数学物理方法与复数特殊函数：英文PDF|Epub|txt|kindle电子书版本下载

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Chapter 1 The Classical Analytical Method of Mathematical Physics1

1.1 Introduction1

1.2 Preliminary Concepts3

1.3 The Basic Properties of Linear Partial Differential Equation4

1.4 The Second-order Linear Partial Differential Equation and their Classification5

1.5 The Fourier Series5

1.5.1 The Fourier Series with respect to Single Variable6

1.5.2 The Double Fourier Series7

1.5.3 The Triple Fourier Series7

1.6 The Power Series8

1.7 Gamma Function10

1.8 The Method of Separation of Variables11

1.8.1 The Vibrations of a String with Fixed Ends12

1.8.2 Two Dimensional Steady-State Isotropic Heat Conduction in Rectangular Region14

1.8.3 Two Dimensional Time-Dependent Isotropic Heat Conduction in Rectangular Region16

1.9 Bessel Function19

1.9.1 Introduction19

1.9.2 The Recurrence Formulas of Bessel Function21

1.9.3 Bessel series expansion22

1.9.4 Modified Bessel Function25

1.9.5 Kelvin function25

1.9.6 Spherical Bessel function26

1.9.7 Modified spherical Bessel function27

1.10 Legendre Polynomial28

1.10.1 Laplace's equation in spherical coordinates28

1.10.2 The solution in Real Power Series for Legendre Equation30

1.10.3 Legendre polynomial31

1.10.4 Investigation into Legendre Polynomial33

1.10.5 Associated Legendre Functions35

1.11 Sturm-Liouville Theory35

1.12 The Hankel Transform37

Chapter 2 Partial Differential Equation in Cartesian and Skew Coordinates and Method of Separation of Complex Variable39

2.1 Introduction39

2.2 Method of Separation of Complex Variables in Cartesian Coordinates39

2.2.1 The Transverse Bending for Anisotropic Rectangular Plate39

2.2.2 The Steady-State Anisotropic Heat Conduction in Rectangular Domain45

2.3 Method of Separation of Complex Variables in Skew Coordinates48

2.3.1 The Transverse Bending for Anisotropic Skew Plate48

2.3.2 The Steady-State Temperature in Anisotropic Skew Domain52

2.4 The Real Principle in Mathematical Physics53

Chapter 3 Time-dependent Heat Conduction in Curve-Typed Anisotropic Cylinder——Complex Cylindrical Polynomial and Complex Cylindrical Function54

3.1 Governing Partial Differential Equation of Modeling the Time-Dependent Temperature Field in Cylindrical Coordinates55

3.2 Zip Differential Equation,Complex Cylindrical Polynomial(Function)and the Analytical Solution56

3.2.1 Zip Differential Equation,Complex Cylindrical Polynomial and Function56

3.2.2 Analytical solution for Time-Dependent Heat Conduction in Solid Circular Cylinder with one Boundary Specialized Temperature60

3.2.3 Investigation into Complex Polynomial of the First Kind62

3.2.4 The Computational Procedure63

3.2.5 Numerical Experiments64

3.2.6 The Simplification of the Solution of Complex Cylindrical Function69

3.3 Complex Cylindrical Function Expansion Theorem and Investigation into Complex Cylindrical Function71

3.3.1 Complex Cylindrical Function Expansion Theorem71

3.3.2 The relationship between Zipl,n(x)polynomial and Bessel polynomial74

3.3.3 The Differential Formulas,Recurrent Formulas and Integral Formulas of Zip(x)75

3.4 Complex Cylindrical Polynomial of the Second Kind and corresponding Recurrence Formulas81

3.5 Complex Cylindrical Polynomial of the Third Kind and corresponding Recurrence Formula83

3.6 Asymtotics for Complex Cylindrical Polynomial85

3.7 Integral Representations for Complex Cylindrical Polynomial86

3.8 Investigation into Time-Dependent Heat Conduction in Anisotropic Circular Domain with other Boundary Condition87

3.8.1 Solid Anisotropic Circular Domain with Insulated Circumference Boundary Condition87

3.8.2 Solid Anisotropic Circular Domain with Heat Exchanged with Surrounding Medium Circumference Boundary Condition88

3.9 Time-Dependent Heat Conduction in Anisotropic Annular Region and Complex Cylindrical-annular Function Expansion Theorem89

3.9.1 Anisotropic Annular Region with Specified Value at Inner and Outer Circumference89

3.9.2 Theorem of Complex Cylindrical-annular Function Expansion over Annular Region91

3.9.3 Time-Dependent Heat Conduction in Anisotropic Annular Region with other Boundary Condition93

3.10 Analytical Solution for Steady-state Temperature in Anisotropic Circular region94

Chapter 4 Radially Symmetric Steady-state Heat Conduction in Anisotropic Solid Cylinder——Conplex Radically Symmetric Cylindrical Function97

4.1 Governing Equation in Cylindrical Coordinates and Complex Radically Symmetric Cylindrical Function97

4.2 The Steady-State Heat Conduction in Solid Anisotropic Cylinder with Specified Temperature in Lateral Boundary Condition102

4.3 The Steady-State Heat Conduction in Solid Anisotropic Cylinder with Insulated Lateral Boundary107

4.4 The Steady-State Heat Conduction in Solid Anisotropic Cylinder with Heat Exchange with Surrounding Medium in Lateral Boundary109

4.5 The Steady-State Heat Conduction in Anisotropic Hollow Cylinder with Zero Temperature in Lateral Boundaries110

4.6 The Steady-State Heat Conduction in Anisotropic Hollow Cylinder with Insulated Inner and Outer Lateral Boundaries111

4.7 The Steady-State Heat Conduction in Anisotropic Hollow Cylinder with Heat Exchange with Surrounding Medium112

4.8 Radically Symmetric Solution of Steady-State Heat Conductionin Isotropic Solid Cylinder with Zero Lateral Boundary112

Chapter 5 Partial Differential Equation for Three Dimensional114

5.1 Governing Equation in Cylindrical Coordinates114

5.2 Modified Zip Differential Equation,Modified Complex Cylindrical Polynomial and the Analytical solution of A-typed Anisotropic Cylinder115

5.2.1 Modified Zip Differential Equation,Modified Complex Cylindrical Polynomial and Function116

5.2.2 The Solution of Steady-State Heat Conduction in A-typed Anisotropic Solid Cylinderwith Zero Temperature in the Lateral Boundary118

5.2.3 The Steady-State Heat Conduction in A-typed Anisotropic Solid Cylinder with InsulatedLateral Boundary124

5.2.4 The Steady-State Heat Conduction in A-typed Anisotropic Solid Cylinder With Heat Exchanged with Surrounding Medium Lateral Boundary125

5.2.5 The Steady-State Heat Conduction in Anisotropic Hollow Cylinder with Zero Temperature in Inner and Outer Lateral Boundaries125

5.2.6 The Steady-State Heat Conduction in Anisotropic Hollow Cylinder with Insulated Inner and Outer Lateral Boundaries126

5.2.7 The Steady State Heat Conduction in Anisotropic Hollow Cylinder with Heat Exchange with Surrounding Medium in Inner and Outer Lateral Boundaries127

5.3 Investigation into Modified Complex Cylindrical Polynomial128

5.3.1 The relationship between ？ip(x)polynomial and modified Bessel polynomial128

5.3.2 The Differential Formulas,Recurrent Formulas and Integral Formulas of ？ip(x)128

5.4 Modified Complex Cylindrical Polynomial of the Second Kind and corresponding Recurrence Formulas133

5.5 Modified Complex Cylindrical Polynomial of the Third Kind139

5.6 Integral Formulas for Modified Complex Cylindrical Polynomial139

5.7 Investigation into the steady-state heat conduction in B-typed three dimensional anisotropic cylinder——B-typed cylindrical polynomial141

5.7.1 B-typed Anisotropic Cylindrical Equation and B-typed Anisotropic Cylindrical Polynomial142

5.7.2 The Solution of Steady-State Heat Conduction in B-typed Anisotropic Solid Cylinder with Zero Temperature in the Lateral Boundary144

5.7.3 The Steady-State Heat Conduction in B-typed Anisotropic Solid Cylinder with Insulated Lateral Boundary148

5.7.4 The Steady-State Heat Conduction in B-typed Anisotropic Solid Cylinder with Heat Exchanged with Surrounding Medium Lateral Boundary149

5.7.5 The Steady-State Heat Conduction in B-typed Anisotropic Hollow Cylinder with Zero Temperature in Inner and Outer Lateral Boundaries149

5.7.6 The Steady-State Heat Conduction in B-typed Anisotropic Hollow Cylinder with Insulated Inner and Outer Lateral Boundaries150

5.7.7 The Steady-State Heat Conduction in B-typed Anisotropic Hollow Cylinder with Heat Exchange with Surrounding Medium in Inner and Outer Lateral Boundaries151

5.8 Investigation into the Steady-State Heat Conduction in C-typedAnisotropic Three Dimensional Cylinder——C-typed Cylindrical Polynomial152

5.8.1 C-typed Anisotropic Cylindrical Equation and C-typed Anisotropic Cylindrical Polynomial153

5.8.2 The Steady-State Heat Conduction in C-typed Anisotropic Solid Cylinder with Zero Temperature in the Lateral Boundary155

5.8.3 The Steady-State Heat Conduction in C-typed Anisotropic Solid Cylinder with Insulated Lateral Boundary159

5.8.4 The Steady-State Heat Conduction in C-typed Anisotropic Solid Cylinder with Heat Exchanged with Surrounding Medium Lateral Boundary160

5.8.5 The Steady-State Heat Conduction in C-typed Anisotropic Hollow Cylinder with Zero Temperature in Inner and Outer Lateral Boundaries160

5.8.6 The Steady-State Heat Conduction in C-typed Anisotropic Hollow Cylinder with Insulated Inner and Outer Lateral Boundaries161

5.8.7 The Steady-State Heat Conduction in C-typed Anisotropic Hollow Cylinder with Heat Exchange with Surrounding Medium in Inner and Outer Lateral Boundaries162

5.9 Investigation into the Steady-State Heat Conduction in General Anisotropic Three Dimensional Cylinder——General Cylindrical162

5.9.1 General Anisotropic Cylindrical Equation and General Anisotropic Cylindrical Polynomial163

5.9.2 The Solution of Steady-State Heat Conduction in General Anisotropic Solid Cylinder with Zero Temperature in the Lateral Boundary165

5.9.3 The Steady-State Heat Conduction in General Anisotropic Solid Cylinder with Insulated Lateral Boundary166

5.9.4 The Steady-State Heat Conduction in General Anisotropic Solid Cylinder with Heat Exchanged with Surrounding Medium Lateral Boundary166

5.9.5 The Steady-State Heat Conduction in General Anisotropic Hollow Cylinder with Zero Temperature in Inner and Outer Lateral Boundaries167

5.9.6 The Steady-State Heat Conduction in General Anisotropic Hollow Cylinder with Insulated Inner and Outer Lateral Boundaries167

5.9.7 The Steady-State Heat Conduction in General Anisotropic Hollow Cylinder with Heat Exchange with Medium in Inner and Outer Lateral Boundaries168

5.10 The Relationship between the General Anisotropic Cylindrical Equation and the Confluent Hypergeometric Function169

Chapter 6 Partial Differential Equation for Three Dimensional Time-dependent AnisotropicHeat Conduction in Cylindrical Coordinates——Complex Cylinder Function170

6.1 Three Dimensional Governing Equation in Cylindrical Coordinates and its Analytical Solution170

6.2 Theorem of Complex Cylinder Function Expansion176

6.2.1 The Orthogonality of the Series of einθeikπ/LZZipl,n(r/Rμ0l,n,j)176

6.2.2 Theorem of Complex Cylinder Function Expansion177

6.3 Solving Procedure and Numerical Experiments178

6.4 Investigation into Time-Dependent Heat Conduction in A-typed Anisotropic Cylinder with other Boundary Condition178

6.4.1 The Solid A-typed Anisotropic Cylinder with the Insulated Lateral Boundary and the Upper,Bottom Boundaries of Zero Temperature178

6.4.2 The Solid A-typed Anisotropic Cylinder with the Lateral Boundary with Heat Exchanged with Surrounding Medium and the Upper,Bottom Boundaries of Zero Temperature179

6.4.3 The A-typed Anisotropic Hollow Cylinder with Zero Temperature in Inner and Outer Lateral Boundaries and the Upper,Bottom Boundaries of Zero Temperature179

6.4.4 The A-typed Anisotropic Hollow Cylinder with Insulated Inner and Outer Lateral Boundaries and the Upper,Bottom Boundaries of Zero Temperature180

6.4.5 The A-typed Anisotropic Hollow Cylinder with Both Inner and Outer Lateral Boundaries with Heat Exchanged with Surrounding Medium and the Upper,Bottom Boundary of Zero Temperature181

6.5 Determine the Eigenvalue λ182

Chapter 7 Analytical Solution of Steady-state Conduction in Thin Curve-typed Anisotropic Circular Plate with Surface Heat Exchange——Modified complex spherical cylindrical polynomial184

7.1 Governing Partial Differential Equation in Polar Coordinates184

7.2 Modified Spherical Zip Equation,Modified Complex Spherical Cylindrical Polynomial and the Analytical Solution185

7.2.1 Modified Spherical Zip equation,Modified Complex Spherical Cylindrical Polynomial and Function185

7.2.2 The Analytical Solution189

7.2.3 Investigation into Polynomials of Y(l)n(x)and ？Cipl,n(x)191

7.2.4 Numerical Experiments193

7.3 The Simplification of the Series Solution in einθ？Cipl,n(x)196

7.4 Investigation into ？Cipl,n(x)polynomial197

7.4.1 The Relationship Between ？Cipl,n(x)Polynomial and Modified Spherical Bessel Function197

7.4.2 The Differential Formulas、Recurrent Formulas and Integral Formulas of ？Cipl,n(x)199

7.5 Modified Complex Spherical Cylindrical Polynomial of the Second Kind and Corresponding Formulas202

Chapter 8 Analytical Solution of time-dependent Conduction in Thin Curve-typed Anisotropic Circular Plate——Complex Spherical Cylindrical Polynomial206

8.1 Governing Partial Differential Equation in Polar Coordinates206

8.2 Spherical Zip Differential Equation,Complex Spherical Cylindrical Polynomial and the Analytical Solution208

8.2.1 Spherical Zip Differential Equation and Complex Spherical Cylindrical Polynomial208

8.2.2 Time-Dependent Heat Conduction in Thin Solid Circular Plate with one Boundary Specialized Temperature211

8.2.3 Investigation into the Complex Polynomial Cipl,n(x)213

8.2.4 The Computational Procedure216

8.2.5 Numerical results216

8.3 The simplification of Complex Spherical-Cylindrical Function220

8.4 Complex Spherical Cylindrical Expansion Theorem221

8.5 The Differential Formulas,Recurrent Formulas and Integral Formulas of Cip(x)224

8.5.1 The differential formulas of Cip(x)224

8.5.2 The recurrent formulas of Cip(x)225

8.5.3 The integral formulas of Cip(x)226

8.6 Complex Spherical Cylindrical Polynomial of the Second Kindand Corresponding Formulas227

8.7 Complex Spherical Cylindrical Polynomial of the Third Kind and Corresponding Formulas229

8.8 Investigation into Time-Dependent Heat Conduction in Anisotropic Circular Thin Plate with other Boundary Condition231

8.8.1 Solid Anisotropic Circular Thin Plate with Insulated Circumference Boundary Condition231

8.8.2 Solid Anisotropic Circular Plate with Heat Exchanged with Surrounding Medium Circumference Boundary Condition232

8.8.3 Time-Dependent Heat Conduction in Anisotropic Thin Annular Plate and Complex Spherical Cylindrical-Annular Function Expansion over Annular Region Theorem232

Chapter 9 Complex Cylindrical Surface Polynomial and Function in the Anisotropic Heat Conduction Partial Differential Equation239

9.1 Introduction239

9.2 The Solution for Steady-State Anisotropic Heat Conduction in Cylindrical Surface240

9.2.1 General Complex Cylindrical Surface Function and the Analytical Solution240

9.2.2 The Solving Procedure242

9.3 The Solution of Time-Dependent Problem in Thin Cylindrical Shell243

9.3.1 Governing Equation and Solving Procedure243

9.3.2 Time-Dependent Heat Conduction in Finite Length Circular Shell with Two Boundary Specialized Temperature246

9.4 Numerical Experiments246

9.5 Complex Cylindrical Surface Function Expansion Theorem248

Chapter 10 Parametric Complex Cylindrical Surface Polynomial and Function with application in Thin Anisotropic Cylindrical Shell Exchanging Heat with Surrounding Medium251

10.1 Introduction251

10.2 The Solution for Steady-State Anisotropic Heat Conduction in Cylindrical Surface Exchanging Heat with the Surrounding Medium251

10.2.1 The Solving Procedure252

10.2.2 The Analytical Solution253

10.3 The Solution of Time-Dependent Heat Conduction in Thin Cylindrical Shell Exchanging Heat with the Surrounding Medium255

10.3.1 Governing Equation and Solving Procedure255

10.3.2 Time-Dependent Heat Conduction in Finite Length Cylindrical Shell with Two Boundary Specialized Temperature258

10.4 Numerical Experiments258

10.5 Parametric Complex Cylindrical Surface Function Expansion Theorem260

Chapter 11 Heat Conduction in Thin Anisotropic Conical Shell263

11.1 Introduction263

11.2 Analytical Solution to Steady-State Heat Conduction in Thin Anisotropic Conical Shell263

11.2.1 Governing Equation in Spherical Coordinates263

11.2.2 The General Analytical Solution264

11.2.3 Numerical Experiments266

11.2.4 The Proof of the Simplification of Complex Series267

11.3 The Analytical Solution to Steady-State Conduction in Thin Anisotropic Conical Shell with Surface Heat Exchange268

11.3.1 Governing Equation in Spherical Coordinates268

11.3.2 The Analytical Procedure269

11.3.3 Numerical Experiments271

11.4 The Analytical Solution to Time-Dependent Conduction in Thin Anisotropic Conical Shell272

11.4.1 Governing Equation in Spherical Coordinates272

11.4.2 Analytical Procedure273

11.4.3 Numerical Results276

Chapter 12 The Series of Complex Cylindrical Function Transforms279

12.1 Complex Cylindrical Function Integral Transform279

12.1.1 Basic Formulas279

12.1.2 The Properties of Complex Cylindrical Function Integral Transform280

12.1.3 Complex Cylindrical Function Integral Transform Table281

12.2 The Finite Complex Cylindrical Function Integral Transform282

12.2.1 Basic Formulas282

12.2.2 The finite complex cylindrical function integral transform Table282

12.3 Complex Spherical-Cylindrical Function Integral Transform283

12.3.1 Basic Formulas283

12.3.2 The Properties of Complex Spherical-Cylindrical Function Integral Transform284

12.4 The finite complex spherical-cylindrical function integral transform285

12.5 Modified Complex Cylindrical Function Integral Transform286

12.5.1 Basic Formulas286

12.5.2 The Properties of Modified Complex Cylindrical Function Integral Transform287

12.6 The Finite Modified Complex Cylindrical Function Integral Transform288

12.7 Modified Complex Spherical-Cylindrical Function Integral Transform288

12.7.1 Basic Formulas288

12.7.2 The Properties of Modified Complex Spherical-Cylindrical Function Integral Transform290

12.8 The Finite Modified Complex Spherical-Cylindrical Function Integral Transform290

12.9 Other Multi-Dimensional Complex Cylindrical Function Integral Transform290

12.9.1 Two-Dimensional Complex Cylindrical Function Integral Transform290

12.9.2 Two-Dimensional Complex Spherical-Cylindrical Function Integral Transform291

12.9.3 Two-Dimensional Modified Complex Cylindrical Function Integral Transform291

12.9.4 Two-Dimensional Modified Complex Spherical-Cylindrical Function Integral Transform292

12.9.5 Three-dimensional complex cylindrical function integral transform293

Chapter 13 Steady-State Anisotropic Heat Conduction in the Spherical Zone——Complex Spherical Polynomial and Function294

13.1 Introduction294

13.2 Governing Equation in Spherical Coordinates294

13.3 Zis Differential Equation,Complex Spherical Function and the General Analytical Solution295

13.3.1 Zis Differential Equation,Complex Spherical Polynomial and Function295

13.3.2 The General Analytical Solution298

13.4 Studies on the Steady-State Isotropic Heat Conduction in the Spherical Zone299

13.5 Numerical Experiments300

13.6 Summary of the Complex Spherical Function Expansion304

13.7 The Recurrence Formulas305

13.8 The Simplification of Complex Spherical Function306

Chapter 14 Steady-State Anisotropic Heat Conduction in Spherical Zone Exchanging Heat with Surrounding Medium——Parametric Complex Spherical Polynomial and Complex Spherical Function309

14.1 Introduction309

14.2 Governing Equation in Spherical Coordinates309

14.3 The Parameter Form of Zis Differential Equation,Parametric Complex Spherical Function and the Analytical Solution310

14.3.1 The Parameter Form of Zis Differential Equation,Parametric Complex Spherical Polynomial and Function310

14.3.2 The General Analytical Solution314

14.4 Numerical Experiments315

14.5 The Recurrence Relation317

14.6 The Proof of the Real Principle for the Complex Spherical Function318

Chapter 15 Partial Differential Equation for Steady-state Anisotropic Heat Conduction in the Spherical Surface——Associated Complex Spherical Polynomial Function321

15.1 Introduction321

15.2 Governing Equation in Spherical Coordinates322

15.3 Associated Zis Differential Equation,Associated Complex Spherical Function and the Analytical Solution323

15.3.1 Associated Zis Differential Equation,Associated Complex Spherical Polynomial and Function323

15.3.2 The General Analytical Solution326

15.4 Studies on the Solution of Isotropic Heat Conduction in Spherical Surface327

15.5 Numerical Experiments328

15.6 The Recurrence Relation332

Chapter 16 Steady-State Anisotropic Heat Conduction in Spherical Surface Exchanging Heat with Surrounding Medium——Parametric Associated Complex Spherical Polynomial and Function334

16.1 Introduction334

16.2 Governing Equation in Spherical Coordinates334

16.3 Associated Zis Differential Equation,Parametric Associated Complex Spherical Function and the Analytical Solution336

16.3.1 Associated Zis differential equation,Parametric Associated Complex Spherical Polynomial and Function336

16.3.2 The General Analytical Solution339

16.4 Numerical Experiments340

16.5 The Recurrence Relation345

Chapter 17 Complex Spherical Zonal Function with Its Application In Partial Differential Equation for Time-dependent Anisotropic Heat Conduction in Spherical Zone346

17.1 Introduction346

17.2 Partial Differential Equation for Time-Dependent Anisotropic Heat Conduction in Spherical Zone347

17.3 The Solving Procedure348

17.4 The z-axis Symmetric Solution for Time-Dependent Heat Conduction in the Global Anisotropic Spherical Surface352

17.5 The Complex Spherical Zonal Polynomial and Analytical Solution for Time-Dependent Heat Conduction353

17.6 Complex Spherical Zonal Function Expansion Theorem355

Chapter 18 Complex Spherical Zonal Function with Its Application In Partial Differential Equation for Steady-state Anisotropic Heat Conduction in Three-dimensional Sphere359

18.1 Partial Differential Equation for Steady-State Anisotropic Heat Conduction in Three-dimensional Sphere359

18.2 The Analytical Procedure360

18.3 The z-axis Symmetric Solution in the Global Anisotropic Sphere365

18.4 The Analytical Solution in the Anisotropic Sphere366

Chapter 19 Complex Sphere Function in Partial Differential Equation for Time-dependent Anisotropic Heat Conduction in Three-dimensional Sphere368

19.1 Governing Equation in Spherical Coordinates368

19.2 The Analytical Procedure369

19.3 The Product Solution for the Anisotropic Solid Sphere375

19.4 Theorem of Complex Sphere Function Expansion378

Chapter 20 Associated Complex Spherical Zonal Function with Its Application in Time-dependent Anisotropic Heat Conduction in Spherical Zone381

20.1 Partial Differential Equation for Time-Dependent AnisotropieHeat Conduction in Spherical zone381

20.2 The Solving Procedure382

20.3 The Product Solution386

20.4 Associated Complex Spherical Zonal Polynomial/function and the Analytical Solution387

20.5 Associated Complex Zonal Spherical Function Expansion Theorem390

Chapter 21 The Analytical Solution to Partial Differential Equation for Steady-state Anisotropic Heat Conduction in Three-dimensional Sphere394

21.1 Partial Differential Equation for Steady-State Anisotropic Heat Conduction in Three-dimensional Sphere394

21.2 The Solving Procedure395

21.3 The Product Solution400

Chapter 22 Partial Differential Equation for Time-dependent Anisotropic Heat Conduction in Three-dimensional Sphere Associated Complex Sphere Function403

22.1 Partial Differential Equation for Time-dependent Anisotropic Heat Conduction in Three-dimensional Sphere403

22.2 The Solving Procedure405

22.3 The Analytical Solution408

22.4 Theorem of Associated Complex Sphere Function Expansion413

Chapter 23 Analytical Solution to the Anisotropic Wave Equations416

23.1 The two dimensional Anisotropic Wave Equation in Cylinder in Cylindrical Coordinates416

23.2 The three-dimensional Anisotropic Wave Equation in Cylindrical Coordinates421

23.3 The two-dimensional Anisotropic Wave Equation in Spherical Membrane in Spherical Coordinates426

23.4 The three-dimensional Anisotropic Wave Equation in Spherical Coordinates430

23.5 The two-dimensional Anisotropic Wave Equation in Circular Membrane in Polar Coordinates436

23.6 Analytical Solution to the Two-dimensional Anisotropic Wave Equation in Spherical Membrane by the Method of Associated Complex Spherical Zonal Function440