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Chapter 1 The Classical Analytical Method of Mathematical Physics 1

1.1 Introduction 1

1.2 Preliminary Concepts 3

1.3 The Basic Properties of Linear Partial Differential Equation 4

1.4 The Second-order Linear Partial Differential Equation and their Classification 5

1.5 The Fourier Series 5

1.5.1 The Fourier Series with respect to Single Variable 6

1.5.2 The Double Fourier Series 7

1.5.3 The Triple Fourier Series 7

1.6 The Power Series 8

1.7 Gamma Function 10

1.8 The Method of Separation of Variables 11

1.8.1 The Vibrations of a String with Fixed Ends 12

1.8.2 Two Dimensional Steady-State Isotropic Heat Conduction in Rectangular Region 14

1.8.3 Two Dimensional Time-Dependent Isotropic Heat Conduction in Rectangular Region 16

1.9 Bessel Function 19

1.9.1 Introduction 19

1.9.2 The Recurrence Formulas of Bessel Function 21

1.9.3 Bessel series expansion 22

1.9.4 Modified Bessel Function 25

1.9.5 Kelvin function 25

1.9.6 Spherical Bessel function 26

1.9.7 Modified spherical Bessel function 27

1.10 Legendre Polynomial 28

1.10.1 Laplace's equation in spherical coordinates 28

1.10.2 The solution in Real Power Series for Legendre Equation 30

1.10.3 Legendre polynomial 31

1.10.4 Investigation into Legendre Polynomial 33

1.10.5 Associated Legendre Functions 35

1.11 Sturm-Liouville Theory 35

1.12 The Hankel Transform 37

Chapter 2 Partial Differential Equation in Cartesian and Skew Coordinates and Method of Separation of Complex Variable 39

2.1 Introduction 39

2.2 Method of Separation of Complex Variables in Cartesian Coordinates 39

2.2.1 The Transverse Bending for Anisotropic Rectangular Plate 39

2.2.2 The Steady-State Anisotropic Heat Conduction in Rectangular Domain 45

2.3 Method of Separation of Complex Variables in Skew Coordinates 48

2.3.1 The Transverse Bending for Anisotropic Skew Plate 48

2.3.2 The Steady-State Temperature in Anisotropic Skew Domain 52

2.4 The Real Principle in Mathematical Physics 53

Chapter 3 Time-dependent Heat Conduction in Curve-Typed Anisotropic Cylinder——Complex Cylindrical Polynomial and Complex Cylindrical Function 54

3.1 Governing Partial Differential Equation of Modeling the Time-Dependent Temperature Field in Cylindrical Coordinates 55

3.2 Zip Differential Equation,Complex Cylindrical Polynomial(Function)and the Analytical Solution 56

3.2.1 Zip Differential Equation,Complex Cylindrical Polynomial and Function 56

3.2.2 Analytical solution for Time-Dependent Heat Conduction in Solid Circular Cylinder with one Boundary Specialized Temperature 60

3.2.3 Investigation into Complex Polynomial of the First Kind 62

3.2.4 The Computational Procedure 63

3.2.5 Numerical Experiments 64

3.2.6 The Simplification of the Solution of Complex Cylindrical Function 69

3.3 Complex Cylindrical Function Expansion Theorem and Investigation into Complex Cylindrical Function 71

3.3.1 Complex Cylindrical Function Expansion Theorem 71

3.3.2 The relationship between Zipl,n(x)polynomial and Bessel polynomial 74

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 Formulas 81

3.5 Complex Cylindrical Polynomial of the Third Kind and corresponding Recurrence Formula 83

3.6 Asymtotics for Complex Cylindrical Polynomial 85

3.7 Integral Representations for Complex Cylindrical Polynomial 86

3.8 Investigation into Time-Dependent Heat Conduction in Anisotropic Circular Domain with other Boundary Condition 87

3.8.1 Solid Anisotropic Circular Domain with Insulated Circumference Boundary Condition 87

3.8.2 Solid Anisotropic Circular Domain with Heat Exchanged with Surrounding Medium Circumference Boundary Condition 88

3.9 Time-Dependent Heat Conduction in Anisotropic Annular Region and Complex Cylindrical-annular Function Expansion Theorem 89

3.9.1 Anisotropic Annular Region with Specified Value at Inner and Outer Circumference 89

3.9.2 Theorem of Complex Cylindrical-annular Function Expansion over Annular Region 91

3.9.3 Time-Dependent Heat Conduction in Anisotropic Annular Region with other Boundary Condition 93

3.10 Analytical Solution for Steady-state Temperature in Anisotropic Circular region 94

Chapter 4 Radially Symmetric Steady-state Heat Conduction in Anisotropic Solid Cylinder——Conplex Radically Symmetric Cylindrical Function 97

4.1 Governing Equation in Cylindrical Coordinates and Complex Radically Symmetric Cylindrical Function 97

4.2 The Steady-State Heat Conduction in Solid Anisotropic Cylinder with Specified Temperature in Lateral Boundary Condition 102

4.3 The Steady-State Heat Conduction in Solid Anisotropic Cylinder with Insulated Lateral Boundary 107

4.4 The Steady-State Heat Conduction in Solid Anisotropic Cylinder with Heat Exchange with Surrounding Medium in Lateral Boundary 109

4.5 The Steady-State Heat Conduction in Anisotropic Hollow Cylinder with Zero Temperature in Lateral Boundaries 110

4.6 The Steady-State Heat Conduction in Anisotropic Hollow Cylinder with Insulated Inner and Outer Lateral Boundaries 111

4.7 The Steady-State Heat Conduction in Anisotropic Hollow Cylinder with Heat Exchange with Surrounding Medium 112

4.8 Radically Symmetric Solution of Steady-State Heat Conductionin Isotropic Solid Cylinder with Zero Lateral Boundary 112

Chapter 5 Partial Differential Equation for Three Dimensional 114

5.1 Governing Equation in Cylindrical Coordinates 114

5.2 Modified Zip Differential Equation,Modified Complex Cylindrical Polynomial and the Analytical solution of A-typed Anisotropic Cylinder 115

5.2.1 Modified Zip Differential Equation,Modified Complex Cylindrical Polynomial and Function 116

5.2.2 The Solution of Steady-State Heat Conduction in A-typed Anisotropic Solid Cylinderwith Zero Temperature in the Lateral Boundary 118

5.2.3 The Steady-State Heat Conduction in A-typed Anisotropic Solid Cylinder with InsulatedLateral Boundary 124

5.2.4 The Steady-State Heat Conduction in A-typed Anisotropic Solid Cylinder With Heat Exchanged with Surrounding Medium Lateral Boundary 125

5.2.5 The Steady-State Heat Conduction in Anisotropic Hollow Cylinder with Zero Temperature in Inner and Outer Lateral Boundaries 125

5.2.6 The Steady-State Heat Conduction in Anisotropic Hollow Cylinder with Insulated Inner and Outer Lateral Boundaries 126

5.2.7 The Steady State Heat Conduction in Anisotropic Hollow Cylinder with Heat Exchange with Surrounding Medium in Inner and Outer Lateral Boundaries 127

5.3 Investigation into Modified Complex Cylindrical Polynomial 128

5.3.1 The relationship between ？ip(x)polynomial and modified Bessel polynomial 128

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 Formulas 133

5.5 Modified Complex Cylindrical Polynomial of the Third Kind 139

5.6 Integral Formulas for Modified Complex Cylindrical Polynomial 139

5.7 Investigation into the steady-state heat conduction in B-typed three dimensional anisotropic cylinder——B-typed cylindrical polynomial 141

5.7.1 B-typed Anisotropic Cylindrical Equation and B-typed Anisotropic Cylindrical Polynomial 142

5.7.2 The Solution of Steady-State Heat Conduction in B-typed Anisotropic Solid Cylinder with Zero Temperature in the Lateral Boundary 144

5.7.3 The Steady-State Heat Conduction in B-typed Anisotropic Solid Cylinder with Insulated Lateral Boundary 148

5.7.4 The Steady-State Heat Conduction in B-typed Anisotropic Solid Cylinder with Heat Exchanged with Surrounding Medium Lateral Boundary 149

5.7.5 The Steady-State Heat Conduction in B-typed Anisotropic Hollow Cylinder with Zero Temperature in Inner and Outer Lateral Boundaries 149

5.7.6 The Steady-State Heat Conduction in B-typed Anisotropic Hollow Cylinder with Insulated Inner and Outer Lateral Boundaries 150

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 Boundaries 151

5.8 Investigation into the Steady-State Heat Conduction in C-typedAnisotropic Three Dimensional Cylinder——C-typed Cylindrical Polynomial 152

5.8.1 C-typed Anisotropic Cylindrical Equation and C-typed Anisotropic Cylindrical Polynomial 153

5.8.2 The Steady-State Heat Conduction in C-typed Anisotropic Solid Cylinder with Zero Temperature in the Lateral Boundary 155

5.8.3 The Steady-State Heat Conduction in C-typed Anisotropic Solid Cylinder with Insulated Lateral Boundary 159

5.8.4 The Steady-State Heat Conduction in C-typed Anisotropic Solid Cylinder with Heat Exchanged with Surrounding Medium Lateral Boundary 160

5.8.5 The Steady-State Heat Conduction in C-typed Anisotropic Hollow Cylinder with Zero Temperature in Inner and Outer Lateral Boundaries 160

5.8.6 The Steady-State Heat Conduction in C-typed Anisotropic Hollow Cylinder with Insulated Inner and Outer Lateral Boundaries 161

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 Boundaries 162

5.9 Investigation into the Steady-State Heat Conduction in General Anisotropic Three Dimensional Cylinder——General Cylindrical 162

5.9.1 General Anisotropic Cylindrical Equation and General Anisotropic Cylindrical Polynomial 163

5.9.2 The Solution of Steady-State Heat Conduction in General Anisotropic Solid Cylinder with Zero Temperature in the Lateral Boundary 165

5.9.3 The Steady-State Heat Conduction in General Anisotropic Solid Cylinder with Insulated Lateral Boundary 166

5.9.4 The Steady-State Heat Conduction in General Anisotropic Solid Cylinder with Heat Exchanged with Surrounding Medium Lateral Boundary 166

5.9.5 The Steady-State Heat Conduction in General Anisotropic Hollow Cylinder with Zero Temperature in Inner and Outer Lateral Boundaries 167

5.9.6 The Steady-State Heat Conduction in General Anisotropic Hollow Cylinder with Insulated Inner and Outer Lateral Boundaries 167

5.9.7 The Steady-State Heat Conduction in General Anisotropic Hollow Cylinder with Heat Exchange with Medium in Inner and Outer Lateral Boundaries 168

5.10 The Relationship between the General Anisotropic Cylindrical Equation and the Confluent Hypergeometric Function 169

Chapter 6 Partial Differential Equation for Three Dimensional Time-dependent AnisotropicHeat Conduction in Cylindrical Coordinates——Complex Cylinder Function 170

6.1 Three Dimensional Governing Equation in Cylindrical Coordinates and its Analytical Solution 170

6.2 Theorem of Complex Cylinder Function Expansion 176

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 Expansion 177

6.3 Solving Procedure and Numerical Experiments 178

6.4 Investigation into Time-Dependent Heat Conduction in A-typed Anisotropic Cylinder with other Boundary Condition 178

6.4.1 The Solid A-typed Anisotropic Cylinder with the Insulated Lateral Boundary and the Upper,Bottom Boundaries of Zero Temperature 178

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 Temperature 179

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 Temperature 179

6.4.4 The A-typed Anisotropic Hollow Cylinder with Insulated Inner and Outer Lateral Boundaries and the Upper,Bottom Boundaries of Zero Temperature 180

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 Temperature 181

6.5 Determine the Eigenvalue λ 182

Chapter 7 Analytical Solution of Steady-state Conduction in Thin Curve-typed Anisotropic Circular Plate with Surface Heat Exchange——Modified complex spherical cylindrical polynomial 184

7.1 Governing Partial Differential Equation in Polar Coordinates 184

7.2 Modified Spherical Zip Equation,Modified Complex Spherical Cylindrical Polynomial and the Analytical Solution 185

7.2.1 Modified Spherical Zip equation,Modified Complex Spherical Cylindrical Polynomial and Function 185

7.2.2 The Analytical Solution 189

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

7.2.4 Numerical Experiments 193

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

7.4 Investigation into ？Cipl,n(x)polynomial 197

7.4.1 The Relationship Between ？Cipl,n(x)Polynomial and Modified Spherical Bessel Function 197

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 Formulas 202

Chapter 8 Analytical Solution of time-dependent Conduction in Thin Curve-typed Anisotropic Circular Plate——Complex Spherical Cylindrical Polynomial 206

8.1 Governing Partial Differential Equation in Polar Coordinates 206

8.2 Spherical Zip Differential Equation,Complex Spherical Cylindrical Polynomial and the Analytical Solution 208

8.2.1 Spherical Zip Differential Equation and Complex Spherical Cylindrical Polynomial 208

8.2.2 Time-Dependent Heat Conduction in Thin Solid Circular Plate with one Boundary Specialized Temperature 211

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

8.2.4 The Computational Procedure 216

8.2.5 Numerical results 216

8.3 The simplification of Complex Spherical-Cylindrical Function 220

8.4 Complex Spherical Cylindrical Expansion Theorem 221

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 Formulas 227

8.7 Complex Spherical Cylindrical Polynomial of the Third Kind and Corresponding Formulas 229

8.8 Investigation into Time-Dependent Heat Conduction in Anisotropic Circular Thin Plate with other Boundary Condition 231

8.8.1 Solid Anisotropic Circular Thin Plate with Insulated Circumference Boundary Condition 231

8.8.2 Solid Anisotropic Circular Plate with Heat Exchanged with Surrounding Medium Circumference Boundary Condition 232

8.8.3 Time-Dependent Heat Conduction in Anisotropic Thin Annular Plate and Complex Spherical Cylindrical-Annular Function Expansion over Annular Region Theorem 232

Chapter 9 Complex Cylindrical Surface Polynomial and Function in the Anisotropic Heat Conduction Partial Differential Equation 239

9.1 Introduction 239

9.2 The Solution for Steady-State Anisotropic Heat Conduction in Cylindrical Surface 240

9.2.1 General Complex Cylindrical Surface Function and the Analytical Solution 240

9.2.2 The Solving Procedure 242

9.3 The Solution of Time-Dependent Problem in Thin Cylindrical Shell 243

9.3.1 Governing Equation and Solving Procedure 243

9.3.2 Time-Dependent Heat Conduction in Finite Length Circular Shell with Two Boundary Specialized Temperature 246

9.4 Numerical Experiments 246

9.5 Complex Cylindrical Surface Function Expansion Theorem 248

Chapter 10 Parametric Complex Cylindrical Surface Polynomial and Function with application in Thin Anisotropic Cylindrical Shell Exchanging Heat with Surrounding Medium 251

10.1 Introduction 251

10.2 The Solution for Steady-State Anisotropic Heat Conduction in Cylindrical Surface Exchanging Heat with the Surrounding Medium 251

10.2.1 The Solving Procedure 252

10.2.2 The Analytical Solution 253

10.3 The Solution of Time-Dependent Heat Conduction in Thin Cylindrical Shell Exchanging Heat with the Surrounding Medium 255

10.3.1 Governing Equation and Solving Procedure 255

10.3.2 Time-Dependent Heat Conduction in Finite Length Cylindrical Shell with Two Boundary Specialized Temperature 258

10.4 Numerical Experiments 258

10.5 Parametric Complex Cylindrical Surface Function Expansion Theorem 260

Chapter 11 Heat Conduction in Thin Anisotropic Conical Shell 263

11.1 Introduction 263

11.2 Analytical Solution to Steady-State Heat Conduction in Thin Anisotropic Conical Shell 263

11.2.1 Governing Equation in Spherical Coordinates 263

11.2.2 The General Analytical Solution 264

11.2.3 Numerical Experiments 266

11.2.4 The Proof of the Simplification of Complex Series 267

11.3 The Analytical Solution to Steady-State Conduction in Thin Anisotropic Conical Shell with Surface Heat Exchange 268

11.3.1 Governing Equation in Spherical Coordinates 268

11.3.2 The Analytical Procedure 269

11.3.3 Numerical Experiments 271

11.4 The Analytical Solution to Time-Dependent Conduction in Thin Anisotropic Conical Shell 272

11.4.1 Governing Equation in Spherical Coordinates 272

11.4.2 Analytical Procedure 273

11.4.3 Numerical Results 276

Chapter 12 The Series of Complex Cylindrical Function Transforms 279

12.1 Complex Cylindrical Function Integral Transform 279

12.1.1 Basic Formulas 279

12.1.2 The Properties of Complex Cylindrical Function Integral Transform 280

12.1.3 Complex Cylindrical Function Integral Transform Table 281

12.2 The Finite Complex Cylindrical Function Integral Transform 282

12.2.1 Basic Formulas 282

12.2.2 The finite complex cylindrical function integral transform Table 282

12.3 Complex Spherical-Cylindrical Function Integral Transform 283

12.3.1 Basic Formulas 283

12.3.2 The Properties of Complex Spherical-Cylindrical Function Integral Transform 284

12.4 The finite complex spherical-cylindrical function integral transform 285

12.5 Modified Complex Cylindrical Function Integral Transform 286

12.5.1 Basic Formulas 286

12.5.2 The Properties of Modified Complex Cylindrical Function Integral Transform 287

12.6 The Finite Modified Complex Cylindrical Function Integral Transform 288

12.7 Modified Complex Spherical-Cylindrical Function Integral Transform 288

12.7.1 Basic Formulas 288

12.7.2 The Properties of Modified Complex Spherical-Cylindrical Function Integral Transform 290

12.8 The Finite Modified Complex Spherical-Cylindrical Function Integral Transform 290

12.9 Other Multi-Dimensional Complex Cylindrical Function Integral Transform 290

12.9.1 Two-Dimensional Complex Cylindrical Function Integral Transform 290

12.9.2 Two-Dimensional Complex Spherical-Cylindrical Function Integral Transform 291

12.9.3 Two-Dimensional Modified Complex Cylindrical Function Integral Transform 291

12.9.4 Two-Dimensional Modified Complex Spherical-Cylindrical Function Integral Transform 292

12.9.5 Three-dimensional complex cylindrical function integral transform 293

Chapter 13 Steady-State Anisotropic Heat Conduction in the Spherical Zone——Complex Spherical Polynomial and Function 294

13.1 Introduction 294

13.2 Governing Equation in Spherical Coordinates 294

13.3 Zis Differential Equation,Complex Spherical Function and the General Analytical Solution 295

13.3.1 Zis Differential Equation,Complex Spherical Polynomial and Function 295

13.3.2 The General Analytical Solution 298

13.4 Studies on the Steady-State Isotropic Heat Conduction in the Spherical Zone 299

13.5 Numerical Experiments 300

13.6 Summary of the Complex Spherical Function Expansion 304

13.7 The Recurrence Formulas 305

13.8 The Simplification of Complex Spherical Function 306

Chapter 14 Steady-State Anisotropic Heat Conduction in Spherical Zone Exchanging Heat with Surrounding Medium——Parametric Complex Spherical Polynomial and Complex Spherical Function 309

14.1 Introduction 309

14.2 Governing Equation in Spherical Coordinates 309

14.3 The Parameter Form of Zis Differential Equation,Parametric Complex Spherical Function and the Analytical Solution 310

14.3.1 The Parameter Form of Zis Differential Equation,Parametric Complex Spherical Polynomial and Function 310

14.3.2 The General Analytical Solution 314

14.4 Numerical Experiments 315

14.5 The Recurrence Relation 317

14.6 The Proof of the Real Principle for the Complex Spherical Function 318

Chapter 15 Partial Differential Equation for Steady-state Anisotropic Heat Conduction in the Spherical Surface——Associated Complex Spherical Polynomial Function 321

15.1 Introduction 321

15.2 Governing Equation in Spherical Coordinates 322

15.3 Associated Zis Differential Equation,Associated Complex Spherical Function and the Analytical Solution 323

15.3.1 Associated Zis Differential Equation,Associated Complex Spherical Polynomial and Function 323

15.3.2 The General Analytical Solution 326

15.4 Studies on the Solution of Isotropic Heat Conduction in Spherical Surface 327

15.5 Numerical Experiments 328

15.6 The Recurrence Relation 332

Chapter 16 Steady-State Anisotropic Heat Conduction in Spherical Surface Exchanging Heat with Surrounding Medium——Parametric Associated Complex Spherical Polynomial and Function 334

16.1 Introduction 334

16.2 Governing Equation in Spherical Coordinates 334

16.3 Associated Zis Differential Equation,Parametric Associated Complex Spherical Function and the Analytical Solution 336

16.3.1 Associated Zis differential equation,Parametric Associated Complex Spherical Polynomial and Function 336

16.3.2 The General Analytical Solution 339

16.4 Numerical Experiments 340

16.5 The Recurrence Relation 345

Chapter 17 Complex Spherical Zonal Function with Its Application In Partial Differential Equation for Time-dependent Anisotropic Heat Conduction in Spherical Zone 346

17.1 Introduction 346

17.2 Partial Differential Equation for Time-Dependent Anisotropic Heat Conduction in Spherical Zone 347

17.3 The Solving Procedure 348

17.4 The z-axis Symmetric Solution for Time-Dependent Heat Conduction in the Global Anisotropic Spherical Surface 352

17.5 The Complex Spherical Zonal Polynomial and Analytical Solution for Time-Dependent Heat Conduction 353

17.6 Complex Spherical Zonal Function Expansion Theorem 355

Chapter 18 Complex Spherical Zonal Function with Its Application In Partial Differential Equation for Steady-state Anisotropic Heat Conduction in Three-dimensional Sphere 359

18.1 Partial Differential Equation for Steady-State Anisotropic Heat Conduction in Three-dimensional Sphere 359

18.2 The Analytical Procedure 360

18.3 The z-axis Symmetric Solution in the Global Anisotropic Sphere 365

18.4 The Analytical Solution in the Anisotropic Sphere 366

Chapter 19 Complex Sphere Function in Partial Differential Equation for Time-dependent Anisotropic Heat Conduction in Three-dimensional Sphere 368

19.1 Governing Equation in Spherical Coordinates 368

19.2 The Analytical Procedure 369

19.3 The Product Solution for the Anisotropic Solid Sphere 375

19.4 Theorem of Complex Sphere Function Expansion 378

Chapter 20 Associated Complex Spherical Zonal Function with Its Application in Time-dependent Anisotropic Heat Conduction in Spherical Zone 381

20.1 Partial Differential Equation for Time-Dependent AnisotropieHeat Conduction in Spherical zone 381

20.2 The Solving Procedure 382

20.3 The Product Solution 386

20.4 Associated Complex Spherical Zonal Polynomial/function and the Analytical Solution 387

20.5 Associated Complex Zonal Spherical Function Expansion Theorem 390

Chapter 21 The Analytical Solution to Partial Differential Equation for Steady-state Anisotropic Heat Conduction in Three-dimensional Sphere 394

21.1 Partial Differential Equation for Steady-State Anisotropic Heat Conduction in Three-dimensional Sphere 394

21.2 The Solving Procedure 395

21.3 The Product Solution 400

Chapter 22 Partial Differential Equation for Time-dependent Anisotropic Heat Conduction in Three-dimensional Sphere Associated Complex Sphere Function 403

22.1 Partial Differential Equation for Time-dependent Anisotropic Heat Conduction in Three-dimensional Sphere 403

22.2 The Solving Procedure 405

22.3 The Analytical Solution 408

22.4 Theorem of Associated Complex Sphere Function Expansion 413

Chapter 23 Analytical Solution to the Anisotropic Wave Equations 416

23.1 The two dimensional Anisotropic Wave Equation in Cylinder in Cylindrical Coordinates 416

23.2 The three-dimensional Anisotropic Wave Equation in Cylindrical Coordinates 421

23.3 The two-dimensional Anisotropic Wave Equation in Spherical Membrane in Spherical Coordinates 426

23.4 The three-dimensional Anisotropic Wave Equation in Spherical Coordinates 430

23.5 The two-dimensional Anisotropic Wave Equation in Circular Membrane in Polar Coordinates 436

23.6 Analytical Solution to the Two-dimensional Anisotropic Wave Equation in Spherical Membrane by the Method of Associated Complex Spherical Zonal Function 440