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《物理学中的数学方法（英文版）》是为从事物理学与其他理工科研究人员和非数学类研究生与本科高年级学生撰写的数学方面的基础理论读物和参考书。对于物理和其他理工学科做研究工作时所必须要用到的数学知识做了比较详细和全面的介绍，包括变分法，希尔伯特空间，二阶常微分方程，贝塞尔函数，狄拉克函数，格林函数，范数与测度，积分方程，数论在物理逆问题中的应用，高维空间中的基本方程等内容。《物理学中的数学方法（英文版）》的写作力求概念说明清楚，公式推导详尽，内容深入浅出。便于读者学习。

Contents

Introduction

Preface

Chapter I Variational Method 1

1.1. Functional and Its Extremal Problems 1

1.1.1. The conception of functional 1

1.1.2. The extremes of functionals 3

1.2. The Variational of Functionals and the Simplest Euler Equation 7

1.2.1. The variational of functionals 7

1.2.2. The simplest Euler equation 12

1.3. The Cases of Multifunctions and Multivariates 16

1.3.1. Multifunctions 16

1.3.2. Multivariates 19

1.4. Functional Extremes under Certain Conditions 22

1.4.1. Isoperimetric problem 22

1.4.2. Geodesic problem 26

1.5. Natural Boundary Conditioiis 29

1.6. Variational Principle 33

1.6.1. Variational principle of classical mechanics 34

1.6.2. Variational principle of quantum mechanics 40

1.7. The Applications of the Variational Method in Physics 41

1.7.1. The applications in classical physics 44

1.7.2. The applications in quantum mechanics 48

Exercises 50

Chapter 2 Hilbert Space 55

2.1. Linear Space， Inner Product Space and Hilbert Space 55

2.1.1. Linear space 55

2.1.2. Inner product space 63

2.1.3. Hilbert space 71

2.2. Operators in Inner Product Spaces 74

2.2.1. Operators and adjoint operators 74

2.2.2. Selfadjoint operators 84

2.2.3. The alternative theorem for the solutions of linear algebraic equations 94

2.3. Complete Set of Orthonormal Functions 96

2.3.1. Three kinds of convergences 96

2.3.2. The completeness of a set of functions 98

2.3.3. Ndimensional space and Hilbert function space 101

2.3.4. Orthogonal polynomials 103

2.4. Polynomial Approximation 109

2.4.1. Weierstrass theorem 109

2.4.2. Polynomial approximation 112

Exercises 120

Chapter 3 Linear Ordinary Differential Equations of Second Order 127

3.1. General Theory 127

3.1.1. The existence and uniqueness of solutions 127

3.1.2. The struct.ure of solutions of homogeneous equations 130

3.1.3. The solutions of inhomogeneous equations 137

3.2. SturmLiouville Eigenvalue Problem 140

3.2.1. The form of SturmLiouville equations 140

3.2.2. The boundary conditions of SturmLiouville equations 142

3.2.3. SturmLiouville eigenvalue problem 144

3.3. The Polynomial Solutions of SturmLiouville Equations 151

3.3.1. Possible forms of kernel and weight functions 151

3.3.2. The expressions in series and in derivatives of the polynomials 158

3.3.3. Generating functions 165

3.3.4. The completeness theorem of orthogonal polynomials as SturmLiouville solutions 169

3.3.5. Applications in numerical integrations 171

3.4. Equations and Functions that Relate to the Polynomial Solutions 174

3.4.1. Laguerre functions 175

3.4.2. Legendre functions 179

3.4.3. Chebyshev functions 185

3.4.4. Hermite functions 190

3.5. Complex Analysis Theory of the Ordinary Differential Equations of Second Order 196

3.5.1. Solutions of homogeneous equations 196

3.5.2. Ordinary differential equations of second order 216

3.6. NonSelfAdjoint Ordinary Differential Equations of Second Order 224

3.6.1. Adjoint equations of ordinary differential equations 224

3.6.2. SturmLiouville operator 225

3.6.3. Complete set of nonselfadjoint ordinary differential equations of second order 229

3.7. The Conditions under Which Inhomogeneous Equations have Solutions 231

Exercises 236

Appendix 3A Generalization of SturmLiouville Theorem to Dirac Equation 244

Chapter 4 Bessel Functions 247

4.1. Bessel Equation 247

4.1.1. Bessel equat.ion and its solutioiis 247

4.1.2. Bessel functions of the first and second kinds 255

4.2. Fundamental Properties of Bessel Functions 258

4.2.1. Recurrence relations of Bessel functions 258

4.2.2. Asymptotic formulas of Bessel functions 261

4.2.3. Zeros of Bessel functions 262

4.2.4. Wronskian 264

4.3. Bessel Functions of Integer Orders 266

4.3.1. Parity and the values at certain points 266

4.3.2. Generating function of Bessel functions of integer orders 267

4.4. Bessel Functions of HalfInteger Orders 273

4.5. Bessel Functions of the Third Kind and Spherical Bessel Functions 275

4.5.1. Bessel functions of the third kind 275

4.5.2. Spherical Bessel functions 280

4.6. Modified Bessel Functions 288

4.6.1. Modified Bessel functions of the first and second kinds 288

4.6.2. Modified Bessel functions of integer orders 293

4.7. Bessel Functions with Real Arguments 294

4.7.1. Eigenvalue problem of Bessel equation 294

4.7.2. Properties of eigenfunctions 297

4.7.3. Eigenvalue problem of spherical Bessel equation 301

Exercises 302

Chapter 5 The Dirac Delta Function 311

5.1. Definition and Properties of the Delta Function 311

5.1.1. Definition of the delta function 311

5.1.2. The delta function is a generalized function 312

5.1.3. The Fourier and Laplace transformations of the delta function 314

5.1.4. Derivative and integration of generalized functions 315

5.1.5. Complex argument

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