Introduction to Methods of Approximation in Physics and Astronomy

<p>Preface</p>

<p>Part I Preliminaries</p>

<p>1. Complex numbers</p>

<p>1.1 Quotients of complex numbers</p>

<p>1.2 Roots of complex numbers</p>

<p>1.3&nbsp;Sequences and Euler's constant&nbsp;</p>

<p>1.4&nbsp;Power series and radius of convergence&nbsp;</p>

<p>1.5&nbsp;Minkowski spacetime</p>

<p>1.6&nbsp;The logarithm and winding number&nbsp;</p>

<p>1.7&nbsp;Branch cuts for z</p>

<p>1.8 Branch cuts for z 1/p</p>

<p>1.9 Exercises&nbsp;</p>

<p>2. Complex function theory</p>

2.1 Analytic functions&nbsp;<p></p>

<p>2.2 Cauchy's Integral Formula&nbsp;</p>

<p>2.3 Evaluation of a real integral&nbsp;</p>

2.4 Residue theorem&nbsp;<p></p>

<p>2.5 Morera's theorem</p>

<p>2.6 Liouville's theorem</p>

<p>2.7 Poisson kernel</p>

<p>2.8 Flux and circulation</p>

2.9 Examples of potential flows<p></p>

<p>2.10Exercises</p>

<p>3. Vectors and linear algebra</p>

3.1 Introduction&nbsp;<p></p>

<p>3.2 Inner and outer products</p>

<p>3.3 Angular momentum vector</p>

<p>3.4 Elementary transformations in the plane</p>

<p>3.5&nbsp;Matrix algebra&nbsp;</p>

<p>3.6&nbsp;Eigenvalue problems</p>

<p>3.7 Unitary matrices and invariants&nbsp;</p>

<p>3.8 Hermitian structure of Minkowski spacetime</p>

3.9 Eigenvectors of Hermitian matrices&nbsp;<p></p>

<p>3.10QR factorization</p>

<p>3.11Exercises&nbsp;</p>

<p>4. Linear partial differential equations</p>

<p>4.1 Hyperbolic equations</p>

<p>4.2 Diffusion equation</p>

<p>4.3 Elliptic equations</p>

<p>4.4 Characteristic of hyperbolic systems&nbsp;</p>

<p>4.5 Weyl equation</p>

<p>4.6 Exercises&nbsp;</p>

<p>Part II Methods of approximation</p>

<p>5. Projections and minimal distances</p>

<p>5.1 Vectors and distances&nbsp;</p>

<p>5.2 Projections of vectors&nbsp;</p>

<p>5.3 Snell’s law and Fermat’s principle</p>

<p>5.4 Fitting data by least squares&nbsp;</p>

<p>5.5 Gauss-Legendre quadrature</p>

<p>5.6 Exercises</p>

<p>6. Spectral methods and signal analysis</p>

<p>6.1 Basis functions</p>

<p>6.2 Expansion in Legendre polynomials 6.3 Fourier expansion</p>

<p>6.4 The Fourier transform</p>

<p>6.5 Fourier series</p>

<p>6.6 Chebychev polynomials</p>

<p>6.7 Weierstrass approximation theorem</p>

<p>6.8 Detector signals in the presence of noise</p>

<p>6.9 Signal detection by FFT using&nbsp;Maxima</p>

6.10GPU-Butterfly filter in (f, f)<p></p>

<p>6.11Exercises</p>

<p>7. Root finding&nbsp;</p>

<p>7.1 Solving for √2 and π<br> 7.2 Convergence in Newton's method</p>

<p>7.3 Contraction mapping</p>

<p>7.4 Newton's method in two dimensions</p>

<p>7.5 Basins of attraction</p>

<p>7.6 Root finding in higher dimensions</p>

<p>7.7 Exercises</p>

<p>8. Finite differencing: differentiation and integration</p>

8.1 Vector fields<p></p>

<p>8.2 Gradient operator</p>

<p>8.3 Integration of ODE’s</p>

<p>8.4 Numerical integration of ODE's&nbsp;</p>

<p>8.5 Examples of ordinary differential equations</p>

<p>8.6 Exercises&nbsp;</p>

<p>9. Perturbation theory, scaling and turbulence</p>

<p>9.1 Roots of a cubic equation&nbsp;</p>

9.2 Damped pendulum&nbsp;<p></p>

<p>9.3 Orbital motion</p>

<p>9.4 Inertial and viscous fluid motion</p>

<p>9.5 Kolmogorov scaling of homogeneous turbulence</p>

<p>9.6 Exercises&nbsp;</p>

<p>Part III Selected topics</p>

<p>10. Thermodynamics of N-body systems</p>

<p>10.1 The action principle</p>

<p>10.2 Momentum in Euler-Lagragne equations</p>

<p>10.3 Legendre transformation</p>

<p>10.4 Hamiltonian formulation</p>

<p>10.5 Globular clusters</p>

<p>10.6 Coefficients of relaxation</p>

<p>10.7 Exercises</p>

<p>11. Accretion flows onto black holes</p>

<p>11.1 Bondi accretioin</p>

<p>11.2 Hoyle-Lyttleton accretion</p>

<p>11.3 Accretion disks</p>

<p>11.4 Gravitational wave emission</p>

<p>11.5 Mass transfer in binaries</p>

<p>11.6 Exercises</p>

<p>12. Rindler observers in astrophysics and cosmology</p>

12.1 The moving mirror problem<p></p>

<p>12.2 Implications for dark matter</p>

<p>12.3 Exercises </p>

<p>A. Some units and constant</p>

<p>B.&nbsp;Г(z) and&nbsp;Ϛ(z) functions</p>