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Analytic Methods in Physics

Analytic Methods in Physics

Charlie Harper

ISBN: 978-3-527-60307-7 January 2005 324 Pages

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This book presents a self-contained treatment of invaluable analytic methods in mathematical physics. It is designed for undergraduate students and it contains more than enough material for a two semester (or three quarter) course in mathematical methods of physics. With the appropriate selection of material, one may use the book for a one semester or a one quarter course. The prerequisites or corequisites are general physics, analytic mechanics, modern physics, and a working knowledge of differential an integral calculus.
1. Vector Analysis (The Cartesian Coordinate System;
Differentiation of Vector Functions;
Orthogonal Curvilinear Coordinates;
Appedix I: SI Units;
Appendix II: Determinants)
2. Modern Algebraic Methods in Physics (Matrix Analysis;
Essentials of Vector Spaces;
Essential Algebraic Structures;
3. Functions of a Complex Variable (Complex Variables and Their Representations;
The de Moivre Theorem;
Analytic Functions of a Complex Variable;
Contour Integrals;
The Taylor Series and Zeros of f(z);
The Laurent Expansion;
Appendix: Series)
4. Calculus of Residues (Isolated Singular Points;
Evaluation of Residues;
The Cauchy Residue Theorem;
The Cauchy Principal Value;
Evaluation of Definite Integrals;
Dispersion Relations;
Conformal Transformations;
Multi-Valued Functions;
5. Fourier Series (The Fourier Cosine and Sine Series;
Change of Interval;
Complex Form of the Fourier Series;
Generalized Fourier Series and the Dirac Delta Function;
Summation of the Fourier Series;
The Gibbs Phenomenon;
Summary of Some Properties of Fourier Series;
6. Fourier Transforms (Cosine and Sine Transforms;
The Transforms of Derivatives;
The Convolution Theorem;
Parseval´s Relation;
7. Ordinary Differential Equations (First-Order Linear Differential Equations;
The Bernoulli Differential Equation;
Second-Order Linear Differential Equations;
Some Numerical Methods;
8. Partial Differential Equations (The Method of Separation of Variables;
Green´s Functions in Potential Theory;
Some Numerical Methods;
9. Special Functions (The Sturm-Liouville Theory;
The Hermite Polynomials;
The Helmholtz Differential Equation in Spherical Coordinates and in Cylindrical Coordinates;
The Hypergeometric Function;
The Confluent Hypergeometric Function;
Other Special Functions Used in Physics;
Worksheet 9.1: The Quantum Mechanical Linear Harmonic Oscillator;
Workshheet 9.2: The Legendre Differential Equation;
Worksheet 9.3: The Laguerre Differential Equation;
Workshheet 9.4: The Bessel Differential Equation;
Workshheet 9.5: The Hypergeometric Differential Equation)
10. Integral Equations (Integral Equations with Separable Kernels and with Displacement Kernels;
The Neumann Series Method;
The Abel Problem;
11. Applied Functional Analysis (Stationary Values of Certain Functions and Functionals;
Hamilton´s Variational Principle in Mechanics;
Formulation of Hamiltonian Mechanics;
Continous Media and Fields;
Transitions to Quantum Mechanics;
12. Geometric Methods in Physics (Transformation of Coordinates in Linear Spaces;
Contravariant and Covariant Tensors;
Tensor Algebra;
The Line Element;
Tensor Calculus;
Equation of the Geodesic Line;
Special Equations involving the Metric Tensor;
Exterior Differential Forms;