Complex numbers

Alessio Mangoni

• Éditeur: Alessio Mangoni
• Année: 2020
• ISBN:
  9788835877646
• Sujet: Matemáticas
• Sous-sujet: Análisis complejo

This book is dedicated to complex numbers. In the first part we expose the theory, where the main arguments are: complex numbers in cartesian, polar and exponential forms, the complex conjugate, the modulus, properties and theorems about complex numbers, polynomial equations. In the second part we propose and solve some original exercise on the topics covered in the first part, to fix the concepts.

Part 1

Theory

 

General form

The general form of a complex number,



is



where x and y are real numbers



and they are called the real part and the imaginary part of the complex number z, respectively. The imaginary unit



has the following property



The real and imaginary parts are denoted with



and



The form of Eq. (1.1) is called the cartesian form of z, but there are also other forms to express a complex number, as we will see later.

If the imaginary part is null, y=0, the number z=x is real, while if the real part is null, x=0, the number z=iy is called a purely imaginary number.

Elementary operations

Addition

Consider the two generic complex numbers ...