物理学中的数学方法（英文版）（Mathematics for Physicists） 电子版 免费 pdf 下载 azw3 fb2 caj mobi

《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 ***st Euler equation 12

1.3. The Cases of Multifuncti*** and Multivariates 16

1.3.1. Multifuncti*** 16

1.3.2. Multivariates 19

1.4. Functional Extremes under Certain Conditi*** 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 Applicati*** of the Variational Method in Physics 41

1.7.1. The applicati*** in classical physics 44

1.7.2. The applicati*** 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. ***s in Inner Product Spaces 74

2.2.1. ***s and adjoint operators 74

2.2.2. Selfadjoint operators 84

2.2.3. The alternative theorem for the soluti*** of linear algebraic equati*** 94

2.3. Complete Set of Orthonormal Functi*** 96

2.3.1. Three kinds of convergences 96

2.3.2. The completeness of a set of functi*** 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 Equati*** of Second Order 127

3.1. General Theory 127

3.1.1. The existence and uniqueness of soluti*** 127

3.1.2. The struct.ure of soluti*** of homogeneous equati*** 130

3.1.3. The soluti*** of inhomogeneous equati*** 137

3.2. SturmLiouville Eigenvalue Problem 140

3.2.1. The form of SturmLiouville equati*** 140

3.2.2. The boundary conditi*** of SturmLiouville equati*** 142

3.2.3. SturmLiouville eigenvalue problem 144

3.3. The Polynomial Soluti*** of SturmLiouville Equati*** 151

3.3.1. Possible forms of kernel and weight functi*** 151

3.3.2. The expressi*** in series and in derivatives of the polynomials 158

3.3.3. Generating functi*** 165

3.3.4. The completeness theorem of orthogonal polynomials as SturmLiouville soluti*** 169

3.3.5. Applicati*** in numerical integrati*** 171

3.4. Equati*** and Functi*** that Relate to the Polynomial Soluti*** 174

3.4.1. Laguerre functi*** 175

3.4.2. Legendre functi*** 179

3.4.3. Chebyshev functi*** 185

3.4.4. Hermite functi*** 190

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

3.5.1. Soluti*** of homogeneous equati*** 196

3.5.2. Ordinary differential equati*** of second order 216

3.6. NonSelfAdjoint Ordinary Differential Equati*** of Second Order 224

3.6.1. Adjoint equati*** of ordinary differential equati*** 224

3.6.2. SturmLiouville operator 225

3.6.3. Complete set of n***elfadjoint ordinary differential equati*** of second order 229

3.7. The Conditi*** under Which Inhomogeneous Equati*** have Soluti*** 231

Exercises 236

Appendix 3A Generalization of SturmLiouville Theorem to Dirac Equation 244

Chapter 4 Bessel Functi*** 247

4.1. Bessel Equation 247

4.1.1. Bessel equat.ion and its solutioiis 247

4.1.2. Bessel functi*** of the first and second kinds 255

4.2. Fundamental Properties of Bessel Functi*** 258

4.2.1. Recurrence relati*** of Bessel functi*** 258

4.2.2. Asymptotic formulas of Bessel functi*** 261

4.2.3. Zeros of Bessel functi*** 262

4.2.4. Wr***kian 264

4.3. Bessel Functi*** of Integer Orders 266

4.3.1. Parity and the values at certain points 266

4.3.2. Generating function of Bessel functi*** of integer orders 267

4.4. Bessel Functi*** of HalfInteger Orders 273

4.5. Bessel Functi*** of the Third Kind and Spherical Bessel Functi*** 275

4.5.1. Bessel functi*** of the third kind 275

4.5.2. Spherical Bessel functi*** 280

4.6. Modified Bessel Functi*** 288

4.6.1. Modified Bessel functi*** of the first and second kinds 288

4.6.2. Modified Bessel functi*** of integer orders 293

4.7. Bessel Functi*** with Real Arguments 294

4.7.1. Eigenvalue problem of Bessel equation 294

4.7.2. Properties of eigenfuncti*** 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 transformati*** of the delta function 314

5.1.4. Derivative and integration of generalized functi*** 315

5.1.5. Complex argument

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