Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

(a) Using Euler's relation (2.76), prove that any complex number \(z=x+\) iy can be written in the form \(z=r e^{i \theta},\) where \(r\) and \(\theta\) are real. Describe the significance of \(r\) and \(\theta\) with reference to the complex plane. (b) Write \(z=3+4 i\) in the form \(z=r e^{i \theta} .\) (c) Write \(z=2 e^{-i \pi / 3}\) in the form \(x+i y.\)

Short Answer

Expert verified
Any complex number can be expressed as \( z = re^{i\theta} \). For \( z = 3+4i \), \( z = 5e^{i0.927} \), and for \( z = 2e^{-i\pi/3} \), \( z = 1 - i\sqrt{3} \).

Step by step solution

01

Understanding Euler's Relation

Euler's relation is given by the formula \( e^{i\theta} = \cos\theta + i\sin\theta \). This relation connects complex exponentials with trigonometric functions.
02

Expressing a Complex Number

To express any complex number \( z = x + iy \) in the form \( z = re^{i\theta} \), we first need to identify \( r \) as the modulus of \( z \) and \( \theta \) as the argument. The modulus is \( r = \sqrt{x^2 + y^2} \), and the argument \( \theta \) can be found using \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).
03

Applying to Specific Values

(a) For any complex number \( z = x + iy \), substituting into Euler's formula gives \( z = r(\cos\theta + i\sin\theta) = re^{i\theta} \). This shows that any complex number can be expressed in polar form using Euler's relation.
04

Transforming z = 3+4i

For the complex number \( z = 3 + 4i \):- Calculate \( r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 \).- Calculate \( \theta = \tan^{-1}\left(\frac{4}{3}\right) \).Therefore, \( z = 5 e^{i \theta} \), where \( \theta \approx 0.927 \) radians.
05

Transforming Polar to Rectangular

(b) For \( z = 2 e^{-i \pi / 3} \), the rectangular form is found using \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \):- Here \( r = 2 \), \( \theta = -\pi/3 \).- \( x = 2 \cos(-\pi/3) = 2 \times 1/2 = 1 \).- \( y = 2 \sin(-\pi/3) = -\sqrt{3} \).Thus, \( z = 1 - i\sqrt{3} \).

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 Vaia!

Key Concepts

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

Euler's Relation
Euler's Relation is an important formula in mathematics that relates complex numbers and trigonometry. It is expressed as \( e^{i\theta} = \cos\theta + i\sin\theta \). This elegantly bridges real and imaginary components of a complex number through exponential functions.
In Euler’s form, any complex number can be represented as \( re^{i\theta} \), where \( r \) is the modulus of the number, and \( \theta \) is the argument. This can be visualized by placing the complex number on a unit circle in the complex plane.
Understanding Euler's relation is crucial because it simplifies complex multiplication and powers by transforming them into exponential operations. This transformation opens the door to advanced topics in math and physics by providing a simple yet powerful way to handle complex numbers.
Polar Form of Complex Numbers
The Polar Form provides a way to express complex numbers emphasizing their magnitude and angle, rather than their algebraic form of \( x + iy \). In the polar form, a complex number \( z \) is written as \( r e^{i\theta} \).
  • \( r \) is the modulus or magnitude of the complex number.
  • \( \theta \) is the argument, which is the angle formed with the positive real axis.
To convert a complex number from its algebraic form to polar form:
  • Calculate the modulus \( r \) using \( r = \sqrt{x^2 + y^2} \).
  • Find the argument \( \theta \) using \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).
This form is particularly useful in simplifying calculations involving powers and roots of complex numbers. It provides an intuitive insight into the geometric representation of complex numbers on the Argand diagram.
Modulus and Argument of Complex Numbers
The modulus and argument are fundamental components when working with complex numbers, providing depth to their geometric and mathematical analysis.
  • Modulus: The modulus \( r \) of a complex number \( z = x + iy \) is its distance from the origin in the complex plane, computed as \( r = \sqrt{x^2 + y^2} \).
  • Argument: The argument \( \theta \) is the angle measured between the positive real axis and the line representing the complex number, given by \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).
Together, these components allow complex numbers to be represented in their polar form, bridging algebraic and geometric perspectives. Knowing the modulus and argument facilitates advanced operations, such as multiplication and division, which become much more manageable in polar form.

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

Consider the complex number \(z=e^{i \theta}=\cos \theta+i \sin \theta .\) (a) By evaluating \(z^{2}\) two different ways, prove the trig identities \(\cos 2 \theta=\cos ^{2} \thetaQEDQED-\sin ^{2} \theta\) and \(\sin 2 \theta=2 \sin \theta \cos \theta .\) (b) Use the same technique to find corresponding identities for \(\cos 3 \theta\) and \(\sin 3 \theta\).

A gun can fire shells in any direction with the same speed \(v_{\mathrm{o}}\). Ignoring air resistance and using cylindrical polar coordinates with the gun at the origin and \(z\) measured vertically up, show that the gun can hit any object inside the surface $$z=\frac{v_{0}^{2}}{2 g}-\frac{g}{2 v_{0}^{2}} \rho^{2}$$ Describe this surface and comment on its dimensions.

For each of the following two pairs of numbers compute \(z+w, z-w, z w,\) and \(z / w\) (a) \(z=6+8 i\) and \(w=3-4 i \quad\) (b) \(z=8 e^{i \pi / 3}\) and \(w=4 e^{i \pi / 6}\) Notice that for adding and subtracting complex numbers, the form \(x+i y\) is more convenient, but for multiplying and especially dividing, the form \(r e^{i \theta}\) is more convenient. In part (a), a clever trick for finding \(z / w\) without converting to the form \(r e^{i \theta}\) is to multiply top and bottom by \(w^{*}\); try this one both ways.

Consider an object that is coasting horizontally (positive \(x\) direction) subject to a drag force \(f=-b v-c v^{2} .\) Write down Newton's second law for this object and solve for \(v\) by separating variables. Sketch the behavior of \(v\) as a function of \(t .\) Explain the time dependence for \(t\) large. (Which force term is dominant when \(t\) is large?)

I kick a puck of mass \(m\) up an incline (angle of slope \(=\theta\) ) with initial speed \(v_{\mathrm{o}}\). There is no friction between the puck and the incline, but there is air resistance with magnitude \(f(v)=c v^{2} .\) Write down and solve Newton's second law for the puck's velocity as a function of \(t\) on the upward journey. How long does the upward journey last?

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free