Problem 12 - Advanced Engineering Mathematics

Question

Express the given number in the form \(a+i b\). $$ \frac{e^{2+3 \pi i}}{e^{-3+\pi i / 2}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(0 - i e^5\).

Step 1: Simplify the Exponentials

Recall the exponential property \( \frac{e^x}{e^y} = e^{x-y} \). Apply this to the given expression: \[ \frac{e^{2+3 \pi i}}{e^{-3+\pi i / 2}} = e^{(2+3\pi i) - (-3+\pi i / 2)} \]Simplify the exponent:\[ e^{2 + 3\pi i + 3 - \pi i / 2} = e^{5 + \pi(6 - \frac{1}{2})i} = e^{5 + \frac{11\pi}{2} i} \]

Step 2: Use Euler's Formula

Recall Euler's formula: \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \). Apply Euler's formula to the complex exponent:\[ e^{5 + \frac{11\pi}{2}i} = e^5 (\cos(\frac{11\pi}{2}) + i\sin(\frac{11\pi}{2})) \]

Step 3: Simplify the Trigonometric Functions

Recall that \( \cos(\theta) \) and \( \sin(\theta) \) are periodic with period \( 2\pi \). Therefore, \( \frac{11\pi}{2} \) corresponds to \( \frac{11\pi}{2} - 2\pi \times 2 = \frac{3\pi}{2}\):\( \cos(\frac{3\pi}{2}) = 0 \) and \( \sin(\frac{3\pi}{2}) = -1\). So we have:\[ e^5 (0 + i(-1)) = e^5 (0 - i) = -ie^5 \]

Step 4: Express in the Form \(a + ib\)

The final expression is \(-ie^5\), which can be written as \(a + ib\) by recognizing that \(a = 0\) and \(b = -e^5\). Hence, the answer is:\[ 0 - i e^5 \]

Key Concepts

Euler's Formula

Euler's Formula is a fascinating bridge between trigonometry and complex exponential functions. It's succinctly given by the expression \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \), where \( i \) is the imaginary unit, satisfying \( i^2 = -1 \).
This formula enables the representation of complex numbers in exponential form, seamlessly connecting exponential functions to trigonometric ones.
When applying Euler's formula in mathematics, we convert complex exponential expressions to their respective polar form. This is achieved by defining a complex number's magnitude and direction using trigonometric functions:

• \( \cos(\theta) \) describes the x-coordinate on the unit circle, which represents the real part of the complex number.
• \( i\sin(\theta) \) represents the y-coordinate, which defines the imaginary part.

For instance, in our exercise, the part \( e^{\frac{11\pi}{2}i} \) was transformed into \( \cos(\frac{11\pi}{2}) + i\sin(\frac{11\pi}{2}) \). Using the periodic properties of cosine and sine, which repeat every \( 2\pi \), \( \frac{11\pi}{2} \) simplifies to \( \frac{3\pi}{2} \), leading to the real part being zero and the imaginary part \(-1\). This understanding of direction and magnitude is powerful in complex analysis.

Exponential Functions

Exponential functions, represented as \( e^x \), describe growth or decay processes, distinct from algebraic ones. In the context of complex numbers, they assume an intriguing dimension through the form \( e^{x + yi} \).
Here, \( x \) determines the growth or decay of the function, while \( y \) introduces rotational dynamics in the complex plane. This becomes apparent in the expression treated in the original solution: \( e^{5 + \frac{11\pi}{2}i} \).
The '5' denotes the exponential growth of the function's magnitude, whereas \( \frac{11\pi}{2} \) affects the function's position via Euler's formula.

• Multiplying the exponential part initially simplifies complex expressions, owing to properties such as \( \frac{e^x}{e^y} = e^{x-y} \).
• Converting \( e^{5 + \frac{11\pi}{2}i} \) into polar form illustrates both the growth with \( e^5 \) and orientation in the complex plane, namely \(-ie^5 \).

This connection implies that exponential functions effectively manage transformations involving both scaling and rotation, which is crucial in fields counting on complex calculus.

Polar Form of Complex Numbers

The Polar Form of Complex Numbers is a versatile technique used to represent these numbers by linking their modulus and argument. A complex number \( z = a + ib \) can be expressed in polar form as \( re^{i\theta} \), where:

• \( r \) is the modulus \( \sqrt{a^2 + b^2} \), representing the distance from the origin on the complex plane.
• \( \theta \) is the argument, the angle subtended from the positive x-axis to the line segment representing the complex number.

This form is particularly useful when multiplying or dividing complex numbers, as one can simply multiply or add their moduli and angles respectively.
In our example, \( e^{5 + \frac{11\pi}{2}i} \) naturally lends itself to polar representation by showcasing both magnitude \( e^5 \) and direction \( \frac{11\pi}{2} \) or \( \frac{3\pi}{2} \).
Consider the final expression \(-ie^5 \):

• Magnitude: \( e^5 \)
• Direction: As we saw from \( i\sin(\frac{3\pi}{2}) = -1 \), indicating a downward vertical alignment on the complex plane.

Utilizing polar forms capitalizes on the inherent cyclical nature of trigonometric functions, enabling seamless transitions and operations with complex numbers.