Noble J.V. Mathematical techniques of theoretical physics

Julian V. Noble, 2001. — 411 p.The physicist's description of the universe seems to be most naturally couched in the language of mathematics. Therefore it behooves the physicist to learn as much of his mother tongue as he can absorb|sub jects that seem arcane to one generation become the routine mathematics of the next. Physicists' need for mathematical pro ciency has called forth myriad texts on \Mathematical Methods of Physics" or \Applied Mathematics for Physics and Engineering".
Most such books cover applied analysis (what used to be called multi-dimensional di erential and integral calculus); the theory of functions of a complex variable; areas of advanced algebra such as linear equations and group theory; and sometimes di erential geometry and tensor analysis. Some books of this genre cover topics in numerical mathematics and approximate methods such as perturbations, asymptotic approximations,
The changing needs of new generations of students demand careful selection of the materials covered in a one- or two semester course of mathematical methods of physics. In particular the advent of powerful yet inexpensive general-purpose computers has impacted the subject in two ways.
First, the widespread availability of generalized programs for computeraided mathematical manipulations (Maple R, Mathematica R, muMathR , MACSYMA R , etc.) has convinced many students|and sadly, some of their professors and deans|that graduate study in physics no longer requires mastering the language of mathematics because \the computer can do whatever I need".
But this is false. Computers cannot replace human knowledge and insight.
The plan of the book is as follows:
Review of in nite series, in nite products and improper (in nite) integrals.
Divergent series and their uses. Review of uni- and multi-variate calculus, vector operators, Gauss's and Stokes's Theorems, locally orthogonal curvilinear coordinate systems.
De nition of complex numbers and analytic functions. Proof of Cauchy's Theorem and some consequences. Classification of singularities of functions. Taylor and Laurent Theorems.
Application of Cauchy's theorem to evaluation of de nite integrals. Inverse functions and reversion of series. Analytic continuation. Dispersion relations. Ordinary di erential equations. Special cases. Heaviside operational calculus and Laplace transform methods. Power series expansions. Classi cation of singularities. Special functions. Generating functions. Riemann P-symbol. Hypergeometric, conuent hypergeometric, and Mathieu functions. Asymptotic methods. Steepest descents and stationary phase approximations.
JWKB approximation.
Linear equations, matrices, determinants and linear transformations.
Linear vector spaces. The Hilbert space `2
Hilbert spaces. The space L2
Complete orthonormal systems in L2
Linear operators on Hilbert space. Boundedness. Compact operators.
Integral equations. Resolvent operator. Functions of operators. Neumann series. Volterra equations. Transform methods. Fredholm integral equations. Schmidt-Hilbert method. Fredholm series.
Physical origins of standard partial di erential equations. Separable coordinate systems in three dimensions. Separation of variables.
Boundary conditions.
Integral transform methods. The Fourier transform. The Laplace transform. Partial di erential equations.
Perturbation theory. Rayleigh-Schrödinger method. Brillouin-Wigner method. Singular perturbations. Degenerate perturbations.