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 π
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 consta
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.
