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(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, \( z = 1 - i\sqrt{3} \).

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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.

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Most popular questions from this chapter

2.16 \(\star\) A golfer hits his ball with speed \(v_{\mathrm{o}}\) at an angle \(\theta\) above the horizontal ground. Assuming that the angle \(\theta\) is fixed and that air resistance can be neglected, what is the minimum speed \(v_{\mathrm{o}}(\min )\) for which the ball will clear a wall of height \(h\), a distance \(d\) away? Your solution should get into trouble if the angle \(\tan \theta < h / d .\) Explain. What is \(v_{\mathrm{o}}(\min )\) if \(\theta=25^{\circ}, d=50 \mathrm{m},\) and \(h=2 \mathrm{m} ?\)

A charged particle of mass \(m\) and positive charge \(q\) moves in uniform electric and magnetic fields, \(\mathbf{E}\) and \(\mathbf{B}\), both pointing in the \(z\) direction. The net force on the particle is \(\mathbf{F}=q(\mathbf{E}+\mathbf{v} \times \mathbf{B})\). Write down the equation of motion for the particle and resolve it into its three components. Solve the equations and describe the particle's motion.

Use the series definition (2.72) of \(e^{z}\) to prove that \(^{12} d e^{z} / d z=e^{z}\).

There are certain simple one-dimensional problems where the equation of motion (Newton's second law) can always be solved, or at least reduced to the problem of doing an integral. One of these (which we have met a couple of times in this chapter) is the motion of a one-dimensional particle subject to a force that depends only on the velocity \(v\), that is, \(F=F(v)\). Write down Newton's second law and separate the variables by rewriting it as \(m d v / F(v)=d t .\) Now integrate both sides of this equation and show that $$t=m \int_{v_{0}}^{v} \frac{d v^{\prime}}{F\left(v^{\prime}\right)}$$ Provided you can do the integral, this gives \(t\) as a function of \(v\). You can then solve to give \(v\) as a function of \(t .\) Use this method to solve the special case that \(F(v)=F_{\mathrm{o}},\) a constant, and comment on your result. This method of separation of variables is used again in Problems 2.8 and \(2.9 .\)

Consider a projectile launched with velocity \(\left(v_{x 0}, v_{y_{0}}\right)\) from horizontal ground (with \(x\) measured horizontally and \(y\) vertically up). Assuming no air resistance, find how long the projectile is in the air and show that the distance it travels before landing (the horizontal range) is \(2 v_{x_{0}} v_{y_{0}} / g.\)

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