A Guide to Mathematical Methods for Physicists

Advanced Topics and Applications

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.

Contents:

• Complex Analysis:

  • Introduction
  • Mapping Properties of Holomorphic Functions
  • Laplace Transform
  • Asymptotic Expansions

• Differential Equations:

  • Introduction
  • The Cauchy Problem for Differential Equations
  • Boundary Value Problems
  • Green Functions
  • Power Series Methods

• Hilbert Spaces:

  • Introduction
  • Compact Operators and Integral Equations
  • Hilbert Spaces and Quantum Mechanics

• Appendices:

  • Review of Basic Concepts
  • Solutions of the Exercises

Readership: Students and professionals in the field.
Key Features:

• This book treats the chosen topics in depth, always keeping in mind physical applications and practical aspects
• The book contains many examples and exercises with solutions. This is of course of practical use for the students, and it is also meant to stress the main purpose of relating concrete physical examples with more formal mathematical aspects