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Synopsis
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 functions2.2 Cauchy's Integral Formula
2.3 Evaluation of a real integral
2.4 Residue theorem2.5 Morera's theorem
2.6 Liouville's theorem
2.7 Poisson kernel
2.8 Flux and circulation
2.9 Examples of potential flows2.10Exercises
3. Vectors and linear algebra
3.1 Introduction3.2 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 matrices3.10QR 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 fields8.2 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 pendulum9.3 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 problem12.2 Implications for dark matter
12.3 Exercises
A. Some units and consta
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