Guide To Mathematical Methods For Physicists, A: With Problems And Solutions

by Michela Petrini, Gianfranco Pradisi, Alberto Zaffaroni

Overview

Mathematics plays a fundamental role in the formulation of physical theories. This textbook provides a self-contained and rigorous presentation of the main mathematical tools needed in many fields of Physics, both classical and quantum. It covers topics treated in mathematics courses for final-year undergraduate and graduate physics programmes, including complex function: distributions, Fourier analysis, linear operators, Hilbert spaces and eigenvalue problems. The different topics are organised into two main parts — complex analysis and vector spaces — in order to stress how seemingly different mathematical tools, for instance the Fourier transform, eigenvalue problems or special functions, are all deeply interconnected. Also contained within each chapter are fully worked examples, problems and detailed solutions. A companion volume covering more advanced topics that enlarge and deepen those treated here is also available.

Product Details

ISBN-13:	9781786343437
Publisher:	World Scientific Publishing Europe Ltd
Publication date:	08/23/2017
Series:	Essential Textbooks In Physics
Pages:	340
Product dimensions:	5.90(w) x 9.10(h) x 1.20(d)

Table of Contents

Preface v

Part I Complex Analysis 1

1 Holomorphic Functions 3

1.1 Complex Functions 3

1.2 Holomorphic Functions 5

1.3 Singularities of Holomorphic Functions 11

1.4 The Riemann Sphere and the Point at Infinity 21

1.5 Elementary Functions 25

1.6 Exercises 30

2 Integration 33

2.1 Curves in the Complex Plane 33

2.2 Line Integral of a Function Along a Curve 37

2.3 Cauchy's Theorem 41

2.4 Primitive of a Holomorphic Function 45

2.5 Holomorphic Functions, Differential Forms and Vector Fields 48

2.6 Cauchy's Integral Formula 50

2.7 Morera's Theorem for a Simply Connected Domain 52

2.8 Other Properties of Holomorphic Functions 53

2.9 Harmonic Functions 56

2.10 Exercises 58

3 Taylor and Laurent Series 61

3.1 Power Series 61

3.2 Taylor Series 64

3.3 Laurent Series 69

3.4 Analytic Continuation 75

3.5 Exercises 85

4 Residues 87

4.1 Residue of a Function at an Isolated Singularity 87

4.2 Residue Theorem 90

4.3 Evaluation of Integrals by Residue Method 94

4.4 Cauchy's Principal Value 104

4.5 Exercises 107

Part II Functional Spaces 111

5 Vector Spaces 113

5.1 Vector Spaces 113

5.2 Metric, Norm and Scalar Product 116

5.3 Complete Spaces 124

5.4 Finite-Dimensional Hilbert Spaces 125

5.5 Hilbert Spaces 128

5.6 The Orthogonal Complement 132

5.7 Exercises 134

6 Spaces of Functions 137

6.1 Different Norms and Different Notions of Convergence 138

6.2 The Space L¹_ω(Ω) 140

6.3 The Hilbert Space L²_ω(Ω) 144

6.4 Hilbert Basis and Fourier Expansion 146

6.5 Exercises 151

7 Distributions 153

7.1 Test Functions 153

7.2 Distributions 156

7.3 Limits of Distributions 163

7.4 Operations on Distributions 165

7.5 Exercises 173

8 Fourier Analysis 175

8.1 Fourier Series 176

8.2 The Fourier Transform 184

8.3 Exercises 197

9 Linear Operators in Hilbert Spaces I: The Finite-Dimensional Case 201

9.1 Linear Operators in Finite Dimension 201

9.2 Spectral Theory 207

9.3 Exercises 220

10 Linear Operators in Hilbert Spaces II: The Infinite-Dimensional Case 223

10.1 Operators in Normed Spaces 224

10.2 Operators in Hilbert Spaces 229

10.3 Eigenvalues and Spectral Theory 237

10.4 Exercises 253

Part III Appendices 257

Appendix A Complex Numbers, Series and Integrals 259

A.1 A Quick Review of Complex Numbers 259

A.2 Notions of Topology, Sequences and Series 261

A.3 The Lebesgue Integral 264

Appendix B Solutions of the Exercises 269

Bibliography 321

Index 323