Physics and Mathematical Tools

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.

Analyticity and Linear Response: Definition of the Susceptibility; Analyticity of the Susceptibility; Kramers-Kronig Relations; Applications and Examples: Admittance of a RLC Circuit; Absorption and Dispersion in a Dielectric; Oscillating Flow in a Capillary; Response of a Plasma within the Vlasov Approximation; Conductivity and Kubo Formula;; Static Green's Functions: Definition and Properties of Green's Functions; Operator Point of View; Laplace Operator; Helmholtz Operator; Applications and Examples: Origin of the Method of Images; Ball in Uniform Motion in a Fluid; Density of States of a Quantum Particle; Scattering by a Repulsive Potential;; Time-Dependent Green's Functions: Green's Functions and Causality; Diffusion Equation; Schrödinger Equation; Wave Equation; Applications and Examples: Diffusion on a Segment; Fraunhofer Diffraction; Emission of Sound Waves; Wavefront in Supersonic Regime; Propagation of Heat; Polarizability of the Hydrogen Atom;; Saddle Point Method: Simple Integrals; Integral on a Path in the Complex Plane; Case of Multiple Integral; Applications and Examples: Equivalence of Canonical and Micro-Canonical Ensembles; Harmonic Crystal at Low Temperature; Ising Model; Semi-Classical Approximation