Mathematics For Physicists

Introduction
Preface
Contents
Chapter 1. Variational Method
	1.1. Functional and Its Extremal Problems
		1.1.1. The conception of functional
		1.1.2. The extremes of functionals
	1.2. The Variational of Functionals and the Simplest Euler Equation
		1.2.1. The variational of functionals
		1.2.2. The simplest Euler equation
	1.3. The Cases of Multifunctions and Multivariates
		1.3.1. Multifunctions
		1.3.2. Multivariates
	1.4. Functional Extremes under Certain Conditions
		1.4.1. Isoperimetric problem
		1.4.2. Geodesic problem
	1.5. Natural Boundary Conditions
	1.6. Variational Principle
		1.6.1. Variational principle of classical mechanics
		1.6.2. Variational principle of quantum mechanics
	1.7. The Applications of the Variational Method in Physics
		1.7.1. The applications in classical physics
		1.7.2. The applications in quantum mechanics
	Exercises
Chapter 2. Hilbert Space
	2.1. Linear Space, Inner Product Space and Hilbert Space
		2.1.1. Linear space
		2.1.2. Inner product space
		2.1.3. Hilbert space
	2.2. Operators in Inner Product Spaces
		2.2.1. Operators and adjoint operators
		2.2.2. Self-adjoint operators
		2.2.3. The alternative theorem for the solutions of linear algebraic equations
	2.3. Complete Set of Orthonormal Functions
		2.3.1. Three kinds of convergences
		2.3.2. The completeness of a set of functions
		2.3.3. N-dimensional space and Hilbert function space
		2.3.4. Orthogonal polynomials
	2.4. Polynomial Approximation
		2.4.1. Weierstrass theorem
		2.4.2. Polynomial approximation
	Exercises
Chapter 3. Linear Ordinary Differential Equations of Second Order
	3.1. General Theory
		3.1.1. The existence and uniqueness of solutions
		3.1.2. The structure of solutions of homogeneous equations
		3.1.3. The solutions of inhomogeneous equations
	3.2. Sturm-Liouville Eigenvalue Problem
		3.2.1. The form of Sturm-Liouville equations
		3.2.2. The boundary conditions of Sturm-Liouville equations
		3.2.3. Sturm-Liouville eigenvalue problem
	3.3. The Polynomial Solutions of Sturm-Liouville Equations
		3.3.1. Possible forms of kernel and weight functions
		3.3.2. The expressions in series and in derivatives of the polynomials
		3.3.3. Generating functions
		3.3.4. The completeness theorem of orthogonal polynomials as Sturm-Liouville solutions
		3.3.5. Applications in numerical integrations
	3.4. Equations and Functions that Relate to the Polynomial Solutions
		3.4.1. Laguerre functions
		3.4.2. Legendre functions
		3.4.3. Chebyshev functions
		3.4.4. Hermite functions
	3.5. Complex Analysis Theory of the Ordinary Differential Equations of Second Order
		3.5.1. Solutions of homogeneous equations
		3.5.2. Ordinary differential equations of second order
	3.6. Non-Self-Adjoint Ordinary Differential Equations of Second Order
		3.6.1. Adjoint equations of ordinary differential equations
		3.6.2. Sturm-Liouville operator
		3.6.3. Complete set of non-self-adjoint ordinary differential equations of second order
	3.7. The Conditions under Which Inhomogeneous Equations have Solutions
	Exercises
	Appendix 3A. Generalization of Sturm-Liouville Theorem to Dirac Equation
Chapter 4. Bessel Functions
	4.1. Bessel Equation
		4.1.1. Bessel equation and its solutions
		4.1.2. Bessel functions of the first and second kinds
	4.2. Fundamental Properties of Bessel Functions
		4.2.1. Recurrence relations of Bessel functions
		4.2.2. Asymptotic formulas of Bessel functions
		4.2.3. Zeros of Bessel functions
		4.2.4. Wronskian
	4.3. Bessel Functions of Integer Orders
		4.3.1. Parity and the values at certain points
		4.3.2. Generating function of Bessel functions of integer orders
	4.4. Bessel Functions of Half-Integer Orders
	4.5. Bessel Functions of the Third Kind and Spherical Bessel Functions
		4.5.1. Bessel functions of the third kind
		4.5.2. Spherical Bessel functions
	4.6. Modified Bessel Functions
		4.6.1. Modified Bessel functions of the first and second kinds
		4.6.2. Modified Bessel functions of integer orders
	4.7. Bessel Functions with Real Arguments
		4.7.1. Eigenvalue problem of Bessel equation
		4.7.2. Properties of eigenfunctions
		4.7.3. Eigenvalue problem of spherical Bessel equation
	Exercises
Chapter 5. The Dirac Delta Function
	5.1. Definition and Properties of the Delta Function
		5.1.1. Definition of the delta function
		5.1.2. The delta function is a generalized function
		5.1.3. The Fourier and Laplace transformations of the delta function
		5.1.4. Derivative and integration of generalized functions
		5.1.5. Complex argument in the delta function
	5.2. The Delta Function as Weak Convergence Limits of Ordinary Functions
	5.3. The Delta Function in Multidimensional Spaces
		5.3.1. Cartesian coordinate system
		5.3.2. The transform from Cartesian coordinates to curvilinear coordinates
	5.4. Generalized Fourier Series Expansion of the Delta Function
	Exercises
Chapter 6. Green's Function
	6.1. Fundamental Theory of Green's Function
		6.1.1. Definition of Green's function
		6.1.2. Properties of Green's function
		6.1.3. Methods of obtaining Green's function
		6.1.4. Physical meaning of Green's function
	6.2. The Basic Solution of Laplace Operator
		6.2.1. Three-dimensional space
		6.2.2. Two-dimensional space
		6.2.3. One-dimensional space
	6.3. Green's Function of a Damped Oscillator
		6.3.1. Solution of homogeneous equation
		6.3.2. Obtaining Green's function
		6.3.3. Generalized solution of the equation
		6.3.4. The case without damping
		6.3.5. The influence of boundary conditions
	6.4. Green's Function of Ordinary Differential Equations of Second Order
		6.4.1. The symmetry of Green's function
		6.4.2. Solutions of boundary value problem of ordinary differential equations of second order
		6.4.3. Modified Green's function
		6.4.4. Examples of solving boundary value problem of ordinary differential equations of second order
	6.5. Green's Function in Multi-dimensional Spaces
		6.5.1. Ordinary differential equations of second order and Green's function
		6.5.2. Examples in two-dimensional space
	6.6. Green's Function of Ordinary Differential Equation of First Order
		6.6.1. Boundary value problem of inhomogeneous equations
		6.6.2. Boundary value problem of homogeneous equations
		6.6.3. Inhomogeneous equations and Green's function
		6.6.4. General solutions of boundary value problem
	6.7. Green's Function of Non-Self-Adjoint Equations
		6.7.1. Adjoint Green's function
		6.7.2. Solutions of inhomogeneous equations
	Exercises
Chapter 7. Norm
	7.1. Banach Space
		7.1.1. Banach space
		7.1.2. Hölder inequality
		7.1.3. Minkowski inequality
	7.2. Vector Norms
		7.2.1. Vector norms
		7.2.2. Equivalence between vector norms
	7.3. Matrix Norms
		7.3.1. Matrix norms
		7.3.2. Spectral norm and spectral radius of matrices
	7.4. Operator Norms
		7.4.1. Operator norms
		7.4.2. Adjoint operators
		7.4.3. Projection operators
	Exercises
Chapter 8. Integral Equations
	8.1. Fundamental Theory of Integral Equations
		8.1.1. Definition and classification of integral equations
		8.1.2. Relations between integral equations and differential equations
		8.1.3. Theory of homogeneous integral equations
	8.2. Iteration Technique for Linear Integral Equations
		8.2.1. The second kind of Fredholm integral equations
		8.2.2. The second kind of Volterra integral equations
	8.3. Iteration Technique of Inhomogeneous Integral Equations
		8.3.1. Iteration procedure
		8.3.2. Lipschitz condition
		8.3.3. Use of contraction
		8.3.4. Anharmonic vibration of a spring
	8.4. Fredholm Linear Equations with Degenerated Kernels
		8.4.1. Separable kernels
		8.4.2. Kernels with a finite rank
		8.4.3. Expansion of kernel in terms of eigenfunctions
	8.5. Integral Equations of Convolution Type
		8.5.1. Fredholm integral equations of convolution type
		8.5.2. Volterra integral equations of convolution type
	8.6. Integral Equations with Polynomials
		8.6.1. Fredholm integral equations with polynomials
		8.6.2. Generating function method
	Exercises
Chapter 9. Application of Number Theory in Inverse Problems in Physics
	9.1. Chen-Möbius Transformation
		9.1.1. Introduction
		9.1.2. Möbius transformation
		9.1.3. Chen-Möbius transformation
	9.2. Inverse Problem in Phonon Density of States in Crystals
		9.2.1. Inversion formula
		9.2.2. Low-temperature approximation
		9.2.3. High-temperature approximation
	9.3. Inverse Problem in the Interaction Potential between Atoms
		9.3.1. One-dimensional case
		9.3.2. Two-dimensional case
		9.3.3. Three-dimensional case
	9.4. Additive Möbius Inversion and Its Applications
		9.4.1. Additive Möbius inversion of functions and its applications
		9.4.2. Additive Möbius inversion of series and its applications
	9.5. Inverse Problem in Crystal Surface Relaxation and Interfacial Potentials
		9.5.1. Pair potentials between an isolated atom and atoms in a semi-infinite crystal
		9.5.2. Relaxation of atoms at a crystal surface
		9.5.3. Inverse problem of interfacial potentials
	9.6. Construction of Biorthogonal Complete Function Sets
	Exercises
	Appendix 9A. Some Values of Riemann ζ Function
	Appendix 9B. Calculation of Reciprocal Coefficients
Chapter 10. Fundamental Equations in Spaces with Arbitrary Dimensions
	10.1. Euclid Spaces with Arbitrary Dimensions
		10.1.1. Cartesian coordinate system and spherical coordinates
		10.1.2. Gradient, divergence and Laplace operator
	10.2. Green's Functions of the Laplace Equation and Helmholtz Equation
		10.2.1. Green's function of the Laplace equation
		10.2.2. Green's function of the Helmholtz equation
	10.3. Radial Equations under Central Potentials
		10.3.1. Radial equation under a central potential in multidimensional spaces
		10.3.2. Helmholtz equation
		10.3.3. Infinitely deep spherical potential
		10.3.4. Finitely deep spherical potential
		10.3.5. Coulomb potential
		10.3.6. Harmonic potential
		10.3.7. Molecular potential with both negative powers
		10.3.8. Molecular potential with positive and negative powers
		10.3.9. Attractive potential with exponential decay
		10.3.10. Conditions that the radial equation has analytical solutions
	10.4. Solutions of Angular Equations
		10.4.1. Four-dimensional space
		10.4.2. Five-dimensional space
		10.4.3. N-dimensional space
	10.5. Pseudo Spherical Coordinates
		10.5.1. Pseudo coordinates in four-dimensional space
		10.5.2. Solutions of Laplace equation
		10.5.3. Five- and six-dimensional spaces
	10.6. Non-Euclidean Space
		10.6.1. Metric tensor
		10.6.2. Five-dimensional Minkowski space and four-dimensional de Sitter space
		10.6.3. Maxwell equations in de Sitter spacetime
	Exercises
	Appendix 10A. Hypergeometric Equation and Hypergeometric Functions
References
15
Answers of Selected Exercises
	Chapter 1
	Chapter 2
	Chapter 3
	Chapter 4
	Chapter 5
	Chapter 6
	Chapter 7
	Chapter 8
Author Index
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Subject Index
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