工程师与科学家必备：详解4000线性偏微分方程手册

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《工程师与科学家手册：线性偏微分方程》是一本专门针对科技工作者的权威参考书籍，第二版由Andrei D. Polyanin和Vladimir E. Nazaikinskii共同编撰。该手册汇集了近4000个线性偏微分方程，包括一阶、二阶、三阶乃至高阶的单个和系统的方程，涵盖了各种类型，如椭圆型、双曲型、抛物型和混合型等。书中不仅提供了详尽的解析解，还包括符号计算（如Maple、Mathematica）和数值方法（如MATLAB R）的运用，帮助读者理解和解决实际问题。本书强调了新发现的线性方程、精确解、变换和构建解的有效公式，通过具体实例展示了书中方法的应用。它还深入探讨了边值问题和特征值问题，这些是解决许多工程和科学领域问题的关键步骤。值得注意的是，书中提及的MATLAB R是在作者许可下的使用，但The MathWorks并不对书中内容或练习的准确性提供保证，其使用并不意味着对该软件的特定教学方法或用途的支持。《工程师与科学家手册：线性偏微分方程》的第二版由CRC Press出版，隶属于Taylor & Francis Group，地址位于佛罗里达州博卡拉顿。版权信息表明，尽管书中信息来源于权威且受人尊敬的来源，但作者和出版社仍尽力确保数据和信息的可靠性，但无法对所有内容的准确性做出完全担保。此外，该书的电子版国际标准书号(ISBN)为1-4665-8149-4，版本日期为2015年11月19日。《工程师与科学家手册：线性偏微分方程》是一部极其丰富的资源，对于研究者、工程师以及对线性偏微分方程感兴趣的学者来说，它提供了一个全面的工具箱，涵盖了理论分析、数值模拟以及实践应用的方方面面，是现代科学技术领域不可或缺的一份指南。

xiv CONTENTS

12.15 Nondissipative Thermoelasticity Equations (the Green–Naghdi Model) . . . . . 1103

12.15.1 Vector Form of the Nondissipative Thermoelasticity Equations . . . . 1103

12.15.2 Decompositions of the Nondissipative Thermoelasticity Equations

with f = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104

12.15.3 Decompositions of Thermoelasticity Equations with f 6= 0 . . . . . . . . 1105

12.16 Viscoelasticity Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106

12.16.1 Vector Form of Viscoelasticity Equations . . . . . . . . . . . . . . . . . . . . . . . 1106

12.16.2 Decompositions of Viscoelasticity Equations with f = 0 . . . . . . . . . . 1106

12.16.3 Various Forms of Decompositions for Viscoelasticity Equations with

f6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108

12.17 Maxwell Equations (Electromagnetic Field Equations) . . . . . . . . . . . . . . . . . . . 1108

12.17.1 Maxwell Equations in a Medium and Constitutive Relations . . . . . . 1108

12.17.2 Some Transformations and Solutions of the Maxwell Equations . . . 1109

12.18 Vector Equations of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1110

12.18.1 Vector Equations Containing O perators div and ∇ . . . . . . . . . . . . . . . 1110

12.18.2 Decompositions of the Homogeneous Vector Equation . . . . . . . . . . . 1111

12.18.3 Decompositions of the Nonhomogeneous Vector Equation . . . . . . . . 1112

12.18.4 Vector Equations Containing More General Operators . . . . . . . . . . . . 1113

12.19 General Systems Involving Vector and Scalar Equations: Part I . . . . . . . . . . . . 1114

12.19.1 Systems Containing Operators div and ∇ . . . . . . . . . . . . . . . . . . . . . . . 1114

12.19.2 Decompositions of Systems with Homogeneous Vector Equation . . 1115

12.19.3 Decompositions of Systems with Nonhomogeneous Vector

Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115

12.19.4 Equations for u and p. Reduction to One Vector Equation . . . . . . . . . 1116

12.19.5 Systems Containing More General Operators . . . . . . . . . . . . . . . . . . . 1117

12.20 General Systems Involving Vector and Scalar Equations: Part II . . . . . . . . . . . 1118

12.20.1 Class of Systems Considered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118

12.20.2 Asymmetric Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118

12.20.3 Symmetric Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119

Part II Analytical Methods 1121

13 Methods for First-Order Linear PDEs 1123

13.1 Linear PDEs with Two Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123

13.1.1 Special First-Order Linear PDEs with Two Independent Variables . . . 1123

13.1.2 General First-Order Linear PDE with Two Independent Variables . . . . 1126

13.2 First-Order Linear PDEs with Three or More Independent Variables . . . . . . . . 1129

13.2.1 Characteristic System. General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 1129

13.2.2 Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1131

14 S econd-Order Linear PDEs: Classiﬁcation, Problems, Particular Solutions 1133

14.1 Classiﬁcation of Second-Order Linear Partial Differential Equations . . . . . . . . 1133

14.1.1 Equations with Two Independent Variables . . . . . . . . . . . . . . . . . . . . . . . 1133

14.1.2 Equations with Many Independent Variables . . . . . . . . . . . . . . . . . . . . . . 1138

14.2 Basic Problems of Mathematical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1140

14.2.1 Initial and Boundary Conditions. Cauchy Problem. Boundary Value

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1140

14.2.2 First, Second, Third, and Mixed Boundary Value Problems . . . . . . . . . 1142

CONTENTS xv

14.3 Properties and Particular Solutions of Linear Equations . . . . . . . . . . . . . . . . . . . . 1144

14.3.1 Homogeneous Linear Equations. Basic Properties of Particular

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144

14.3.2 Separable Solutions. Solutions in the Form of Inﬁnite Series . . . . . . . . 1147

14.3.3 Nonhomogeneous Linear Equations and Their Properties . . . . . . . . . . . 1150

14.3.4 General Solutions of Some Hyperbolic Equations . . . . . . . . . . . . . . . . . . 1150

15 S ep aration of Variables and Integral Transform Methods 1153

15.1 Separation of Variables (Fourier Method) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1153

15.1.1 Description of Separation of Variables. General Stage of Solution . . . 1153

15.1.2 Problems for Parabolic Equations: Final Stage of Solution . . . . . . . . . . 1157

15.1.3 Problems for Hyperbolic Equations: Final Stage of Solution . . . . . . . . 1159

15.1.4 Solution of Boundary Value Problems for Elliptic Equations . . . . . . . . 1160

15.1.5 Solution of Boundary Value Problems for Higher-Order Equations . . . 1163

15.2 Integral Transform Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165

15.2.1 Laplace Transform and Its Application in Mathematical Physics . . . . . 1165

15.2.2 Fourier Transform and Its Application in Mathematical Physics . . . . . 1170

15.2.3 Fourier Sine and Cosine Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173

15.2.4 Mellin, Hankel, and Other Integral Transforms . . . . . . . . . . . . . . . . . . . . 1177

16 Cauchy Problem. Fundamental Solutions 1181

16.1 Dirac Delta Function. Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1181

16.1.1 Dirac Delta Function and Its Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 1181

16.1.2 Fundamental Solutions. Constructing Particular Solutions . . . . . . . . . . 1182

16.2 Representation of the Solution of the Cauchy Problem via the Fundamental

Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185

16.2.1 Cauchy Problem for Ordinary Differential Equations . . . . . . . . . . . . . . . 1185

16.2.2 Cauchy Problem for Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . 1187

16.2.3 Cauchy Problem for Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . 1190

16.2.4 Higher-O rder Linear PDEs. Generalized Cauchy P roblem . . . . . . . . . . 1193

17 Boundary Value Problems. Green’s Fu nction 1199

17.1 Boundary Value Problems for Parabolic Equations with One Space Variable.

Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1199

17.1.1 Representation of Solutions via the Green’s Function . . . . . . . . . . . . . . 1199

17.1.2 Problems for Equation s(x)

∂w

∂t

∂

∂x



p(x)

∂w

∂x



−q(x)w +Φ(x, t) . . . 1202

17.2 Boundary Value Problems for Hyperbolic Equations with One Space Variable.

Green’s Function. Goursat Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205

17.2.1 Representation of Solutions via the Green’s Function . . . . . . . . . . . . . . 1205

17.2.2 Problems for Equation s(x)

∂

∂t

∂

∂x



p(x)

∂w

∂x



−q(x)w +Φ(x, t) . . . 1207

17.2.3 Problems for Equation

∂

∂t

+ a(t)

∂w

∂t

b(t)



∂

∂x



p(x)

∂w

∂x



−q(x)w



+Φ(x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1208

17.2.4 Generalized Cauchy Problem with Initial Conditions Set along a

Curve. Riemann Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1210

17.2.5 Goursat Problem (a Problem with Initial Data on Characteristics) . . . . 1212

17.3 Boundary Value Problems for Elliptic Equations with Two Space Variables . . 1214

17.3.1 Problems and the Green’s Functions for Equation

a(x)

∂

∂x

∂

∂y

+b(x)

∂w

∂x

+c(x)w = −Φ(x, y) . . . . . . . . . . . . . . . . . . . . 1214

17.3.2 Representation of Solutions of Boundary Value Problems via Green’s

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216

xvi CONTENTS

17.4 Boundary Value Problems with Many Space Variables. Green’s Function . . . . 1218

17.4.1 Problems for Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218

17.4.2 Problems for Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1220

17.4.3 Problems for Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1221

17.4.4 Comparison of the Solution Structures for Boundary Value Problems

for Equations of Various Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1222

17.5 Construction of the Green’s Functions. G eneral Formulas and Relations . . . . . 1223

17.5.1 Green’s Functions of Boundary Value Problems for Equations of

Various Types in Bounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223

17.5.2 Green’s Functions Admitting Incomplete Separation of Variables . . . . 1224

17.5.3 Construction of Green’s Functions via Fundamental Solutions . . . . . . . 1227

18 Duhamel’s Principles. Some Transformations 1233

18.1 Duhamel’s Principles in Nonstationary Problems . . . . . . . . . . . . . . . . . . . . . . . . . 1233

18.1.1 Problems for Homogeneous Linear Equations . . . . . . . . . . . . . . . . . . . . . 1233

18.1.2 Problems for Nonhomogeneous Linear Equations . . . . . . . . . . . . . . . . . . 1235

18.2 Transformations Simplifying Initial and Boundary Conditions . . . . . . . . . . . . . . 1237

18.2.1 Transformations That Lead to Homogeneous B oundary Conditions . . 1237

18.2.2 Transformations That Lead to Homogeneous Initial and Boundary

Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1238

19 S ystems of Linear Coupled PDEs. Decomposition Methods 1239

19.1 Asymmetric and Symmetric Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239

19.1.1 Asymmetric Decomposition. Order of Decomposition . . . . . . . . . . . . . . 1239

19.1.2 Symmetric Decomposition. Invariant Transformations . . . . . . . . . . . . . 1242

19.2 First-Order Decompositions. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244

19.2.1 Systems of Linear PDEs without Mass Forces (f = 0) . . . . . . . . . . . . . . 1244

19.2.2 Systems of Linear PDEs with Mass Forces . . . . . . . . . . . . . . . . . . . . . . . . 1249

19.3 Higher-Order Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1251

19.3.1 Decomposition of Systems Consisting of One Vector Equation . . . . . . 1251

19.3.2 Decomposition of Systems Consisting of a Vector Equation and a

Scalar Equation (the First Approach) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1252

19.3.3 Decomposition of Systems Consisting of a Vector Equation and a

Scalar Equation (the Second Approach) . . . . . . . . . . . . . . . . . . . . . . . . . . 1253

20 S ome Asymptotic Results and Formulas 1255

20.1 Regular Perturbation Theory Formulas for the Eigenvalues . . . . . . . . . . . . . . . . . 1256

20.1.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256

20.1.2 Formulas for the Coefﬁcients of the E xpansion . . . . . . . . . . . . . . . . . . . . 1256

20.2 Singular Perturbation T heory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257

20.2.1 Cauchy Problem for the Schr¨odinger Equation . . . . . . . . . . . . . . . . . . . . 1257

20.2.2 Stationary Phase Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1261

20.2.3 Fourier Transform with a Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264

21 Elements of Theory of Generalized F unctions 1265

21.1 Generalized Functions of One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265

21.1.1 Important Terminological Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265

21.1.2 Test Function Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265

21.1.3 Distribution Space. Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . 1266

21.1.4 Derivatives of Distributions. Some Formulas . . . . . . . . . . . . . . . . . . . . . . 1267

21.1.5 Operations on Distributions and Some Transformations . . . . . . . . . . . . 1269

CONTENTS xvii

21.1.6 Tempered Distributions and Fourier Transform . . . . . . . . . . . . . . . . . . . . 1270

21.1.7 Generalized Solutions of Linear Ordinary Differential Equations . . . . 1271

21.2 Generalized Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1271

21.2.1 Some Deﬁnitions. Partial Derivatives. Direct Product. Linear

Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1271

21.2.2 Dirac Delta Function. Generalized Solutions of Linear PDEs . . . . . . . . 1273

Part III Symbolic and Numerical Solutions with Maple, Mathematica,

and MATLAB



1275

22 Linear Partial Differential Equations with Maple 1277

22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277

22.1.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277

22.1.2 Brief Introduction to Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1279

22.1.3 Maple Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1280

22.2 Analytical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1282

22.2.1 Constructing Analytical Solutions in Terms of Predeﬁned Functions . 1282

22.2.2 Constructing General Solutions via the Method of Characteristics . . . . 1289

22.2.3 Constructing General Solutions via Transformations to Canonical

Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1291

22.2.4 Constructing Analytical Solutions of C auchy P roblems . . . . . . . . . . . . . 1293

22.2.5 Constructing Analytical Solutions of B oundary Value Problems . . . . . 1297

22.2.6 Constructing Analytical Solutions of Initial-Boundary Value Problems 1298

22.2.7 Constructing Analytical Solutions of Systems of Linear PDEs . . . . . . . 1299

22.3 Analytical Solutions of Mathematical Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 1301

22.3.1 Constructing Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1301

22.3.2 Constructing Analytical Solutions via Integral Transform Methods . . . 1305

22.3.3 Constructing Analytical Solutions in Terms of Green’s Functions . . . . 1306

22.4 Numerical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1309

22.4.1 Constructing Numerical Solutions in Terms of Predeﬁned Functions . 1310

22.4.2 Numerical Methods Embedded in Maple . . . . . . . . . . . . . . . . . . . . . . . . . 1312

22.4.3 Numerical Solutions of Initial-Boundary Value Problems . . . . . . . . . . . 1318

22.4.4 Numerical Solutions of Boundary Value Problems . . . . . . . . . . . . . . . . . 1322

22.4.5 Numerical Solutions of Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . . . 1323

22.4.6 Numerical Solutions of Systems of Linear PDEs . . . . . . . . . . . . . . . . . . 1325

23 Linear Partial Differential Equations with Mathematica 1327

23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327

23.1.1 Some Notational Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327

23.1.2 Brief Introduction to Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327

23.1.3 Mathematica Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329

23.1.4 Dynamic Computation and Visualization in Mathematica Notebook . . 1332

23.2 Analytical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1333

23.2.1 Constructing Analytical Solutions in Terms of Predeﬁned Functions . 1333

23.2.2 Constructing General Solutions via the Method of Characteristics . . . . 1335

23.2.3 Constructing General Solutions via Conversion to Canonical Forms . . 1338

23.2.4 Constructing Analytical Solutions of C auchy P roblems . . . . . . . . . . . . . 1340

23.2.5 Constructing Analytical Solutions of B oundary Value Problems . . . . . 1342

23.2.6 Constructing Analytical Solutions of Initial-Boundary Value Problems 1344

23.2.7 Constructing Analytical Solutions of Systems of Linear PDEs . . . . . . . 1345

xviii CONTENTS

23.3 Analytical Solutions of Mathematical Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 1347

23.3.1 Constructing Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347

23.3.2 Constructing Analytical Solutions via Integral Transform Methods . . . 1350

23.3.3 Constructing Analytical Solutions in Terms of Green’s Functions . . . . 1352

23.4 Numerical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356

23.4.1 Constructing Numerical Solutions in Terms of Predeﬁned Functions . 1356

23.4.2 Numerical Methods Embedded in Mathematica . . . . . . . . . . . . . . . . . . . 1358

23.4.3 Numerical Solutions of Initial-Boundary Value Problems . . . . . . . . . . . 1359

23.4.4 Numerical Solutions of Boundary Value Problems . . . . . . . . . . . . . . . . . 1363

23.4.5 Numerical Solutions of Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . . . 1364

23.4.6 Numerical Solutions of Systems of Linear PDEs . . . . . . . . . . . . . . . . . . 1365

24 Linear Partial Differential Equations with MATLA B



1367

24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367

24.1.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367

24.1.2 Brief Introduction to MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368

24.1.3 MATLAB Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1371

24.2 Numerical Solutions of Linear PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374

24.2.1 Constructing Numerical Solutions via Predeﬁned Functions . . . . . . . . . 1375

24.2.2 Numerical Methods Embedded in MATLAB . . . . . . . . . . . . . . . . . . . . . . 1383

24.2.3 Numerical Solutions of Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . . . 1383

24.2.4 Numerical Solutions of Initial-Boundary Value Problems . . . . . . . . . . . 1385

24.2.5 Numerical Solutions of Boundary Value Problems . . . . . . . . . . . . . . . . . 1388

24.3 Constructing Finite Difference Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 1392

24.3.1 Explicit Finite Difference Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1392

24.3.2 Implicit Finite Difference Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395

24.4 Numerical Solutions of Systems of Linear PDEs . . . . . . . . . . . . . . . . . . . . . . . . . 1396

24.4.1 Linear Systems of 1D PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396

24.4.2 Linear Systems of 2D PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1399

Part IV Tables and Supplements 1403

25 Elementary Functions and Their Properties 1405

25.1 Power, Exponential, and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1405

25.1.1 Properties of the Power Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405

25.1.2 Properties of the Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405

25.1.3 Properties of the Logarithmic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406

25.2 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407

25.2.1 Simplest Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407

25.2.2 Reduction Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1408

25.2.3 Relations between Trigonometric Functions of Single Argument . . . . . 1408

25.2.4 Addition and Subtraction of Trigonometric Functions . . . . . . . . . . . . . . 1408

25.2.5 Products of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1409

25.2.6 Powers of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1409

25.2.7 Addition Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1409

25.2.8 Trigonometric Functions of Multiple Arguments . . . . . . . . . . . . . . . . . . 1410

25.2.9 Trigonometric Functions of Half Argument . . . . . . . . . . . . . . . . . . . . . . . 1410

25.2.10 Differentiation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1410

25.2.11 Integration Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1410

25.2.12 Expansion in Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1411

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