer

The expression of \( e^z \) is \( 0 - i e^{-\pi} \).

Step 1: Identify z as Complex Number

The given complex number is \( z = -\pi + \frac{3\pi}{2}i \). This is in the form of a complex number where \( x = -\pi \) and \( y = \frac{3\pi}{2} \).

Step 2: Apply Euler's Formula

We use Euler's formula for complex exponentiation:\[ e^z = e^{x + yi} = e^x (\cos(y) + i \sin(y)) \]. Substitute \( x = -\pi \)and\( y = \frac{3\pi}{2} \)into the formula.

Step 3: Compute the Real Exponential Part

Calculate \( e^{x} = e^{-\pi} \). This is the real exponential component of the solution.

Step 4: Calculate the Trigonometric Parts

Compute \( \cos(y) = \cos\left(\frac{3\pi}{2}\right) \) and \( \sin(y) = \sin\left(\frac{3\pi}{2}\right) \). Since \( \frac{3\pi}{2} \) corresponds to a rotation on the unit circle, we obtain \( \cos\left(\frac{3\pi}{2}\right) = 0 \) and \( \sin\left(\frac{3\pi}{2}\right) = -1 \).

Step 5: Formulate the Final Expression

Substitute the computed values into the Euler's formula: \[ e^z = e^{-\pi} ( 0 + i(-1) ) = -i e^{-\pi} \]. Thus, the expression for \( e^z \) in the form \( a + ib \) is\( 0 + i(-e^{-\pi}) \).

Key Concepts

Euler's Formula

Euler's formula is a fundamental bridge connecting the realms of complex numbers and trigonometry. It beautifully states that for any real number \( y \), the complex exponential function can be expressed as follows: \[ e^{yi} = \cos(y) + i\sin(y) \] This equation is remarkable because it links exponential growth, complex numbers, and trigonometric functions together. Let's break down the components:

• \(e^{yi}\) is a complex number represented on the complex plane.
• \(\cos(y)\) represents the x-coordinate or the horizontal projection on the unit circle.
• \(i\sin(y)\) gives the y-coordinate or the vertical projection.

Euler's formula simplifies the expression of complex numbers raised to exponential powers. It transforms the routinely daunting task into a simple interplay of trigonometric functions. This allows us to easily manipulate and understand exponential complex numbers.

Complex Numbers

Complex numbers are numbers that have both a real and an imaginary component. In general, a complex number \( z \) is expressed as \( z = x + yi \), where \( x \), the real part, is a real number, and \( yi \), the imaginary part, involves the imaginary unit \( i \) (where \( i^2 = -1 \)).

• They are often represented in a plane called the complex plane, where the horizontal axis is the real axis, and the vertical axis is the imaginary axis.
• The number \( x \) affects the position horizontally, while \( yi \) shifts the position vertically.
• Adding or multiplying complex numbers follows specific rules, thanks to the unique nature of the imaginary unit \( i \).

Understanding complex numbers is crucial for many areas of mathematics and physics, such as signal processing and quantum mechanics.
The exercise shows us how a complex number \( z \), expressed as \( x + yi \), can be interpreted and manipulated using Euler's formula.

Trigonometric Functions

Trigonometric functions like \( \cos \) and \( \sin \) are basic yet powerful tools in mathematics. They originate from the study of triangles and their relationships but extend far beyond into various fields, including complex numbers.
In the context of the exercise, when calculating \( \cos\left(\frac{3\pi}{2}\right) \) and \( \sin\left(\frac{3\pi}{2}\right) \), we make use of the fact that these functions provide coordinates on the unit circle:

• \( \cos(y) \) represents the adjacent side in the unit circle as a result of a rotation by angle \( y \).
• \( \sin(y) \) indicates the rotation's opposite side.

For angle \( \frac{3\pi}{2} \), these functions give:

• \( \cos\left(\frac{3\pi}{2}\right) = 0 \), because at this angle, there's no horizontal displacement on the unit circle.
• \( \sin\left(\frac{3\pi}{2}\right) = -1 \), indicating a full rotation to the negative vertical direction.

Thus, trigonometric functions are pivotal in transforming and interpreting the expression of complex exponents, making them simpler to handle.