Problem 12 - Advanced Engineering Mathematics

Question

In Problems 11 and 12, 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 \(0 + ie^5\).

Step 1: Simplify the Exponents

First, simplify the expression by separating the real and imaginary parts in the exponents. We have:\[e^{2+3 heta i} = e^2 e^{3\pi i} \quad \text{and} \quad e^{-3+\pi i / 2} = e^{-3} e^{\pi i / 2}\]

Step 2: Simplify the Quotient

Divide each part separately:\[\frac{e^2 e^{3\pi i}}{e^{-3} e^{\pi i / 2}} = e^{2 - (-3)} \cdot \frac{e^{3\pi i}}{e^{\pi i / 2}}\]This simplifies to:\[e^5 \cdot e^{(3\pi - \pi/2)i} = e^5 \cdot e^{\frac{5\pi}{2}i}\]

Step 3: Calculate Real and Imaginary Parts

Convert the exponential of an imaginary number to trigonometric form using Euler's Formula: \[e^{\theta i} = \cos(\theta) + i\sin(\theta)\]Thus:\[e^{\frac{5\pi}{2}i} = \cos\left(\frac{5\pi}{2}\right) + i\sin\left(\frac{5\pi}{2}\right)\]Note that \(\frac{5\pi}{2} = 2\pi + \pi/2\), which has the same sine and cosine as \(\pi/2\):\[ \cos(\pi/2) = 0, \quad \sin(\pi/2) = 1 \]Thus:\[e^{\frac{5\pi}{2}i} = 0 + i(1) = i\]

Step 4: Multiply and Simplify

Combine the results of previous steps:\[e^5 \cdot e^{\frac{5\pi}{2}i} = e^5 (i)\]So the expression in the form \(a + ib\) is:\[0 + ie^5\]

Key Concepts

Exponential Form of Complex Numbers

Complex numbers can be expressed in various forms, and the exponential form is one of them. It beautifully combines trigonometry with exponential functions, which can simplify many complex operations. In its exponential form, a complex number is written as \( e^{x+yi} \), where:

• \( x \) is the real part
• \( y \) is the imaginary part
• \( e \) is Euler's number, approximately equal to 2.71828

This format is particularly useful in engineering and physics, as it simplifies multiplication and division of complex numbers. By using the property \( e^{a+b} = e^a \cdot e^b \), it's possible to separate the real and imaginary parts to handle them individually. This makes operations with complex numbers much clearer and more straightforward.

Euler's Formula

Euler's Formula provides the bridge between exponential and trigonometric forms of complex numbers. It states that for any real number \( \theta \):\[e^{i\theta} = \cos(\theta) + i\sin(\theta)\]Using this formula, you can convert complex numbers between these two forms seamlessly. Here’s how it helps:

• Transforms an exponential expression into a trigonometric one, revealing the relationship between the angles \( \theta \) and the real and imaginary parts.
• Makes it easier to visualize and work with rotations and oscillations in the complex plane.

For example, in the given problem, the exponential expression \( e^{\frac{5\pi}{2}i} \) is turned into its trigonometric counterpart \( \cos\left(\frac{5\pi}{2}\right) + i\sin\left(\frac{5\pi}{2}\right) \). Euler’s formula becomes a powerful tool, especially for simplifying complex number calculations.

Trigonometric Form of Complex Numbers

The trigonometric form, another way to express complex numbers, emphasizes the role of angles and magnitudes. It is written as \( r(\cos(\theta) + i\sin(\theta)) \), where:

• \( r \) is the magnitude of the complex number.
• \( \theta \) is the argument or angle.

This form employs trigonometric functions to denote the real and imaginary components, making it handy for analyzing problems involving periodic events or waves. Converting from exponential to trigonometric form via Euler's Formula helps in breaking down complex expressions. For instance, when dealing with \( e^{\frac{5\pi}{2}i} \) in its trigonometric form as demonstrated, it simplifies down to \( i \), since the cosine part vanishes. Understanding the trigonometric form is crucial for interpreting how complex numbers position themselves on the plane relative to their angles and magnitudes.