Problem 17 - Advanced Engineering Mathematics

Question

$$\left|e^{z}\right|^{2}=e^{2 x} \cos ^{2} y+e^{2 x} \sin ^{2} y=e^{2 x}\left(\cos ^{2} y+\sin ^{2} y\right)=e^{2 x} \quad \text { implies }\left|e^{z}\right|=e^{x}$$

Step-by-Step Solution

Answer

The expression verifies that \(|e^{z}| = e^{x}\) as required.

Step 1: Understand the Expression

We start with the given expression: \( |e^{z}|^2 = e^{2x} \cos^2 y + e^{2x} \sin^2 y \). This expression needs to be simplified or verified.

Step 2: Factor Out \(e^{2x}\)

Notice that \( e^{2x} \) is a common factor in both terms on the right-hand side. We can factor it out: \( e^{2x} (\cos^2 y + \sin^2 y) \).

Step 3: Use Pythagorean Identity

Recall the Pythagorean identity: \( \cos^2 y + \sin^2 y = 1 \). Substitute this identity into the expression: \( e^{2x} (1) = e^{2x} \).

Step 4: Take Square Root

The equation \( |e^{z}|^2 = e^{2x} \) implies taking the square root on both sides: \( |e^{z}| = \sqrt{e^{2x}} \).

Step 5: Simplify the Square Root

\( \sqrt{e^{2x}} = e^{x} \) because the square root of an exponential function like \( e^{2x} \) simplifies to \( e^{x} \).

Key Concepts

Magnitude of Complex Numbers

Understanding the magnitude of a complex number is a key concept in complex analysis. A complex number is typically expressed in the form \( z = x + yi \), where \( x \) is the real part and \( y \) is the imaginary part. The magnitude (or modulus) of a complex number, denoted as \( |z| \), measures the distance of the complex number from the origin on the complex plane.

The formula to find the magnitude of a complex number \( z \) is:

• \( |z| = \sqrt{x^2 + y^2} \)

This is analogous to finding the hypotenuse in a right triangle, where \( x \) and \( y \) are the triangle's legs.

In the context of exponential functions involving complex numbers, such as \( |e^z| \), the magnitude can be simplified. The original exercise shows that \( |e^{z}|^2 = e^{2x} \). By taking the square root of both sides:

• \( |e^{z}| = e^{x} \)

This tells us that the magnitude of \( e^z \) is determined solely by the exponential part related to \( x \), the real component of the complex number in the exponential function.

Euler's Formula

Euler's Formula is a remarkable relationship in mathematics that links complex exponentials to trigonometric functions. It states:

• \( e^{iy} = \cos y + i\sin y \)

This equation connects exponential functions involving imaginary numbers to sine and cosine functions, showing a deep relationship between these seemingly different mathematical worlds.

It is essential when working with expressions of the form \( e^z \), where \( z = x + yi \). By Euler’s formula, we can express \( e^z \) as:

• \( e^z = e^x(\cos y + i\sin y) \)

This structure helps simplify the calculation of magnitudes and supports complex calculations.

In practice, Euler's Formula provides a powerful tool for simplifying and understanding expressions involving complex exponentials, as it enables the separation of magnitude and phase, allowing easier manipulation of complex numbers.

Pythagorean Identity

The Pythagorean Identity is a fundamental rule in trigonometry, describing the relationship between the squares of the sine and cosine of the same angle. It is expressed as:

• \( \cos^2 y + \sin^2 y = 1 \)

This identity is incredibly useful in scenarios where trigonometric terms appear together.

In the context of the original exercise, this identity is employed to simplify complex expressions. Particularly, \( \cos^2 y + \sin^2 y \) was replaced by 1, thereby simplifying the complex exponential expression \( |e^z|^2 = e^{2x}(\cos^2 y + \sin^2 y) \) to \( e^{2x} \).

By utilizing the Pythagorean Identity, mathematical expressions that are initially broad or complex are reduced to simpler forms. This not only helps in theoretical situations but is also practical in computations involving wave functions, electrical circuits, and other applications where complex numbers and trigonometric functions intertwine.