Table of Contents
Table of Contents
"Exercises of Power, Taylor and Fourier Series"
INTRODUCTION
POWER SERIES
TAYLOR SERIES
FOURIER SERIES
"Exercises of Power, Taylor and Fourier Series"
"Exercises of Power, Taylor and Fourier Series"
SIMONE MALACRIDA
In this book exercises are carried out regarding the following mathematical topics:
power series
developments in Taylor and MacLaurin series
Fourier series
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 – POWER SERIES
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 – TAYLOR SERIES
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
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III - FOURIER SERIES
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
INTRODUCTION
INTRODUCTION
In this workbook, some examples of calculations relating to power series, expansions in Taylor and MacLaurin series and Fourier series are carried out.
Series expansions are a very powerful tool in the context of mathematical analysis.
Power series allow a generalization of any function of real variable and can also be extended to complex analysis.
Taylor series, especially as regards MacLaurin developments, provide a valid support in the calculation of limits and explain, a posteriori, many significant limits.
The Fourier series are part of the more general harmonic analysis, absolutely fundamental in several application fields, such as physics, electromagnetism and telecommunications.
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 exposed in this workbook is generally addressed in the courses of mathematical analysis 1 (for Taylor series) and mathematical analysis 2 for power and Fourier series.
I
POWER SERIES
POWER SERIES
A power series is a particular series of functions that can be expressed with this relation:
These series are generalizations of polynomials and the coefficients can assume real or complex values.
The coefficient c is called the center of the series.
A power series converges for some value of the variable x.
There is a value, called radius of convergence, such that the series converges if this condition is met:
The radius of convergence is calculated by the Cauchy-Hadamard formula:
If the limit exists and is finite, i.e. if the radius of convergence is not infinite, then the previous formula can be simplified according to D'Alembert's formula:
Where the series converges, absolute convergence also occurs, while total and uniform convergence occur for each compact subset.
Abel's theorem states that if the power series converges at a point on the frontier then the series is continuous at that point.
Furthermore, if the power series converges punctually at a point then it converges uniformly in a compact contained within the interval between the magnitude of that point's value.
Abel's theorem allows an estimate of the radius of convergence: in fact, given a series centered in one point and converging in another point, the radius of convergence is greater than or equal to the module of the distance between the two points.
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Given two power series, addition and subtraction are done by adding or subtracting their respective coefficients. The multiplication of two power series is defined as follows:
This product is called the Cauchy
Impressum
Verlag: BookRix GmbH & Co. KG
Tag der Veröffentlichung: 20.04.2023
ISBN: 978-3-7554-3968-4
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