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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

The hyperbolic functions cosh \(z\) and \(\sinh z\) are defined as follows: $$\cosh z=\frac{e^{z}+e^{-z}}{2} \quad \text { and } \quad \sinh z=\frac{e^{z}-e^{-z}}{2}$$ for any \(z,\) real or complex. (a) Sketch the behavior of both functions over a suitable range of real values of \(z\). (b) Show that \(\cosh z=\cos (i z)\). What is the corresponding relation for \(\sinh z ?\) (c) What are the derivatives of cosh \(z\) and \(\sinh z ?\) What about their integrals? ( \(\mathbf{d}\) ) Show that \(\cosh ^{2} z-\sinh ^{2} z=1\) (e) Show that \(\int d x / \sqrt{1+x^{2}}=\operatorname{arcsinh} x\). [Hint: One way to do this is to make the substitution \(x=\sinh z .]\)

where the primes denote successive derivatives of \(f(x)\). (Depending on the function this series may converge for any increment \(\delta\) or only for values of \(\delta\) less than some nonzero "radius of convergence.") This theorem is enormously useful, especially for small values of \(\delta\), when the first one or two terms of the series are often an excellent approximation. \(^{11}\) (a) Find the Taylor series for \(\ln (1+\delta)\). (b) Do the same for cos \(\delta\). (c) Likewise sin \(\delta\). (d) And \(e^{\delta}\).

Prove that \(|z|=\sqrt{z^{*} z}\) for any complex number \(z\).

Problem 2.7 is about a class of one-dimensional problems that can always be reduced to doing an integral. Here is another. Show that if the net force on a one-dimensional particle depends only on position, \(F=F(x),\) then Newton's second law can be solved to find \(v\) as a function of \(x\) given by $$v^{2}=v_{\mathrm{o}}^{2}+\frac{2}{m} \int_{x_{0}}^{x} F\left(x^{\prime}\right) d x^{\prime}$$ [Hint: Use the chain rule to prove the following handy relation, which we could call the " \(v\) dv/d \(x\) rule": If you regard \(v\) as a function of \(x,\) then $$\dot{v}=v \frac{d v}{d x}=\frac{1}{2} \frac{d v^{2}}{d x}$$ Use this to rewrite Newton's second law in the separated form \(m d\left(v^{2}\right)=2 F(x) d x\) and then integrate from \(x_{\mathrm{o}}\) to \(x .\) ] Comment on your result for the case that \(F(x)\) is actually a constant. (You may recognise your solution as a statement about kinetic energy and work, both of which we shall be discussing in Chapter 4.)

[Computer] Use suitable graph-plotting software to plot graphs of the trajectory (2.36) of a projectile thrown at 45 ^above the horizontal and subject to linear air resistance for four different values of the drag coefficient, ranging from a significant amount of drag down to no drag at all. Put all four trajectories on the same plot. [Hint: In the absence of any given numbers, you may as well choose convenient values. For example, why not take \(v_{x \mathrm{o}}=v_{y \mathrm{o}}=1\) and \(g=1 .\) (This amounts to choosing your units of length and time so that these parameters have the value \(1 .\).) With these choices, the strength of the drag is given by the one parameter \(v_{\text {ter }}=\tau,\) and you might choose to plot the trajectories for \(v_{\text {ter }}=0.3,1,3,\) and \(\infty\) (that is, no drag at all), and for times from \(t=0\) to \(3 .\) For the case that \(v_{\text {ter }}=\infty\) you'll probably want to write out the trajectory separately.]

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