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Introduction to Methods of Approximation in Physics and Astronomy

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Table of Contents

Preface Part I Preliminaries 1. Complex numbers 1.1 Quotients of complex numbers 1.2 Roots of complex numbers 1.3 Sequences and Euler's constant 1.4 Power series and radius of convergence 1.5 Minkowski spacetime 1.6 The logarithm and winding number 1.7 Branch cuts for z 1.8 Branch cuts for z 1/p 1.9 Exercises 2. Complex function theory 2.1 Analytic functions 2.2 Cauchy's Integral Formula 2.3 Evaluation of a real integral 2.4 Residue theorem 2.5 Morera's theorem 2.6 Liouville's theorem 2.7 Poisson kernel 2.8 Flux and circulation 2.9 Examples of potential flows 2.10Exercises 3. Vectors and linear algebra 3.1 Introduction 3.2 Inner and outer products 3.3 Angular momentum vector 3.4 Elementary transformations in the plane 3.5 Matrix algebra 3.6 Eigenvalue problems 3.7 Unitary matrices and invariants 3.8 Hermitian structure of Minkowski spacetime 3.9 Eigenvectors of Hermitian matrices 3.10QR factorization 3.11Exercises 4. Linear partial differential equations 4.1 Hyperbolic equations 4.2 Diffusion equation 4.3 Elliptic equations 4.4 Characteristic of hyperbolic systems 4.5 Weyl equation 4.6 Exercises Part II Methods of approximation 5. Projections and minimal distances 5.1 Vectors and distances 5.2 Projections of vectors 5.3 Snell's law and Fermat's principle 5.4 Fitting data by least squares 5.5 Gauss-Legendre quadrature 5.6 Exercises 6. Spectral methods and signal analysis 6.1 Basis functions 6.2 Expansion in Legendre polynomials 6.3 Fourier expansion 6.4 The Fourier transform 6.5 Fourier series 6.6 Chebychev polynomials 6.7 Weierstrass approximation theorem 6.8 Detector signals in the presence of noise 6.9 Signal detection by FFT using Maxima 6.10GPU-Butterfly filter in (f, f) 6.11Exercises 7. Root finding 7.1 Solving for 2 and 7.2 Convergence in Newton's method 7.3 Contraction mapping 7.4 Newton's method in two dimensions 7.5 Basins of attraction 7.6 Root finding in higher dimensions 7.7 Exercises 8. Finite differencing: differentiation and integration 8.1 Vector fields 8.2 Gradient operator 8.3 Integration of ODE's 8.4 Numerical integration of ODE's 8.5 Examples of ordinary differential equations 8.6 Exercises 9. Perturbation theory, scaling and turbulence 9.1 Roots of a cubic equation 9.2 Damped pendulum 9.3 Orbital motion 9.4 Inertial and viscous fluid motion 9.5 Kolmogorov scaling of homogeneous turbulence 9.6 Exercises Part III Selected topics 10. Thermodynamics of N-body systems 10.1 The action principle 10.2 Momentum in Euler-Lagragne equations 10.3 Legendre transformation 10.4 Hamiltonian formulation 10.5 Globular clusters 10.6 Coefficients of relaxation 10.7 Exercises 11. Accretion flows onto black holes 11.1 Bondi accretioin 11.2 Hoyle-Lyttleton accretion 11.3 Accretion disks 11.4 Gravitational wave emission 11.5 Mass transfer in binaries 11.6 Exercises 12. Rindler observers in astrophysics and cosmology 12.1 The moving mirror problem 12.2 Implications for dark matter 12.3 Exercises A. Some units and constant B. (z) and (z) functions

About the Author

MAURICE H. P. M. VAN PUTTEN is a Professor of Physics and Astronomy at Sejong University and an Associate Member of the School of Physics, Korea Institute for Advanced Study. He received his Ph.D. from the California Institute of Technology and held postdoctoral research positions at the Institute for Theoretical Physics at the University of California, Santa Barbara, and the Center for Radiophysics and Space Research at Cornell University. He held faculty positions at the Massachusetts Institute of Technology, Nanjing University and the Institute for Advanced Studies at CNRS-Orleans. His current research focus is on multimessenger emissions from rotating black holes including gravitational radiation from core-collapse supernovae and long gamma-ray bursts, Ultra-High Energy Cosmic Rays (UHECRs) from Seyfert galxies, dynamical dark energy in cosmology and hyperbolic formulations of general relativity and relativistic magneto-hydrodynamics.


"This is a thorough and rigorous introduction to the subject, intended for students at the graduate or very advanced undergraduate level. Van Putten (Sejong Univ., South Korea) presents a collection of problems that most readers will find quite challenging to solve. ... Summing Up: Recommended. With reservations. Upper-division undergraduates through faculty and professionals." (T. Barker, Choice, Vol. 55 (10), June 2018)

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