Introduction to Advanced Mathematical Analysis E-Book

Simone Malacrida

• Herausgeber: BookRix
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2023

Beschreibung

The following mathematical topics are presented in this book: real functions with several variables implicit functions integral calculus for functions of several variables developments in power series, Taylor series and Fourier series analysis in the complex field

Leseprobe

Simone Malacrida

Introduction to Advanced Mathematical Analysis

BookRix GmbH & Co. KG81371 Munich

Table of Contents

Table of Contents

“Introduction to Advanced Mathematical Analysis”

INTRODUCTION

REAL FUNCTIONS WITH MULTIPLE VARIABLES

IMPLICIT FUNCTIONS

INTEGRAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES

DEVELOPMENTS IN SERIES

COMPLEX ANALYSIS

“Introduction to Advanced Mathematical Analysis”

“Introduction to Advanced Mathematical Analysis”

SIMON MALACRIDA

The following mathematical topics are presented in this book:

real functions with several variables

implicit functions

integral calculus for functions of several variables

developments in power series, Taylor series and Fourier series

analysis in the complex field

Simone Malacrida (1977)

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

ANALYTICAL INDEX

––––––––

INTRODUCTION

––––––––

I – REAL FUNCTIONS WITH MULTIPLE VARIABLES

Introduction

Operations

––––––––

II – IMPLICIT FUNCTIONS

––––––––

III - INTEGRAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES

Introduction

Surface and volume integrals

Remarkable theorems

––––––––

IV - DEVELOPMENTS IN SERIES

Convergence criteria for numerical series

Sequence and series of functions

Power series

Taylor and Maclaurin series

Fourier series

––––––––

V - COMPLEX ANALYSIS

Property

Monodromy and polydromy

Complex integration

Euler functions

Complex series

INTRODUCTION

INTRODUCTION

The exposition of mathematical analysis does not stop at the introduction of the concepts of neighbourhood, limit, derivative, integral and at the study of real functions with one variable.

These first notions are only a precondition for other much more advanced concepts and, as such, subsequent not only on a cognitive level but also on an applied level.

Real functions with several variables and implicit functions are a first possible extension, as is integral calculus with several variables.

The two fundamental points, however, are given by the developments in series and by the complex analysis.

The series expansion of a function can be done in many ways and this leads to different mathematical and scientific applications.

Power series, Taylor series, and Fourier series are very powerful and effective symbolisms.

On the other hand, complex analysis makes it possible to extend everything studied in the set of real numbers to that of complex numbers, with considerable benefits in terms of general results.

What is stated in this manual is essential for understanding and solving differential equations and functional analysis problems.

For this reason, the topics presented are generally addressed in advanced mathematical analysis courses (2 and 3).

I

REAL FUNCTIONS WITH MULTIPLE VARIABLES

REAL FUNCTIONS WITH MULTIPLE VARIABLES

Introduction

––––––––

Functions of real variables with several variables are an extension of what has been said for real functions with one variable.

Almost all the properties mentioned for one-variable functions remain valid (such as injectivity, surjectivity and bijectivity), except the ordering property which is not definable.

The domain of a multivariate function is given by the Cartesian product of the domains calculated on the single variables.

A level set , or level curve, is the set of points such that:

The level set with c=0 is used to analyze the sign of the function in the domain.

––––––––

Operations

––––––––