数学公式与积分手册（第四版）——新增解析映射章节

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"Handbook of Mathematical Formulas and Integrals, 4th Edition" 是一本由Alan Jeffery和Hui-Hui Dai合著的数学手册，主要针对工程数学和数学领域的专业人士和学者。这本书在第三版的基础上进行了扩展和更新，特别是在特殊函数、正交多项式和球面调和函数等方面增加了内容。第四版的一个显著新增章节是关于共形映射的讨论，这在之前的版本中是没有的。此外，作者还修正了一些印刷错误，并对第三版的最后四章进行了重新组织，以提供更方便的阅读结构。本书涵盖了广泛的数学公式和积分，包括但不限于： 1. 特殊函数：这些函数在物理、工程和数学中有广泛应用，如伽马函数、贝塞尔函数、厄米特函数、拉盖尔多项式等。第四版增加了这部分内容的深度，使读者能更好地理解和应用这些函数。 2. 正交多项式：这些多项式在数值分析、概率论和统计学中至关重要。书中详细介绍了它们的性质、生成函数和应用，如勒让德多项式和赫尔辛基多项式。 3. 球面调和函数：这是物理学和工程学中处理球面问题的关键工具，特别是在电磁学和量子力学中。更新的描述提供了更深入的理解和计算方法。 4. 共形映射：这是一个在复分析中极其重要的概念，用于研究平面到平面的保角映射。新加入的章节详细解释了共形映射的基本理论、常见映射函数及其应用，对于理解二维流形的几何性质非常有帮助。 5. 积分：书中包含了各种类型的积分，如不定积分、定积分、重积分、曲线积分、曲面积分等，以及相关的积分技巧和公式，对于解决实际问题和理论研究提供了强大的工具。除此之外，这本书还可能包含其他数学主题，如微积分、线性代数、矩阵理论、概率论和数理统计等，旨在为读者提供一个全面的数学参考资源。无论是科研工作还是教学，这本书都能成为研究人员和学生不可或缺的伙伴。由于版权原因，本书的具体内容无法在此处详述，但可以确认的是，"Handbook of Mathematical Formulas and Integrals, 4th Edition" 是一本集实用性和理论性于一体的数学工具书，它通过清晰的表述和丰富的实例，帮助读者掌握复杂的数学概念并应用于实际问题中。

Contents xv

19 Laplace Transformation 337

19.1. Introduction 337

19.1.1. Deﬁnition of the Laplace Transform 337

19.1.2. Basic Properties of the Laplace Transform 338

19.1.3. The Dirac Delta Function δ(x) 340

19.1.4. Laplace Transform Pairs 340

19.1.5. Solving Initial Value Problems by the Laplace

Transform 340

20 Fourier Transforms 353

20.1. Introduction 353

20.1.1. Fourier Exponential Transform 353

20.1.2. Basic Properties of the Fourier Transforms 354

20.1.3. Fourier Transform Pairs 355

20.1.4. Fourier Cosine and Sine Transforms 357

20.1.5. Basic Properties of the Fourier Cosine and Sine

Transforms 358

20.1.6. Fourier Cosine and Sine Transform Pairs 359

21 Numerical Integration 363

21.1. Classical Methods 363

21.1.1. Open- and Closed-Type Formulas 363

21.1.2. Composite Midpoint Rule (open type) 364

21.1.3. Composite Trapezoidal Rule (closed type) 364

21.1.4. Composite Simpson’s Rule (closed type) 364

21.1.5. Newton–Cotes formulas 365

21.1.6. Gaussian Quadrature (open-type) 366

21.1.7. Romberg Integration (closed-type) 367

22 Solutions of Standard Ordinary Diﬀerential

Equations 371

22.1. Introduction 371

22.1.1. Basic Deﬁnitions 371

22.1.2. Linear Dependence and Independence 371

22.2. Separation of Variables 373

22.3. Linear First-Order Equations 373

22.4. Bernoulli’s Equation 374

22.5. Exact Equations 375

22.6. Homogeneous Equations 376

22.7. Linear Diﬀerential Equations 376

22.8. Constant Coeﬃcient Linear Diﬀerential

Equations—Homogeneous Case 377

22.9. Linear Homogeneous Second-Order Equation 381

Preface

This book contains a collection of general mathematical results, formulas, and integrals that

occur throughout applications of mathematics. Many of the entries are based on the updated

ﬁfth edition of Gradshteyn and Ryzhik’s ”Tables of Integrals, Series, and Products,” though

during the preparation of the book, results were also taken from various other reference works.

The material has been arranged in a straightforward manner, and for the convenience of the

user a quick reference list of the simplest and most frequently used results is to be found in

Chapter 0 at the front of the book. Tab marks have been added to pages to identify the twelve

main subject areas into which the entries have been divided and also to indicate the main

interconnections that exist between them. Keys to the tab marks are to be found inside the

front and back covers.

The Table of Contents at the front of the book is suﬃciently detailed to enable rapid location

of the section in which a speciﬁc entry is to be found, and this information is supplemented by

a detailed index at the end of the book. In the chapters listing integrals, instead of displaying

them in their canonical form, as is customary in reference works, in order to make the tables

more convenient to use, the integrands are presented in the more general form in which they

are likely to arise. It is hoped that this will save the user the necessity of reducing a result to a

canonical form before consulting the tables. Wherever it might be helpful, material has been

added explaining the idea underlying a section or describing simple techniques that are often

useful in the application of its results.

Standard notations have been used for functions, and a list of these together with their

names and a reference to the section in which they occur or are deﬁned is to be found at the

front of the book. As is customary with tables of indeﬁnite integrals, the additive arbitrary

constant of integration has always been omitted. The result of an integration may take more

than one form, often depending on the method used for its evaluation, so only the most common

forms are listed.

A user requiring more extensive tables, or results involving the less familiar special functions,

is referred to the short classiﬁed reference list at the end of the book. The list contains works

the author found to be most useful and which a user is likely to ﬁnd readily accessible in a

library, but it is in no sense a comprehensive bibliography. Further specialist references are to

be found in the bibliographies contained in these reference works.

Every eﬀort has been made to ensure the accuracy of these tables and, whenever possible,

results have been checked by means of computer symbolic algebra and integration programs,

but the ﬁnal responsibility for errors must rest with the author.

xix

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