Problem 19 - Advanced Engineering Mathematics

Question

Verify the given result. $$ e^{z+\pi i}=e^{z-\pi i} $$

Step-by-Step Solution

Answer

The equality holds because of the periodic property: \(e^{z+2\pi i} = e^z\).

Step 1: Understanding the Exponential Function

First, recall that the complex exponential function has a periodic property when dealing with imaginary components. Specifically, for any complex number \( z \) and any integer \( n \), the exponential function satisfies: \( e^{z+2\pi ni} = e^z \). This means that adding or subtracting multiples of \( 2\pi i \) to the exponent of \( e \) does not change its value.

Step 2: Analyzing the Given Expression

The given expression is \( e^{z+\pi i} = e^{z-\pi i} \). Notice that \( (z+\pi i) \) and \( (z-\pi i) \) differ by \( 2\pi i \) because: \((z + \pi i) - (z - \pi i) = 2\pi i \).

Step 3: Applying the Periodicity Property

Using the periodic property of the exponential function, \( e^{z+\pi i} = e^{z} \cdot e^{\pi i} \) and \( e^{z-\pi i} = e^z \cdot e^{-\pi i} \). Since \( e^{2\pi i} = 1 \), multiplying \( e^z \cdot e^{\pi i} \) by \( e^{2\pi i} = 1 \) gives \( e^{z-\pi i} \). This confirms that both sides of the equation are indeed equal.

Key Concepts

Periodicity of Exponential Functions

The concept of periodicity in exponential functions is at the heart of understanding how the complex exponential function behaves. Specifically for complex exponential functions with imaginary exponents, periodicity plays a critical role. The key property here is that if you have any complex number \( z \), the expression \( e^{z+2\pi ni} = e^z \) is true for any integer \( n \).

This means that you can add or subtract any multiples of \( 2\pi i \) to the exponent of the base \( e \), and the value of the entire expression remains unchanged. This property is crucial to a wide range of problems and is often encapsulated in the identity \( e^{2\pi i} = 1 \).

• This identity highlights how rotation by \( 2\pi \) in the imaginary plane brings you back to the same point.
• It also simplifies complex expressions involving exponential terms by reducing them based on this periodic nature.

Understanding this principle allows you to simplify and manipulate complex exponential expressions accurately and insightfully.

Complex Numbers

Complex numbers form the foundation for working with the complex exponential function. A complex number is generally expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit, with the property that \( i^2 = -1 \).

The use of complex numbers allows us to extend the idea of the exponential function from real to complex domains. This extension is not merely a mathematical curiosity; it is indispensable in fields like electrical engineering and quantum mechanics.

• Complex numbers allow the representation of oscillations and waveforms in terms of exponential functions.
• They enable the simplification of trigonometric identities into exponential forms, leading to more flexible problem-solving techniques.

Grasping the properties and behaviors of complex numbers expands your mathematical toolbox and enhances your understanding of various phenomena in physics and engineering.

Euler's Formula

Euler's formula is a pivotal connection between exponential functions and trigonometry. Euler's formula states:

\[ e^{ix} = \cos(x) + i\sin(x)\]

This formula demonstrates how the complex exponential function relates to the trigonometric functions cosine and sine. Here, \( x \) is a real number, and the formula shows how exponential functions handle rotations in the complex plane.

• Using Euler's formula, you can express complex exponential functions in terms of their trigonometric counterparts.
• It provides insights into the motifs that underlie some periodic functions, by showing how these functions naturally repeat.

Euler's formula not only bridges the gap between trigonometry and complex numbers but also provides a deeper insight into the interplay of different mathematical domains. It's a powerful tool that reveals the inherent periodic properties of complex exponentials.