Now available as an eBook from:

Preface to the Second Edition xiii

Preface to the First Edition xv

Notation for Special Functions xvii

Chapter 1. Infinite Series, Improper Integrals, and Infinite Products 1

1.1 Introduction 1

1.2 Infinite Series of Constants 2

1.2.1 The Geometric Series 4

1.2.2 Summary of Convergence Tests 6

1.2.3 Operations with Series 11

1.2.4 Factorials and Binomial Coefficients 15

1.3 Infinite Series of Functions 21

1.3.1 Properties of Uniformly Convergent Series 23

1.3.2 Power Series 25

1.3.3 Sums and Products of Power Series 29

1.4 Fourier Trigonometric Series 33

1.4.1 Cosine and Sine Series 36

1.5 Improper Integrals 39

1.5.1 Types of Improper Integrals 39

1.5.2 Convergence Tests 42

1.5.3 Pointwise and Uniform Convergence 43

1.6 Asymptotic Formulas 47

1.6.1 Small Arguments 48

1.6.2 Large Arguments 50

1.7 Infinite Products 55

1.7.1 Associated Infinite Series 56

1.7.2 Products of Functions 57

Chapter 2. The Gamma Function and Related Functions 61

2.1 Introduction 61

2.2 Gamma Function 62

2.2.1 Integral Representations 64

2.2.2 Legendre Duplication Formula 70

2.2.3 Weierstrass' Infinite Product 71

2.3 Applications 77

2.3.1 Miscellaneous Problems 77

2.3.2 Fractional-Order Derivatives 79

2.4 Beta Function 82

2.5 Incomplete Gamma Function 87

2.5.1 Asymptotic Series 88

2.6 Digamma and Polygamma Functions 90

2.6.1 Integral Representations 93

2.6.2 Asymptotic Series 95

2.6.3 Polygamma Functions 100

2.6.4 Riemann Zeta Function 102

Chapter 3. Other Functions Defined by Integrals 109

3.1 Introduction 109

3.2 Error Function and Related Functions 110

3.2.1 Asymptotic Series 112

3.2.2 Fresnel Integrals 113

3.3 Applications 118

3.3.1 Probability and Statistics 118

3.3.2 Heat Conduction in Solids 119

3.3.3 Vibrating Beams 122

3.4 Exponential Integral and Related Functions 126

3.4.1 Logarithmic Integral 128

3.4.2 Sine and Cosine Integrals 129

3.5 Elliptic Integrals 133

3.5.1 Limiting Values and Series Representations 134

3.5.2 The Pendulum Problem 135

Chapter 4. Legendre Polynomials and Related Functions 141

4.1 Introduction 141

4.2 Legendre Polynomials 142

4.2.1 The Generating Function 142

4.2.2 Special Values and Recurrence Formulas 146

4.2.3 Legendre's Differential Equation 151

4.3 Other Representations of the Legendre Polynomials 157

4.3.1 Rodrigues' Formula 157

4.3.2 Laplace Integral Formula 158

4.3.3 Some Bounds on Pn(x) 159

4.4 Legendre Series 162

4.4.1 Orthogonality of the Polynomials 162

4.4.2 Finite Legendre Series 165

4.4.3 Infinite Legendre Series 167

4.5 Convergence of the Series 173

4.5.1 Piecewise Continuous and Piecewise Smooth Functions 174

4.5.2 Pointwise Convergence 175

4.6 Legendre Functions of the Second Kind 181

4.6.1 Basic Properties 184

4.7 Associated Legendre Functions 186

4.7.1 Basic Properties of P_n^m`(x) 189

4.8 Applications 192

4.8.1 Electric Potential due to a Sphere 193

4.8.2 Steady-State Temperatures in a Sphere 197

Chapter 5. Other Orthogonal Polynomials 203

5.1 Introduction 203

5.2 Hermite Polynomials 204

5.2.1 Recurrence Formulas 206

5.2.2 Hermite Series 207

5.2.3 Simple Harmonic Oscillator 209

5.3 Laguerre Polynomials 214

5.3.1 Recurrence Formulas 215

5.3.2 Laguerre Series 217

5.3.3 Associated Laguerre Polynomials 218

5.3.4 The Hydrogen Atom 221

5.4 Generalized Polynomial Sets 226

5.4.1 Gegenbauer Polynomials 226

5.4.2 Chebyshev Polynomials 228

5.4.3 Jacobi Polynomials 231

Chapter 6. Bessel Functions 237

6.1 Introduction 237

6.2 Bessel Functions of the First Kind 238

6.2.1 The Generating Function 238

6.2.2 Bessel Functions of the Nonintegral Order 240

6.2.3 Recurrence Formulas 242

6.2.4 Bessel's Differential Equation 243

6.3 Integral Representations 248

6.3.1 Bessel's Problem 250

6.3.2 Geometric Problems 253

6.4 Integrals of Bessel Functions 256

6.4.1 Indefinite Integrals 256

6.4.2 Definite Integrals 258

6.5 Series Involving Bessel Functions 265

6.5.1 Addition Formulas 265

6.5.2 Orthogonality of Bessel Functions 267

6.5.3 Fourier-Bessel Series 269

6.6 Bessel Functions of the Second Kind 273

6.6.1 Series Expansion for Yn(x) 274

6.6.2 Asymptotic Formulas for Small Arguments 277

6.6.3 Recurrence Formulas 278

6.7 Differential Equations Related to Bessel's Equation 280

6.7.1 The Oscillating Chain 282

Chapter 7. Bessel Functions of Other Kinds 287

7.1 Introduction 287

7.2 Modified Bessel Functions 287

7.2.1 Modified Bessel Functions of the Second Kind 290

7.2.2 Recurrence Formulas 291

7.2.3 Generating Function and Addition Theorems 292

7.3 Integral Relations 298

7.3.1 Integral Representations 298

7.3.2 Integrals of Modified Bessel Functions 299

7.4 Spherical Bessel Functions 302

7.4.1 Recurrence Formulas 305

7.4.2 Modified Spherical Bessel Functions 305

7.5 Other Bessel Functions 308

7.5.1 Hankel Functions 308

7.5.2 Struve Functions 309

7.5.3 Kelvin's Functions 311

7.5.4 Airy Functions 312

7.6 Asymptotic Formulas 316

7.6.1 Small Arguments 316

7.6.2 Large Arguments 317

Chapter 8. Applications Involving Bessel Functions 323

8.1 Introduction 323

8.2 Problems in Mechanics 323

8.2.1 The Lengthening Pendulum 323

8.2.2 Buckling of a Long Column 327

8.3 Statistical Communication Theory 332

8.3.1 Narrowband Noise and Envelope Detection 333

8.3.2 Non-Rayleigh Radar Sea Clutter 336

8.4 Heat Conduction and Vibration Phenomena 339

8.4.1 Radial Symmetric Problems Involving Circles 340

8.4.2 Radial Symmetric Problems Involving Cylinders 343

8.4.3 The Helmholtz Equation 345

8.5 Step-Index Optical Fibers 351

Chapter 9. The Hypergeometric Function 357

9.1 Introduction 357

9.2 The Pochhammer Symbol 358

9.3 The Function F(a,b;c;x) 361

9.3.1 Elementary Properties 362

9.3.2 Integral Representation 364

9.3.3 The Hypergeometric Equation 365

9.4 Relation to Other Functions 370

9.4.1 Legendre Functions 373

9.5 Summing Series and Evaluating Integrals 377

9.5.1 Action-Angle Variables 380

Chapter 10. The Confluent Hypergeometric Functions 385

10.1 Introduction 385

10.2 The Functions M(a;c;x) and U(a;c;x) 386

10.2.1 Elementary Properties of M(a;c;x) 386

10.2.2 Confluent Hypergeometric Equation and U(a;c;x) 388

10.2.3 Asymptotic Formulas 390

10.3 Relation to Other Functions 395

10.3.1 Hermite Functions 397

10.3.2 Laguerre Functions 399

10.4 Whittaker Functions 403

Chapter 11. Generalized Hypergeometric Functions 411

11.1 Introduction 411

11.2 The Set of Functions pFq 412

11.2.1 Hypergeometric-Type Series 413

11.3 Other Generalizations 419

11.3.1 The Meijer G Function 419

11.3.2 The MacRobert E Function 425

Chapter 12. Applications Involving Hypergeometric-Type Functions 429

12.1 Introduction 429

12.2 Statistical Communication Theory 429

12.2.1 Nonlinear Devices 431

12.3 Fluid Mechanics 437

12.3.1 Unsteady Hydrodynamic Flow Past an Infinite Plate 437

12.3.2 Transonic Flow and the Euler-Tricomi Equation 440

12.4 Random Fields 444

12.4.1 Structure Function of Temperature 445

Bibliography 451

Appendix: A List of Special Function Formulas 453

Selected Answers to Exercises 469

Index 473

Publishers' note: This new softcover printing of the Second Edition of Special Functions of Mathematics for Engineers, originally published by McGraw-Hill in 1992, includes known corrections to the text and formulas. Because of the importance of this material in modern engineering, SPIE The International Society for Optical Engineering and Oxford University Press are republishing it to make it available to the engineering, science, and mathematics communities.

Modern engineering and physical science applications demand a more thorough knowledge of applied mathematics particularly special functions than ever before. These functions typically arise in applications such as communication systems, electro-optics, nonlinear wave propagation, electromagnetic theory, electric circuit theory, and quantum mechanics, among others. This book systematically introduces important special functions and explores their properties and applications in engineering and science.

The book is suitable as a classroom textbook in courses dealing with higher mathematical functions or as a reference text for practicing engineers and scientists. The second edition includes numerous applications drawn from a variety of fields, including fiber optics, statistical communication theory, vibration phenomena, and fluid mechanics. Whenever possible, related applications are discussed in the chapter introducing the special function. The volume includes a brief review of calculus concepts, such as infinite series and improper integrals, because of their close association with special functions. Each chapter includes exercises to facilitate learning.

Larry C. Andrews is a professor of mathematics at the University of Central Florida and a member of the Department of Electrical and Computer Engineering. Dr. Andrews is also an associate member of the Center for Research and Education in Optics and Lasers (CREOL). Along with special functions, his research interests include laser beam propagation through random media, detection theory, and signal processing.