Problem 18 - Advanced Engineering Mathematics

Question

Verify the given result. $$ \frac{e^{z_{1}}}{e^{z_{2}}}=e^{z_{1}-z_{2}} $$

Step-by-Step Solution

Answer

The equation \( \frac{e^{z_1}}{e^{z_2}} = e^{z_1 - z_2} \) is verified using properties of exponents.

Step 1: Identify the Given Expression

We need to verify the equation \( \frac{e^{z_1}}{e^{z_2}} = e^{z_1 - z_2} \). This equation involves exponential functions with complex exponents, where \( z_1 \) and \( z_2 \) are complex numbers.

Step 2: Recall the Property of Exponents

Recall the fundamental property of exponents: for any real or complex numbers \( a \), \( b \), and \( c \), \( \frac{a^b}{a^c} = a^{b-c} \). This property indicates that dividing two exponential expressions with the same base is equivalent to subtracting the exponents.

Step 3: Apply the Exponential Property

Apply the property from Step 2 to the given expression: \( \frac{e^{z_1}}{e^{z_2}} = e^{z_1 - z_2} \). Here, we are using \( a = e \), \( b = z_1 \), and \( c = z_2 \). This step confirms that the left-hand side simplifies to the right-hand side.

Step 4: Verify the Result

Since the simplification using the property of exponents holds true under all circumstances for complex numbers, the original equation \( \frac{e^{z_1}}{e^{z_2}} = e^{z_1 - z_2} \) is verified as mathematically accurate.

Key Concepts

Properties of Exponents

Exponentiation is a fundamental mathematical operation and its properties are crucial to understanding both real and complex exponentials. Exponential functions denote repeated multiplication of the same base.

• When you multiply exponential expressions with the same base, you add the exponents: \[a^m \cdot a^n = a^{m+n}\].
• For division, you subtract the exponents: \[\frac{a^m}{a^n} = a^{m-n}\]. This is the key property used in our exercise.
• Raising an exponential expression to another power results in multiplying the exponents: \[(a^m)^n = a^{m \cdot n}\].

These properties help simplify expressions, making them easier to handle and solve. They hold true for both real and complex numbers. Understanding these rules is fundamental before diving into complex numbers and their operations.

Complex Numbers

Complex numbers expand our understanding of numerical systems beyond the real numbers. They have the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

• The imaginary unit \(i\) is defined as \(\sqrt{-1}\).
• Complex conjugates, which are pairs like \(a+bi\) and \(a-bi\), are important for simplification in division.
• Complex numbers can be represented in polar form using the magnitude and angle: \(r \text{cis} \theta\), where \(r\) is the modulus, and \(\theta\) is the argument.

In exponential functions, complex exponents result in values that can be simplified using Euler's formula: \(e^{i\theta} = \cos \theta + i \sin \theta\). This representation connects exponentials and trigonometric functions beautifully.

Verification of Mathematical Equations

Verifying mathematical equations confirms that the equations hold true under all specified conditions. For the equation \(\frac{e^{z_1}}{e^{z_2}} = e^{z_1 - z_2}\), we establish its truth by:

• Identifying the components of the equation. Knowing \(z_1\) and \(z_2\) are complex numbers helps clarify the roles of these variables in calculations.
• Applying known mathematical principles. The properties of exponents simplify and prove the relation between divided exponentials.
• Recognizing that these properties apply universally for all complex numbers, ensuring that the equation holds true in general scenarios.

This process of verification not only solidifies our understanding of mathematical concepts but also assures us of the robustness and consistency of the mathematical principles used.