Exercises of Complex Numbers E-Book

Simone Malacrida

• Herausgeber: Simone Malacrida
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2022

Beschreibung

In this book, exercises are carried out regarding the following mathematical topics:
complex numbers
solving third degree equations
hyperbolic functions and properties
Initial theoretical hints are also presented to make the performance of the exercises understood.

Leseprobe

Table of Contents

"Exercises of Complex Numbers"

INTRODUCTION

COMPLEX NUMBERS

HYPERBOLIC FUNCTIONS

"Exercises of Complex Numbers"

SIMONE MALACRIDA

In this book, exercises are carried out regarding the following mathematical topics:

complex numbers

solving third degree equations

hyperbolic functions and properties

Initial theoretical hints are also presented to make the performance of the exercises understood.

Simone Malacrida (1977)

Engineer and writer, has worked on research, finance, energy policy and industrial plants.

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ANALYTICAL INDEX

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INTRODUCTION

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I – COMPLEX NUMBERS

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Exercise 9

Exercise 10

Exercise 11

Exercise 12

Exercise 13

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II – HYPERBOLIC FUNCTIONS

Exercise 1

Exercise 2

Exercise 3

INTRODUCTION

In this workbook, some examples of calculations relating to complex numbers and hyperbolic functions are carried out.

The definition of complex numbers has led to the overcoming of many barriers in the development of elementary mathematics, from the removal of several conditions of existence to the algebraic resolution of previously unsolvable equations up to the enunciation of the fundamental theorem of algebra.

On the other hand, hyperbolic functions are an extension of both trigonometric and logarithmic and exponential functions.

In order to understand in more detail what is explained in the resolution of the exercises, the theoretical context of reference is recalled at the beginning of each chapter.

What is presented in this workbook is generally covered in basic courses of mathematical analysis at university level.

I

COMPLEX NUMBERS

Complex numbers were introduced to give solutions to polynomial equations in any case , but nowadays they represent an important mathematical tool for solving various concrete problems, from physics to electrical engineering, from signal transmission to mechanics.

The starting point is the definition of an imaginary unit , called i which satisfies this property:

Therefore, with the introduction of complex numbers, the condition of existence given by the radical greater than or equal to zero for roots with even index is removed.

A complex number is defined by taking a real part and an imaginary part, like this:

Where a is the real part, denoted by Re(z), while b is the imaginary part, denoted by Im(z). This notation of complex numbers is called Cartesian form and the complex number is said to be expressed in Cartesian coordinates.

A complex number is an ordered pair of real numbers (a,b) and the real and imaginary parts are obtained simply by setting b=0 or a=0.

In this way every complex number can be written as a linear combination: