Physics and Mathematical Tools

Methods and Examples

This book presents mathematical methods and tools which are useful for physicists and engineers: response functions, Kramers–Kronig relations, Green's functions, saddle point approximation. The derivations emphasize the underlying physical arguments and interpretations without any loss of rigor. General introductions describe the main features of the methods, while connections and analogies between a priori different problems are discussed. They are completed by detailed applications in many topics including electromagnetism, hydrodynamics, statistical physics, quantum mechanics, etc. Exercises are also proposed, and their solutions are sketched. A self-contained reading of the book is favored by avoiding too technical derivations, and by providing a short presentation of important tools in the appendices. It is addressed to undergraduate and graduate students in physics, but it can also be used by teachers, researchers and engineers.

Contents:

• Linear Response and Analyticity

• Static Green's Functions

• Dynamical Green's Functions

• Saddle-Point Method

• Appendices:

  • Functions of a Complex Variable
  • Laplace Transform
  • One-Variable Differential Operators
  • Hilbert Spaces and Dirac Notation
  • Gaussian Integrals
  • Overview of Coordinate Transformations
  • Spherical Harmonics
  • Functional Derivative
  • Usual Green's Functions
  • Solutions of Exercises
  • References

Readership: Graduate students in general physics, electrodynamics, statistical physics, plasma physics, quantum mechanics and theoretical physics.
Key Features:

• Transversal presentation of methods and tools within a physical framework. Emphasis on the universality of the methods related to basic principles of physics
• Illustration of the methods through detailed physical examples in various fields, which should enforce a global understanding of physics, as well as a good expertise of the application of mathematical tools
• There is no similar book available on the market. Of course, when dealing with mathematical methods for physicists, some books immediately come to mind. However, our book is really different from these books. One important difference is that we always start from a physics point of view and keep it