A Guide to Mathematical Methods for Physicists: Advanced Topics and Applications (Advanced Textbooks in Physics)

✍ Scribed by Michela Petrini, Gianfranco Pradisi, Alberto Zaffaroni

• Publisher: WSPC (Europe)
• Year: 2019
• Tongue: English
• Leaves: 306
• Category: Library

✦ Synopsis

This book provides a self-contained and rigorous presentation of the main mathematical tools needed to approach many courses at the last year of undergraduate in Physics and MSc programs, from Electromagnetism to Quantum Mechanics. It complements A Guide to Mathematical Methods for Physicists with advanced topics and physical applications. The different arguments are organised in three main sections: Complex Analysis, Differential Equations and Hilbert Spaces, covering most of the standard mathematical method tools in modern physics.

One of the purposes of the book is to show how seemingly different mathematical tools like, for instance, Fourier transforms, eigenvalue problems, special functions and so on, are all deeply interconnected. It contains a large number of examples, problems and detailed solutions, emphasising the main purpose of relating concrete physical examples with more formal mathematical aspects.

✦ Table of Contents

Contents
Preface
Part I. Complex Analysis
Introduction
1. Mapping Properties of Holomorphic Functions
1.1. Local Behaviour of Holomorphic Functions
1.2. The Riemann Mapping Theorem
1.3. Applications of Conformal Mapping
1.3.1. The Dirichlet problem for the Laplace equation
1.3.2. Fluid mechanics
1.3.3. Electrostatics
1.4. Exercises
2. Laplace Transform
2.1. The Laplace Transform
2.1.1. Properties of the Laplace transform
2.1.2. Laplace transform of distributions
2.2. Integral Transforms and Differential Equations
2.3. Exercises
3. Asymptotic Expansions
3.1. Asymptotic Series
3.2. Laplace and Fourier Integrals
3.3. Laplace’s Method
3.4. Stationary Phase Method
3.5. Saddle-Point Method
3.6. Exercises
Part II. Differential Equations
Introduction
4. The Cauchy Problem for Differential Equations
4.1. Cauchy Problem for Ordinary Differential Equations
4.2. Second-Order Linear Ordinary Differential Equations
4.3. Second-Order Linear Partial Differential Equations
4.4. Exercises
5. Boundary Value Problems
5.1. Boundary Value Problems in One Dimension
5.1.1. Sturm–Liouville problems
5.1.2. Notable examples of Sturm–Liouville problems
5.2. Boundary Value Problems for Partial Differential Equations
5.3. The Dirichlet Problem for the Sphere
5.3.1. Spectrum of the Laplace operator on the sphere
5.3.2. The solution of the Dirichlet problem
5.4. Exercises
6. Green Functions
6.1. Fundamental Solutions and Green Functions
6.2. Linear Ordinary Differential Equations
6.2.1. Fundamental solution for equations with constant coefficients
6.2.2. Green functions for boundary and initial problems
6.3. The Fourier Transform Method
6.4. Linear Partial Differential Equations
6.4.1. Laplace equation
6.4.2. Heat equation
6.4.3. Wave equation
6.4.4. Cauchy–Riemann operator
6.5. Green Functions and Linear Response
6.6. Green Functions and the Spectral Theorem
6.7. Exercises
7. Power Series Methods
7.1. Ordinary and Singular Points of an Ordinary Differential Equation
7.2. Series Solutions for Second-Order Linear Ordinary Differential Equations
7.2.1. Solutions around ordinary points
7.2.2. Solutions around fuchsian points
7.2.3. Solutions around an essential singular point
7.3. Exercises
Part III. Hilbert Spaces
Introduction
8. Compact Operators and Integral Equations
8.1. Compact Operators
8.1.1. Fredholm alternative
8.2. Fredholm Equation of Second Type
8.2.1. Method of iterations
8.2.2. Degenerate kernels
8.3. Exercises
9. Hilbert Spaces and Quantum Mechanics
9.1. The Schrödinger Equation
9.2. Quantum Mechanics and Probability
9.3. Spectrum of the Hamiltonian Operator
9.4. Heisenberg Uncertainty Principle
9.5. Compatible Observables
9.6. Time Evolution for Conservative Systems
9.7. Dirac Notation
9.8. WKB Method
9.9. Exercises
Part IV. Appendices
Appendix A. Review of Basic Concepts
Appendix B. Solutions of the Exercises
Bibliography
Index