Problem 37 - Advanced Engineering Mathematics

Question

Find all values of \(z\) satisfying the given equation. \(e^{z-1}=-i e^{2}\)

Step-by-Step Solution

Answer

The solution is \(z = 3 - \frac{i\pi}{2}\).

Step 1: Understand the Equation

The equation given is \(e^{z-1} = -i e^{2}\). To solve this, recognize that it involves exponential functions and the imaginary number \(i\), which indicates a connection to complex numbers.

Step 2: Convert to Standard Exponential Form

Express the right side of the equation in a standard exponential form. Note that the complex number \(-i\) can be expressed using Euler's formula as \(-i = e^{-i\pi/2}\). Thus, we rewrite the equation as follows: \(-i e^2 = e^2 e^{-i\pi/2} = e^{2 - i\pi/2}\).

Step 3: Equate Exponents

Since the bases of the exponential terms in the equation \(e^{z-1} = e^{2 - i\pi/2}\) are the same, their exponents must be equal. Therefore, set the exponents equal to each other: \(z - 1 = 2 - i\pi/2\).

Step 4: Solve for z

Solve the equation \(z - 1 = 2 - i\pi/2\) by adding 1 to both sides to isolate \(z\):\[z = 3 - \frac{i\pi}{2}.\]

Key Concepts

Exponential Equations

Exponential equations like the one in our problem, which is expressed as \( e^{z-1} = -i e^{2} \), involve finding an unknown variable in the exponent. In these equations, the bases on both sides of the equation are the same. That means we can equate the exponents to solve the equation.

Here are some key aspects to understand exponential equations:

• **Recognizing the Base:** It's important to identify that\( e \) (Euler's number) is commonly the base in such problems when dealing with complex numbers and exponential growth or decay.
• **Changing Forms:** We often need to convert the equation to a standard form. This helps in solving the equation by setting their exponents equal. For instance, the complex number \(-i\) was converted using Euler's formula.
• **Solving for the Variable:** Once the exponents are equal, solving an exponential equation becomes straightforward algebra, turning into an equation like \( z - 1 = 2 - \frac{i\pi}{2} \).

Euler's Formula

Euler's formula is a fundamental bridge between complex numbers and exponential functions. It states that for any real number \( \theta \), \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \). This relation is crucial in transforming complex numbers into exponential form.

In our exercise, we used Euler's formula to convert the imaginary unit \(-i\) into an exponential form. Here's how:

• **Calculation Using Euler's Formula:** Recognize that \(-i\) can be written as \( e^{-i\pi/2} \). This expresses \(-i\) as an exponential expression, which helps in equating the exponents.
• **Form Transformation Advantage:** Using Euler's formula simplifies the handling of complex numbers by allowing their transformation into exponential form, making calculations more manageable.
• **General Use in Complex Analysis:** Beyond solving equations, Euler's formula is also instrumental in understanding oscillations and waves, as it elegantly ties together trigonometric functions and exponential functions.

Imaginary Numbers

Imaginary numbers are an intriguing concept and a critical component of complex numbers. An imaginary number is defined as \( i \), the square root of \(-1\). In our exponential equation, the term \(-i\) signifies a rotation in the complex plane by \( -\frac{\pi}{2} \) radians.

Understanding imaginary numbers involves grasping a few essential concepts:

• **Basic Definition:** \( i \) satisfies the equation \( i^2 = -1 \). This extends the real number line into a complex plane.
• **Complex Numbers:** They are of the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit.
• **Role in Rotations:** Imaginary numbers help represent rotations in the complex plane, like how \( -i \) represents a 90-degree counterclockwise rotation.
• **Usage in Exponential Equations:** They enable the representation of complex numbers in exponential form, using Euler's formula for efficient computations.

By understanding imaginary numbers, students can better grasp the concept of complex numbers and solve complex exponential equations effectively.