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E-raamat: Correct Antidifferentiation: The Change Of Variable Well Done

  • Formaat: 284 pages
  • Ilmumisaeg: 15-Sep-2020
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789811227479
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Correct Antidifferentiation: The Change Of Variable Well DoneSuurem pilt
  • Formaat: 284 pages
  • Ilmumisaeg: 15-Sep-2020
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789811227479
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A typical source of mistakes that frequently lead to a wrong or incomplete solution for the antiderivative of a given real function of one real variable is a misuse of the technique of change of variable. The increasing implementation of software in apparently mechanic tasks such as the calculation of antiderivatives has not improved the situation, yet those software packages issue generic warnings such as "the answer's is not guaranteed to be continuous" or "the solution might be only valid for parts of the function". The practical meaning of those vague machine messages is clearly envisaged in this book, which shows how to handle the technique of change of variable in order to provide correct solutions.This book is monographically focused on elementary antidifferentiation and reasonably self-contained, yet it is written in a "hand-book" style: it has plenty of examples and graphics in an increasing level of difficulty; the most standard changes of variable are studied and the hardest theoretic parts are included in a final Appendix. Each practical chapter has a list of exercises and solutions.This book is intended for instructors and university students of Mathematics of first and second year.
Preface vii
1 Background
1(12)
1.1 Notation and basic concepts
1(5)
1.2 Sequences of real numbers
6(5)
1.3 Compactness in R
11(2)
2 Real-valued functions of a real variable
13(26)
2.1 Limits of functions
13(5)
2.2 Continuous functions
18(5)
2.3 Differentiable functions
23(9)
2.4 Integrable functions
32(7)
3 Elementary real-valued functions
39(22)
3.1 Power functions and root functions
39(2)
3.2 Polynomials
41(6)
3.3 The exponential function and the log function
47(2)
3.4 General exponentiation
49(1)
3.5 The circular functions and their inverse functions
50(7)
3.6 The hyperbolic functions and their inverse functions
57(4)
4 Antidifferentiation
61(24)
4.1 Antiderivatives and primitives
61(5)
4.2 Fundamental theorems of antidifferentiation. Change of variable
66(5)
4.3 Important results concerning a change of variable
71(5)
4.4 A practical method to perform a change of variable
76(3)
4.5 Continuous functions with no elementary antiderivative
79(2)
4.6 Basic primitives
81(4)
5 Antidifferentiation by parts
85(8)
5.1 Antidifferentiating by parts
85(1)
5.2 Examples
85(4)
5.3 Exercises
89(4)
6 Rational functions
93(22)
6.1 Fractions with denominator of degree one and numerator of degree zero
94(1)
6.2 Fractions with irreducible denominator of degree two and numerator of degree zero
94(1)
6.3 Fractions with irreducible denominator of degree two and numerator of degree one
95(1)
6.4 Fractions of negative degree whose denominator only has simple real roots
96(1)
6.5 Fractions of negative degree whose denominator has only simple roots but at least one is not real
97(6)
6.6 Fractions of negative degree whose denominator has only real roots but at least one is multiple
103(4)
6.7 Fractions of negative degree whose denominator has at least one non-real root and some multiple root
107(4)
6.8 Fractions of non-negative degree
111(1)
6.9 Exercises
112(3)
7 Fractions of polynomials over √ax2 + bx + c
115(10)
7.1 Fractions 1/√ax2 + bx + c
116(2)
7.2 Fractions (Ax + B)/√ax2 + bx + c
118(2)
7.3 Fractions P(x)/√ax2 + bx + c for any polynomial P(x)
120(3)
7.4 Exercises
123(2)
8 Fractions 1/(ax + β)p √ax2 + bx + c, p ε IN
125(14)
8.1 The change of variable t = 1/(ax + β)
126(1)
8.2 Primitives of 1/(ax + β)p √ax2 + bx + c, p ε IN
127(8)
8.3 Exercises
135(4)
9 Rational functions of x and of rational powers of √ax + b/cx + d
139(14)
9.1 The domain of R(x, (ax + b/cx + d)P1/q1, ..., (ax + b/cx + d) pn/qn)
139(2)
9.2 The change of variable g(x) = (ax + b/cx + d) 1/q, 1 > q ε IN
141(1)
9.3 The primitive of R(x, (ax + b/cx + d) P1/q1, ..., (ax + b/cx + d) pn/qn)
141(8)
9.4 Exercises
149(4)
10 Binomial Differentials
153(26)
10.1 The domain of a binomial differential
153(2)
10.2 Changes of variable for binomial differentials
155(1)
10.3 Classification of binomial differentials
156(1)
10.4 The primitive of a binomial differential of type I
157(7)
10.5 The primitive of a binomial differential of type II
164(11)
10.6 Exercises
175(4)
11 Rational functions of trigonometric arguments
179(34)
11.1 The domain of R(sin x, cos x)
179(1)
11.2 Trigonometric changes of variable
180(3)
11.3 The primitive of R(sin x, cos x)
183(9)
11.4 The case R(sin x, cos x) = R(- sin x, - cos x)
192(7)
11.5 The case R(- sin x, cos x) = - R(sin x, cos x)
199(5)
11.6 The case R(sin x, - cos x) = - R(sin x, cos x)
204(3)
11.7 Exercises
207(6)
12 Functions R(x, √ax2 + bx + c)
213(12)
12.1 The domain of R(x, √ax2 + bx + c)
213(1)
12.2 Changes of variable associated with R(x, √ax2 + bx + c)
214(1)
12.3 Functions R(x, √1 - x2)
215(3)
12.4 Functions R(x, √1 + x2)
218(2)
12.5 Functions R(x, √x2 - 1)
220(2)
12.6 Exercises
222(3)
13 Tables
225(6)
13.1 Elementary methods of antidifferentiation
225(3)
13.2 Trigonometric identities
228(3)
Appendix A Complements
231(32)
A.1 Some observations on the real-valued n-th root functions in connection with the complex numbers
231(1)
A.2 The role of the fundamental functions in Real Analysis
232(1)
A.3 Notes on the construction of the functions log x and ex
233(2)
A.4 Notes on the construction of the circular functions
235(1)
A.5 Some facts about the relation between differentiability and integrability
236(1)
A.6 Geometrical representation of complex numbers and the n-th complex roots of unity
237(2)
A.7 Long division of polynomials
239(2)
A.8 Synthetic division of polynomials
241(2)
A.9 Completing squares
243(1)
A.10 The Fundamental Theorem of Algebra
244(4)
A.11 Roots of polynomials of degree greater than two
248(2)
A.12 Decomposition of fractions of polynomials
250(11)
A.13 Cardinality of sets
261(2)
Bibliography 263(2)
Index 265
Antonio Martínez-Abejón, MD in Mathematics by the University of Zaragoza (Spain); PhD in Mathematics by the University of Cantabria (Spain); Martínez-Abejón has been a Teaching Assistant in the University of Zaragoza, a Visiting Researcher at the University of Texas at Austin (USA) and currently works as a Profesor Titular at the University of Oviedo (Spain) and member of the AMS. Active researcher in Functional Analysis, mainly in the isomorphic theory of Banach spaces, their local structure and their operators. He has coauthored with Manuel González the book Tauberian operators, published by Birkhäuser.
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