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E-book
Applied Mathematical Methods in Theoretical PhysicsISBN: 978-3-527-60490-6
E-book
11 pages
March 2006
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Introduction.
1 Function Spaces, Linear Operators and Green’s Functions.
1.1 Function Spaces.
1.2 Ortho normal System of Functions.
1.3 Linear Operators.
1.4 Eigen values and Eigen functions.
1.5 The Fredholm Alternative.
1.6 Self-adjoint Operators.
1.7 Green’s Functions for Differential Equations.
1.8 Review of Complex Analysis.
1.9 Review of Fourier Transform.
2 Integral Equations and Green’s Functions.
2.1 Introduction to Integral Equations.
2.2 Relationship of Integral Equations with Differential Equations and Green’s Functions.
2.3 Sturm–Liouville System.
2.4 Green’s Function for Time-Dependent Scattering Problem.
2.5 Lippmann–Schwinger Equation.
2.6 Problems for Chapter 2.
3 Integral Equations of Volterra Type.
3.1 Iterative Solution to Volterra Integral Equation of the Second Kind.
3.2 Solvable cases of Volterra Integral Equation.
3.3 Problems for Chapter 3.
4 Integral Equations of the Fredholm Type.
4.1 Iterative Solution to the Fredholm Integral Equation of the Second Kind.
4.2 Resolvent Kernel.
4.3 Pincherle–Goursat Kernel.
4.4 Fredholm Theory for a Bounded Kernel.
4.5 Solvable Example.
4.6 Fredholm Integral Equation with a Translation Kernel.
4.7 System of Fredholm Integral Equations of the Second Kind.
4.8 Problems for Chapter 4.
5 Hilbert–Schmidt Theory of Symmetric Kernel.
5.1 Real and Symmetric Matrix.
5.2 Real and Symmetric Kernel.
5.3 Bounds on the Eigen values.
5.4 Rayleigh Quotient.
5.5 Completeness of Sturm–Liouville Eigen functions.
5.6 Generalization of Hilbert–Schmidt Theory.
5.7 Generalization of Sturm–Liouville System.
5.8 Problems for Chapter 5.
6 Singular Integral Equations of Cauchy Type.
6.1 Hilbert Problem.
6.2 Cauchy Integral Equation of the First Kind.
6.3 Cauchy Integral Equation of the Second Kind.
6.4 Carleman Integral Equation.
6.5 Dispersion Relations.
6.6 Problems for Chapter 6.
7 Wiener–Hopf Method and Wiener–Hopf Integral Equation.
7.1 The Wiener–Hopf Method for Partial Differential Equations.
7.2 Homogeneous Wiener–Hopf Integral Equation of the Second Kind.
7.3 General Decomposition Problem.
7.4 Inhomogeneous Wiener–Hopf Integral Equation of the Second Kind.
7.5 Toeplitz Matrix and Wiener–Hopf Sum Equation.
7.6 Wiener–Hopf Integral Equation of the First Kind and Dual Integral Equations.
7.7 Problems for Chapter 7.
8 Nonlinear Integral Equations.
8.1 Nonlinear Integral Equation of Volterra type.
8.2 Nonlinear Integral Equation of Fredholm Type.
8.3 Nonlinear Integral Equation of Hammerstein type.
8.4 Problems for Chapter 8.
9 Calculus of Variations: Fundamentals.
9.1 Historical Background.
9.2 Examples.
9.3 Euler Equation.
9.4 Generalization of the Basic Problems.
9.5 More Examples.
9.6 Differential Equations, Integral Equations, and Extremization of Integrals.
9.7 The Second Variation.
9.8 Weierstrass–Erdmann Corner Relation.
9.9 Problems for Chapter 9.
10 Calculus of Variations: Applications.
10.1 Feynman’s Action Principle in Quantum Mechanics.
10.2 Feynman’s Variational Principle in Quantum Statistical Mechanics.
10.3 Schwinger–Dyson Equation in Quantum Field Theory.
10.4 Schwinger–Dyson Equation in Quantum Statistical Mechanics.
10.5 Weyl’s Gauge Principle.
10.6 Problems for Chapter 10.
Bibliography.
Index.
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