Introduction to Methods of Approximation in Physics and Astronomy

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This textbook provides students with a solid introduction to the techniques of approximation commonly used in data analysis across physics and astronomy. The choice of methods included is based on their usefulness and educational value, their applicability to a broad range of problems and their utility in highlighting key mathematical concepts. Modern astronomy reveals an evolving universe rife with transient sources, mostly discovered - few predicted - in multi-wavelength observations. Our window of observations now includes electromagnetic radiation, gravitational waves and neutrinos. For the practicing astronomer, these are highly interdisciplinary developments that pose a novel challenge to be well-versed in astroparticle physics and data-analysis. The book is organized to be largely self-contained, starting from basic concepts and techniques in the formulation of problems and methods of approximation commonly used in computation and numerical analysis. This includes root finding, integration, signal detection algorithms involving the Fourier transform and examples of numerical integration of ordinary differential equations and some illustrative aspects of modern computational implementation. Some of the topics highlighted introduce the reader to selected problems with comments on numerical methods and implementation on modern platforms including CPU-GPU computing. Developed from lectures on mathematical physics in astronomy to advanced undergraduate and beginning graduate students, this book will be a valuable guide for students and a useful reference for practicing researchers. To aid understanding, exercises are included at the end of each chapter. Furthermore, some of the exercises are tailored to introduce modern symbolic computation.

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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 â 2 and Ï 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. Ð (z) and Ï (z) functions

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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