Table of Contents
Table of Contents
“Exercises of Transforms”
INTRODUCTION
THEORETICAL OUTLINE
EXERCISES
“Exercises of Transforms”
“Exercises of Transforms”
SIMONE MALACRIDA
In this book, exercises are carried out regarding the following mathematical topics:
Fourier transform
Laplace transform
zeta transform and discrete transforms
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.
ANALYTICAL INDEX
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INTRODUCTION
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I – THEORETICAL OUTLINE
Introduction and definitions
Fourier integral transform
Laplace integral transform
Other integral transforms
Discreet transforms
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II – EXERCISES
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
Exercise 14
Exercise 15
Exercise 16
Exercise 17
Exercise 18
Exercise 19
Exercise 20
Exercise 21
Exercise 22
Exercise 23
Exercise 24
Exercise 25
Exercise 26
Exercise 27
Exercise 28
Exercise 29
Exercise 30
Exercise 31
Exercise 32
Exercise 33
Exercise 34
Exercise 35
Exercise 36
Exercise 37
Exercise 38
Exercise 39
INTRODUCTION
INTRODUCTION
In this exercise book some examples of calculations relating to the transforms are carried out.
Furthermore, the main theorems used in functional analysis of transforms and their practical use in order to solve problems are presented.
Transforms are a powerful mathematical means for solving a variety of mathematical topics such as differential equations and some notable integrals.
In addition, transforms are absolutely fundamental in the fields of telecommunications, electronics, information technology and mechanics.
In order to understand in more detail what is presented in the resolution of the exercises, the theoretical reference context is recalled in the first chapter.
What is presented in this workbook is generally addressed in advanced mathematical analysis courses (analysis 3) or as a preparatory practice for specific university courses.
I
THEORETICAL OUTLINE
THEORETICAL OUTLINE
Introduction and definitions
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Transforms have been introduced to solve many mathematical problems, especially differential equations.
A first large family of transforms are the integral transforms which are integral applications of a space of functions on another space of functions.
The general form of an integral equation is given by:
Where K(s,t) is the function that characterizes the various transforms and is called kernel.
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Fourier integral transform
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We begin by considering the Fourier transform, which is also the most important integral transform.
The Fourier transform of a function belonging to the Lebesgue space 
is:
We will see that the transform can also be extended to the Hilbert space 
.
Fourier's inversion theorem states that if the function and its transform belong to , then it is possible to write the inversion formula as follows:
This inverse function of the Fourier transform is called the inverse Fourier transform.
The Fourier transform has linearity properties and the following properties:
If a function is real and even, then its Fourier transform is real and even, while if the function is real and odd, then its Fourier transform is imaginary and odd.
Given two functions, one of which has compact support and the other can be integrated according to Lebesgue, then the convolution between the two functions is the following relation:
Impressum
Verlag: BookRix GmbH & Co. KG
Tag der Veröffentlichung: 22.04.2023
ISBN: 978-3-7554-3993-6
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