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E-raamat: Introduction to Methods of Approximation in Physics and Astronomy

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Introduction to Methods of Approximation in Physics and AstronomySuurem pilt
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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.

Arvustused

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)

Part I Preliminaries
1 Complex Numbers
3(32)
1.1 Quotients of Complex Numbers
7(1)
1.2 Roots of Complex Numbers
8(1)
1.3 Sequences and Euler's Constant
9(4)
1.4 Power Series and Radius of Convergence
13(4)
1.5 Minkowski Spacetime
17(7)
1.5.1 Rindler Observers
20(4)
1.6 The Logarithm and Winding Number
24(1)
1.7 Branch Cuts for log Z
25(2)
1.8 Branch Cuts for
27(2)
1.9 Exercises
29(6)
Reference
33(2)
2 Complex Function Theory
35(26)
2.1 Analytic Functions
35(3)
2.2 Cauchy's Integral Formula
38(4)
2.3 Evaluation of a Real Integral
42(1)
2.4 Residue Theorem
43(5)
2.5 Morera's Theorem
48(2)
2.6 Liouville's Theorem
50(1)
2.7 Poisson Kernel
51(1)
2.8 Flux and Circulation
52(3)
2.9 Examples of Potential Flows
55(1)
2.10 Exercises
56(5)
References
60(1)
3 Vectors and Linear Algebra
61(48)
3.1 Introduction
61(2)
3.2 Inner and Outer Products
63(1)
3.3 Angular Momentum Vector
63(11)
3.3.1 Rotations
64(1)
3.3.2 Angular Momentum and Mach's Principle
65(1)
3.3.3 Energy and Torque
66(4)
3.3.4 Coriolis Forces
70(1)
3.3.5 Spinning Top
71(3)
3.4 Elementary Transformations in the Plane
74(3)
3.4.1 Reflection Matrix
74(2)
3.4.2 Rotation Matrix
76(1)
3.5 Matrix Algebra
77(1)
3.6 Eigenvalue Problems
78(5)
3.6.1 Eigenvalues of R(φ)
78(1)
3.6.2 Eigenvalues of a Real-Symmetric Matrix
79(1)
3.6.3 Hermitian Matrices
80(3)
3.7 Unitary Matrices and Invariants
83(3)
3.8 Hermitian Structure of Minkowski Spacetime
86(5)
3.9 Eigenvectors of Hermitian Matrices
91(3)
3.10 QR Factorization
94(7)
3.10.1 Examples of Image and Null Space
96(1)
3.10.2 Dimensions of Image and Null Space
97(3)
3.10.3 QR Factorization by Gram-Schmidt
100(1)
3.11 Exercises
101(8)
References
107(2)
4 Linear Partial Differential Equations
109(26)
4.1 Hyperbolic Equations
109(6)
4.1.1 Inhomogeneous Wave Equation (Duhamel)
113(2)
4.2 Diffusion Equation
115(7)
4.2.1 Photon Diffusion in the Sun
121(1)
4.3 Elliptic Equations
122(3)
4.4 Characteristics of Hyperbolic Systems
125(1)
4.5 Weyl Equation
126(3)
4.6 Exercises
129(6)
References
131(4)
Part II Methods of Approximation
5 Projections and Minimal Distances
135(24)
5.1 Vectors and Distances
135(2)
5.2 Projections of Vectors
137(5)
5.3 Snell's Law and Fermat's Principle
142(6)
5.4 Fitting Data by Least Squares
148(3)
5.5 Gauss-Legendre Quadrature
151(3)
5.6 Exercises
154(5)
References
158(1)
6 Spectral Methods and Signal Analysis
159(38)
6.1 Basis Functions
159(1)
6.2 Expansion in Legendre Polynomials
160(7)
6.2.1 Symbolic Computation in Maxima
163(4)
6.3 Fourier Expansion
167(1)
6.4 The Fourier Transform
167(9)
6.4.1 The Sine Function
169(1)
6.4.2 Plancherel's Theorem
170(1)
6.4.3 Fourier Series by Sampling ƒ(ω)
171(3)
6.4.4 Discrete Fourier Transform (DFT)
174(2)
6.5 Fourier Series
176(1)
6.6 Chebyshev Polynomials
177(2)
6.7 Weierstrass Approximation Theorem
179(1)
6.8 Detector Signals in the Presence of Noise
180(3)
6.8.1 Convolution and Cross-Correlation
181(2)
6.9 Signal Detection by FFT Using Maxima
183(2)
6.10 CPU-Butterfly Filter in (ƒ, ƒ)
185(5)
6.11 Exercises
190(7)
References
196(1)
7 Root Finding
197(16)
7.1 Solving for 2 and π
197(3)
7.2 Convergence in Newton's Method
200(1)
7.3 Contraction Mapping
201(3)
7.4 Newton's Method in Two Dimensions
204(2)
7.5 Basins of Attraction
206(2)
7.6 Root Finding in Higher Dimensions
208(2)
7.7 Exercises
210(3)
References
211(2)
8 Finite Differencing: Differentiation and Integration
213(30)
8.1 Vector Fields
213(7)
8.1.1 Estimating Velocity
215(4)
8.1.2 Estimating Acceleration
219(1)
8.2 Gradient Operator
220(1)
8.3 Integration by Finite Summation
221(2)
8.4 Numerical Integration of ODE's
223(2)
8.5 Examples of Ordinary Differential Equations
225(11)
8.5.1 Osmosis
225(1)
8.5.2 Migration Time of the Moon
226(6)
8.5.3 Cosmological Expansion
232(4)
8.6 Exercises
236(7)
References
241(2)
9 Perturbation Theory, Scaling and Turbulence
243(30)
9.1 Roots of a Cubic Equation
243(2)
9.2 Damped Pendulum
245(2)
9.3 Orbital Motion
247(5)
9.3.1 Perihelion Precession
249(3)
9.4 Inertial and Viscous Fluid Motion
252(11)
9.4.1 Large and Small Reynolds Numbers
255(3)
9.4.2 Vorticity Equation
258(1)
9.4.3 Jeans Instability in Linearized Euler's Equations
259(4)
9.5 Kolmogorov Scaling of Homogeneous Turbulence
263(3)
9.6 Exercises
266(7)
References
269(4)
Part III Selected Topics
10 Thermodynamics of N-body Systems
273(18)
10.1 The Action Principle
273(2)
10.2 Momentum in Euler-Lagrange Equations
275(1)
10.3 Legendre Transformation
276(1)
10.4 Hamiltonian Formulation
277(1)
10.5 Globular Clusters
278(7)
10.6 Coefficients of Relaxation
285(2)
10.7 Exercises
287(4)
References
288(3)
11 Accretion Flows onto Black Holes
291(26)
11.1 Bondi Accretion
293(5)
11.2 Hoyle-Lyttleton Accretion
298(3)
11.3 Accretion Disks
301(3)
11.4 Gravitational Wave Emission
304(3)
11.5 Mass Transfer in Binaries
307(5)
11.6 Exercises
312(5)
References
314(3)
12 Rindler Observers in Astrophysics and Cosmology
317(8)
12.1 The Moving Mirror Problem
317(2)
12.2 Implications for Dark Matter
319(4)
12.3 Exercises
323(2)
References
324(1)
Appendix A Some Units and Constants 325(2)
Appendix B Γ(z) and σ(z) Functions 327(2)
Appendix C Free Fall in Schwarzschild Spacetime 329(10)
Index 339
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.
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