Introduction to Methods of Approximation in Physics and Astronomy

PrefacePart I Preliminaries1. Complex numbers1.1 Quotients of complex numbers1.2 Roots of complex numbers1.3 Sequences and Euler's constant 1.4 Power series and radius of convergence 1.5 Hyperbolic structure of Minkowski spacetime1.6 The logarithm and winding number 1.7 Branch cuts for z1.8 Branch cuts for z 1/p1.9 Exercises 2. Complex function theory2.1 Analytic functions 2.2 Cauchy's Integral Formula 2.3 Evaluation of a real integral 2.4 Residue theorem 2.5 Morera's theorem2.6 Liouville's theorem2.7 Poisson kernel2.8 Flux and circulation2.9 Examples of potential flows2.10Exercises3. Linear algebra 3.1 Introduction 3.2 Elementary transformations in the plane3.3 Matrix algebra 3.4 Eigenvalue problems3.5 Unitary matrices and invariants 3.6 Hermitian structure of Minkowski spacetime3.7 Eigenvectors of Hermitian matrices 3.8 QR factorization3.9 Exercises 4. Linear partial differential equations4.1 Hyperbolic equations4.2 Diffusion equation4.3 Elliptic equations4.4 Characteristic equation of hyperbolic systems 4.5 Weyl equation4.6 Exercises Part II Methods of approximation5. Approximation by projection5.1 Vectors and distances 5.2 Projections of vectors 5.3 Fitting data by least squares 5.4 Gauss-Legendre quadrature5.5 Exercises6. Function approximation and signal analysis6.1 Basis functions6.2 Expansion in Legendre polynomials6.3 Fourier expansion6.4 The Fourier transform6.5 Fourier series6.6 Chebychev polynomials6.7 Weierstrass approximation theorem6.8 Detector signals in the presence of noise6.9 Signal detection by FFT using Maxima6.10Butterfly filter in (f, f)6.11Exercises7. Root finding 7.1 Solving for 2 and pi 7.2 Convergence in Newton's method7.3 Contraction mapping7.4 Newton's method in two dimensions7.5 Basins of attraction7.6 Root finding in higher dimensions7.7 Exercises8. Finite differencing8.1 Vector fields8.2 Numerical integration of ODE's 8.3 Examples of ordinary differential equations8.4 Exercises 9. Perturbation theory and scaling9.1 Roots of a cubic equation 9.2 Damped pendulum 9.3 Perihelion precession9.4 Inertial and viscous fluid motion9.5 Kolmogorov scaling of homogeneous9.6 Exercises 10. Thermodynamics of N-body systems10.1 The action principle10.2 Momentum in Euler-Lagragne equations10.3 Legendre transformation10.4 Hamiltonian formulation10.5 Globular clusters10.6 Coefficients of relaxation10.7 ExercisesA. Some units and constant B. (z) and (z) functions