Problem 37 In Problems 35-38, find all valu... [FREE SOLUTION]

Chapter 17: Problem 37

In Problems 35-38, find all values of $z$ satisfying the given equation. $$ e^{z-1}=-i e^{2} $$

Short Answer

Expert verified

All values of $z$ are $z = 3 + i\left(-\frac{\pi}{2} + 2k\pi\right)$ for any integer $k$.

Step by step solution

Write the Equation in Exponential Form

We start by recognizing that \[ e^{z-1} = -i e^2 \]is an exponential equation. Our goal is to express both sides in a comparable exponential form, which means treating $-i$ as an exponential.

Express $-i$ Using Euler's Formula

Recall that any complex number can be expressed in exponential form using Euler's formula: \[ -i = e^{i\left(-\frac{\pi}{2}\right)} \]Notice that $-i$ is on the negative imaginary axis, corresponding to an angle of $-\frac{\pi}{2}$ or equivalently $-\frac{\pi}{2} + 2k\pi$ for any integer $k$.

Combine Exponential Parts

Substitute $-i$ in exponential form into the original equation:\[ e^{z-1} = e^{i\left(-\frac{\pi}{2} + 2k\pi\right)} e^2 \]which simplifies to:\[ e^{z-1} = e^{2 + i\left(-\frac{\pi}{2} + 2k\pi\right)} \]

Equate Exponents

Since the bases of the exponentials on both sides are the same, the exponents must be equal. Thus, we have:\[ z-1 = 2 + i\left(-\frac{\pi}{2} + 2k\pi\right) \]

Solve for $z$

Rearrange the equation from Step 4 to solve for $z$:\[ z = 2 + 1 + i\left(-\frac{\pi}{2} + 2k\pi\right) \]which simplifies to:\[ z = 3 + i\left(-\frac{\pi}{2} + 2k\pi\right) \]

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Euler's Formula

Euler's Formula is a fundamental bridge between the trigonometric functions and the complex exponential function. It states that for any real number $ x $, \[ e^{ix} = \cos(x) + i\sin(x) \]This remarkable equation allows us to express complex numbers in an exponential form using the angle $ x $ in radians.

If $ x $ is an angle on the unit circle, $ \cos(x) $ gives the x-coordinate and $ \sin(x) $ gives the y-coordinate, forming the real and imaginary parts of the complex number.
Euler's formula is especially useful for integrating exponential functions and trigonometric identities.
In our exercise, Euler's formula helps transform $-i$ into an exponential form, i.e., $-i = e^{i(-\frac{\pi}{2})}$, by finding the angle $-\frac{\pi}{2}$ on the complex plane.

Understanding this principle is key to handling complex exponential equations efficiently. It simplifies expressions and aids in visualizing complex numbers as points or rotations in the complex plane.

Exponential Equations

Exponential equations involve variables in the exponent. In our original problem, the equation \[ e^{z-1} = -i e^2 \]takes this form, where both sides include exponentials.

To solve such equations, a common method is to express both sides with the same base. This allows us to equate the exponents directly, simplifying the problem significantly.
Both real and complex exponential equations have a structure which makes them solveable by transforming all terms into exponential form.
The solution involves balancing the exponentials by extracting and aligning both the real and imaginary components.

Through exponent rules and understanding of logarithms, complex exponential equations can be broken down systematically.

Complex Exponential Form

Complex exponential form is a method of expressing complex numbers as exponentials, particularly useful in solving problems involving rotation and oscillation.

A complex number $ z $ written as $ z = r e^{i\theta} $ combines the modulus $ r $ with the argument $ \theta $, providing a clear depiction of magnitude and direction.
This form utilizes Euler's formula and is integral for simplifying complex multiplication and division.
In our exercise, expressing $-i$ and other terms in this form allowed us to directly equate exponents, decoupling the problem.
These transformations lead to straightforward solutions, where previously complex arithmetic becomes a matter of exponent comparison.

Mastering the complex exponential form facilitates a broader range of problem-solving tactics by boiling down the essence of complex interactions to rotations and scalings on the complex plane.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

In Problems 35-38, find all values of \(z\) satisfying the given equation. $$ e^{z-1}=-i e^{2} $$

Short Answer

Step by step solution

Write the Equation in Exponential Form

Express \(-i\) Using Euler's Formula

Combine Exponential Parts

Equate Exponents

Solve for \(z\)

Key Concepts

Euler's Formula

Exponential Equations

Complex Exponential Form

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity

Scientific Method Physics

Fundamentals of Physics

Modern Physics

Physics of Motion

Quantum Physics

Study anywhere. Anytime. Across all devices.