Atlas for Computing Mathematical Functions: An Illustrated Guide for Practitioners With Programs in C and Mathematica: William J. Thompson: 9780471002604: Book: Technology: eCampus.com

Introducing a comprehensive book and CD-ROM for visualizing and computing special functions

Here is an invaluable reference and learning guide to more than 150 special functions of applied mathematics, statistics, physics, chemistry, computer science, and engineering. The Atlas contains over 700 graphics of the functions, which readers can also create by using the annotated Mathematica files provided. It offers complete, consistent instructions and test values for computing the functions accurately and efficiently with the full ANSI C source programs on the CD-ROM, which is both Windows and Macintosh compatible.

With extensive references and indexing, this integrated package is superbly designed and easy to use-ideal for anyone who works with special functions. Contents include:
* Elementary Transcendental Functions
* Exponential Integrals and Related Functions
* Gamma and Beta Functions
* Combinatorial Functions
* Number Theory Functions
* Probability Distributions
* Error Function, Fresnel, and Dawson Integrals
* Orthogonal Polynomials
* Legendre Functions
* Spheroidal Wave Functions
* Bessel Functions
* Struve, Anger, and Weber Functions
* Hypergeometric Functions and Coulomb Wave Functions
* Elliptic Integrals and Elliptic Functions
* Parabolic Cylinder Functions
* Miscellaneous Functions for Science and Engineering
* Mathematica Notebooks
* C Driver Programs

Preface

xiii

INTRODUCTION

(14)

The Atlas of Functions

(2)

What This Atlas Contains

(1)

How to Use the Atlas

(1)

About the Production of the Atlas

(1)

The Computer Interface

(12)

What the CD-ROM Contains

(1)

How to Locate a Function

(1)

Exploring Functions with Mathematica

(3)

The C Functions: No Assembly Required

(1)

Hints for Fortran and Pascal Programmers

(4)

File Names for PC-Based Systems

(1)

Reliability of Programs: Discliamer

(1)

References on the Computer Interface

(3)

PART I. THE FUNCTIONS

(578)

1 Introduction to the Functions

(2)

How the Function Descriptions Are Organized

(1)

2 A Visual Tour of the Atlas

(8)

3 Computing Strategies

(10)

3.1 General Computing Strategies

(2)

3.2 Iteration and Recursion

(3)

3.3 Continued Fractions and Rational Approximations

(1)

3.4 Using Asymptotic Expansions

(1)

3.5 Euler-Maclaurin Summation Formula

(1)

3.6 Accuracy and Precision of the Functions

(1)

3.7 Mathematical Constants Used in the Atlas

(1)

References on Computing Strategies

(1)

4 Elementary Transcendental Functions

(24)

4.1 Exponential and Logarithmic Functions

(5)

4.1.1 Exponentials

(2)

4.1.2 Logarithms

(2)

4.2 Circular and Inverse Circular Functions

(9)

4.2.1 Circular Functions

(4)

4.2.2 Inverse Circular Functions

(5)

4.3 Hyperbolic and Inverse Hyperbolic Functions

(9)

4.3.1 Hyperbolic Functions

(4)

4.3.2 Inverse Hyperbolic Functions

(5)

References on Elementary Transcendental Functions

(1)

5 Exponential Integrals and Related Functions

(20)

5.1 Exponential and Logarithmic Integrals

(13)

5.1.1 Exponential Integral of the First Kind

(5)

5.1.2 Exponential Integral of the Second Kind

(5)

5.1.3 Logarithmic Integral

(3)

5.2 Cosine and Sine Integrals

(6)

References on Exponential Integrals and Related Functions

(1)

6 Gamma and Beta Functions

(30)

6.1 Gamma Function and Beta Function

(7)

6.1.1 Gamma Function

(5)

6.1.2 Beta Function

(2)

6.2 Psi (Digamma) and Polygamma Functions

(11)

6.2.1 Psi Function

(4)

6.1.2 Polygamma Functions

(6)

6.3 Incomplete Gamma and Beta Functions

(9)

6.3.1 Incomplete Gamma Function

(5)

6.3.2 Incomplete Beta Function

102

(4)

References on Gamma and Beta Functions

106

(3)

7 Combinatorial Functions

109

(24)

7.1 Factorials and Rising Factorials

109

(6)

7.1.1 Factorial Function

110

(3)

7.1.2 Rising Factorial Functions

113

(2)

7.2 Binomial and Multinomial Coefficients

115

(6)

7.2.1 Binomial Coefficients

115

(3)

7.2.2 Multinomial Coefficients

118

(3)

7.3 Stirling Numbers of First and Second Kinds

121

(5)

7.3.1 Stirling Numbers of the First Kind

121

(3)

7.3.2 Stirling Numbers of the Second Kind

124

(2)

7.4 Fibonacci and Lucas Polynomials

126

(4)

7.4.1 Fibonacci Polynomials and Fibonacci Numbers

126

(2)

7.4.2 Lucas Polynomials and Lucas Numbers

128

(2)

References on Combinatorial Functions

130

(3)

8 Number Theory Functions

133

(26)

8.1 Bernoulli Numbers and Bernoulli Polynomials

133

(6)

8.1.1 Bernoulli Numbers

133

(4)

8.1.2 Bernoulli Polynomials

137

(2)

8.2 Euler Numbers and Euler Polynomials

139

(5)

8.2.1 Euler Numbers

139

(3)

8.2.2 Euler Polynomials

142

(2)

8.3 Reimann Zeta Function

144

(3)

8.4 Other Sums of Reciprocal Powers

147

(4)

8.5 Polygarithms

151

(5)

References on Number Theory Functions

156

(3)

9 Proability Distributions

159

(64)

9.1 Overview of Probability Distribution Functions

160

(1)

9.2 Discrete Probability Distributions

160

(15)

9.2.1 Binomial Distribution

162

(2)

9.2.2 Negative Binomial (Pascal) Distribution

164

(2)

9.2.3 Geometric Distribution

166

(2)

9.2.4 Hypergeometric Distribution

168

(3)

9.2.5 Logarithmic Series Distribution

171

(2)

9.2.6 Poisson Distribution

173

(2)

9.3 Normal Probablity Distribution

175

(24)

9.3.1 Gauss (Normal) Probability Function

177

(2)

9.3.2 Bivariate Normal Probability Function

179

(3)

9.3.3 Chi-Square Probability Functions

182

(6)

9.3.4 F-(Variances-Ratio) Distribution Functions

188

(4)

9.3.5 Student's t-Distribution Functions

192

(4)

9.3.6 Lognormal Distribution

196

(3)

9.4 Other Continuous Probability Distributions

199

(22)

9.4.1 Cauchy (Lorentz) Distribution

200

(3)

9.4.2 Exponential Distribution

203

(2)

9.4.3 Pareto Distribution

205

(3)

9.4.4 Weibull Distribution

208

(3)

9.4.5 Logistic Distribution

211

(2)

9.4.6 Laplace Distribution

213

(2)

9.4.7 Kolmogorov-Smirnov Distribution

215

(3)

9.4.8 Beta Distribution

218

(3)

References on Probability Distribution Functions

221

(2)

10 Error Function, Fresnel and Dawson Integrals

223

(16)

10.1 Error Function

223

(3)

10.2 Fresnel Integrals

226

(8)

10.3 Dawson Integrals

234

(4)

References on Error Functions, Fresnel and Dawson Integrals

238

(1)

11 Orthogonal Polynomials

239

(30)

11.1 Overview of Orthogonal Polynomials

239

(5)

11.2 Chebyshev Polynomials

244

(7)

11.2.1 Chebyshev Polynomials of the First Kind

245

(2)

11.2.2 Chebyshev Polynomials of the Second Kind

247

(4)

11.3 Gegenbauer (Ultraspherical) Polynomials

251

(3)

11.4 Hermite Polynomials

254

(3)

11.5 Laguerre Polynomials

257

(3)

11.6 Legendre Polynomials

260

(3)

11.7 Jacobi Polynomials

263

(4)

References on Orthogonal Polynomials

267

(2)

12 Legendre Functions

269

(28)

12.1 Overview of Legendre Functions

269

(9)

12.1.1 Visualizing Legendre Functions of the First Kind

270

(3)

12.1.2 Visualizing Legendre Functions of the Second Kind

273

(4)

12.1.3 Legendre Functions and Coordinate Systems

277

(1)

12.2 Spherical Legendre Functions

278

(3)

12.2.1 Spherical Polar Coordinates

278

(1)

12.2.2 Legendre Functions of the First Kind for Integer m and n

279

(5)

12.2.3 Legendre Functions of the Second Kind for Integer m and n

284

(7)

12.3 Toroidal Legendre Functions

291

(9)

12.3.1 Toroidal Coordinates

291

(2)

12.3.2 Toroidal Functions of the First Kind

293

(3)

12.3.3 Toroidal Functions of the Second Kind

296

(4)

12.4 Conical Legendre Functions

300

(4)

12.4.1 Laplace Equation on a Cone

300

(1)

12.4.2 Conical Functions

301

(3)

References on Legendre Functions

304

(3)

13 Spheriodal Wave Functions

307

(38)

13.1 Overview of Spheroidal Wave Functions

307

(14)

13.1.1 Spheroidal Coordinates

308

(1)

13.1.2 Scalar Wave Equation in Spheroidal Coordinates

309

(1)

13.1.3 Eigenvalues for Spheroidal Equations

310

(10)

13.1.4 Auxiliary Functions for Eigenvalues

320

(1)

13.2 Spheroidal Angular Functions

321

(15)

13.2.1 Expansion Coefficients for Angular Functions

321

(10)

13.2.2 Spheroidal Angular Functions

331

(5)

13.3 Spheroidal Radial Functions

336

(8)

13.3.1 Expansion Coefficients for Radial Functions

336

(4)

13.3.2 Spheroidal Radial Functions

340

(4)

References on Spheroidal Wave Functions

344

(1)

14 Bessel Functions

345

(90)

14.1 Overview of Bessel Functions

345

(5)

14.2 Bessel Functions of Integer Order

350

(25)

14.2.1 Regular Cylindrical Bessel Function

351

(6)

14.2.2 Irregular Cylindrical Bessel Function

357

(4)

14.2.3 Regular Hyperbolic Bessel Function

361

(7)

14.2.4 Irregular Hyperbolic Bessel Function

368

(7)

14.3 Kelvin Functions

375

(19)

14.3.1 Regular Kelvin Functions

375

(10)

14.3.2 Irregular Kelvin Functions

385

(9)

14.4 Bessel Functions of Half-Integer Order

394

(22)

14.4.1 Regular Spherical Bessel Function

394

(7)

14.4.2 Irregular Spherical Bessel Function

401

(4)

14.4.3 Regular Modified Spherical Bessel Function

405

(7)

14.4.4 Irregular Modified Spherical Bessel Function

412

(4)

14.5 Airy Functions

416

(18)

14.5.1 Airy Functions

416

(9)

14.5.2 Derivatives of Airy Functions

425

(9)

References on Bessel Functions

434

(1)

15 Struve, Anger, and Weber Functions

435

(26)

15.1 Struve Functions

435

(12)

15.1.1 Struve Function

435

(7)

15.1.1 Modified Struve Function

442

(6)

15.2 Anger and Weber Functions

448

(10)

15.2.1 Overview of Anger and Weber Functions

448

(1)

15.2.2 Anger Function

449

(6)

Weber Function

455

(3)

References on Struve, Anger, and Weber Functions

458

(3)

16 Hypergeometric Functions and Coulomb Wave Functions

461

(34)

16.1 Hypergeometric Functions

461

(4)

16.2 Confluent Hypergeometric Functions

465

(13)

16.2.1 Regular Function

465

(6)

16.2.2 Irregular Function

471

(7)

16.3 Coulomb Wave Functions

478

(15)

16.3.1 Regular Functions and Derivatives

478

(9)

16.3.2 Irregular Functions and Derivatives

487

(6)

References on Hypergeometric Functions and Coulomb Wave Functions

493

(2)

17 Elliptic Integrals and Elliptic Functions

495

(44)

17.1 Overview of Elliptic Integrals and Elliptic Functions

495

(1)

17.2 Elliptic Integrals

496

(21)

17.2.1 Elliptic Integrals of the First Kind

496

(6)

17.2.2 Elliptic Integrals of the Second Kind

502

(4)

17.2.3 Jacobi Zeta Function

506

(4)

17.2.4 Heuman Lambda Function

510

(3)

17.2.5 Elliptic Integrals of the Third Kind

513

(4)

17.3 Jacobi Elliptic Functions and Theta Functions

517

(19)

17.3.1 Jacobi Elliptic Functions

517

(8)

17.3.2 Theta functions

525

(6)

17.3.3 Logarithmic Derivatives of Theta Functions

531

(5)

References on Elliptic Inte`grals and Elliptic Functions

536

(3)

18 Parabolic Cylinder Functions

539

(12)

18.1 Parabolic Cylinder Coordinates

539

(1)

18.2 Parabolic Cylinder Functions

540

(10)

18.2.1 Parabolic Cylinder Function U

540

(6)

18.2.2 Parabolic Cylinder Functions V

546

(4)

References on Parabolic Cylinder Functions

550

(1)

19 Miscellaneous Functions for Science and Engineering

551

(42)

19.1 Debye Functions

551

(3)

19.2 Sievert Integral

554

(3)

19.3 Abromowitz Function

557

(5)

19.4 Spence Integral

562

(3)

19.5 Clausen Integral

565

(5)

19.6 Voigt (Plasma Dispersion) Function

570

(6)

19.7 Angular Momentum Coupling Coefficients

576

(13)

19.7.1 3-j Coefficients

578

(4)

19.7.2 6-j Coefficients

582

(4)

19.7.3 9-j Coefficients

586

(3)

References on Miscellaneous Functions for Science and Engineering

589

(4)

PART II. THE COMPUTER INTERFACE

593

(286)

20 The Mathematica Notebooks

593

(204)

20.1 Introduction to the Notebooks

593

(1)

20.2 Exploring with the Notebook Cells

594

(1)

20.3 The Annotated Notebooks

594

(1)

20.4 Elementary Transcendental Functions

595

(7)

20.5 Exponential Integrals and Related Functions

602

(5)

20.6 Gamma and Beta Functions

607

(11)

20.7 Combinatorial Functions

618

(9)

20.8 Number Theory Functions

627

(6)

20.9 Probability Distributions

633

(30)

Error Function, Fresnel and Dawson Integrals

663

(4)

20.11 Orthogonal Polynomials

667

(9)

20.12 Legendre Functions

676

(16)

20.13 Spheroidal Wave Functions

692

(16)

20.14 Bessel Functions

708

(39)

20.15 Struve, Anger, and Weber Functions

747

(9)

20.16 Hypergeometric Functions and Coulomb Wave Functions

756

(9)

20.17 Elliptic Integrals and Elliptic Functions

765

(17)

20.18 Parabolic Cylinder Functions

782

(6)

20.19 Miscellaneous Functions for Science and Engineering

788

(9)

21 The C Driver Programs

797

(82)

21.1 Introduction to the C Driver Programs

797

(1)

21.2 How the C Drivers Are Organized

797

(1)

21.3 Annotations to the C Driver Programs

798

(1)

21.4 Elementary Transcendental Functions

798

(4)

21.4.1 Exponential and Logarithmic Functions

798

(1)

21.4.2 Circular and Inverse Circular Functions

799

(1)

21.4.3 Hyperbolic and Inverse Hyperbolic Functions

800

(2)

21.5 Exponential Integrals and Related Function

802

(2)

21.5.1 Exponential and Logarithmic Integrals

802

(1)

21.5.2 Cosine and Sine Integrals

803

(1)

21.6 Gamma and Beta Functions

804

(4)

21.6.1 Gamma Function and Beta Function

804

(1)

21.6.2 Psi (Digamma) and Poygamma Functions

805

(2)

21.6.3. Incomplete Gamme and Beta Functions

807

(1)

21.7 Combinatorial Functions

808

(5)

21.7.1 Factorials and Rising Factorials

808

(1)

21.7.2 Binomial and Multinomial Coefficents

809

(2)

21.7.3 Stirling Numbers of the First and Second Kinds

811

(1)

21.7.4 Fibonacci and Lucas Polynomials

811

(2)

21.8 Number Theory Functions

813

(3)

21.8.1 Bernoulli Numbers and Bernoulli Polynomials

813

(1)

21.8.2 Euler Numbers and Euler Polynomials

814

(1)

21.8.3 Riemann Zeta Function

814

(1)

21.8.4 Other Sums of Reciprocal Powers

815

(1)

21.8.5 Polylogarithms

816

(1)

21.9 Probability Distributions

816

(14)

21.9.1 Organization of the PDFs

816

(1)

21.9.2 Discrete Probability Distributions

816

(4)

21.9.3 Normal Probability Distributions

820

(5)

21.9.4 Other Continuous Probability Distributions

825

(5)

21.10 Error Function, Fresnel and Dawson Integrals

830

(3)

21.10.1 Error Functions

830

(1)

21.10.2 Fresnel Integrals

831

(1)

21.10.3 Dawson Integral

832

(1)

21.11 Orthogonal Polynomials

833

(4)

21.11.1 Orthogonal Polynomial Functions

833

(1)

21.11.2 Chebyshev Polynomials

833

(1)

21.11.3 Gegenbauer (Ultraspherical) Polynomials

834

(1)

21.11.4 Hermite Polynomials

835

(1)

21.11.5 Laguerre Polynomials

835

(1)

21.11.6 Legendre Polynomials

836

(1)

21.11.7 Jacobi Polynomials

837

(1)

21.12 Legendre Functions

837

(5)

21.12.1 Overview of Legendre Functions

838

(1)

21.12.2 Spherical Legendre Functions

838

(2)

21.12.3 Toroidal Legendre Functions

840

(1)

21.12.4 Conical Legendre Functions

841

(1)

21.13 Spheroidal Wave Functions

842

(4)

21.13.1 Overview of Spheroidal Wave Functions

842

(1)

21.13.2 Spheroidal Angular Functions

843

(1)

21.13.3 Spheroidal Radial Functions

844

(2)

21.14 Bessel Functions

846

(11)

21.14.1 Overview of Bessel Functions

846

(1)

21.14.2 Bessel Functions of Integer Order

846

(4)

21.14.3 Kelvin Functions

850

(2)

21.14.4 Bessel Functions of Half-Integer Order

852

(4)

21.14.5 Airy Functions

856

(1)

21.15 Struve, Anger, and Wave Functions

857

(4)

21.15.1 Struve Functions

857

(2)

21.15.2 Anger and Weber Functions

859

(2)

21.16 Hypergeometric Functions and Coulomb Wave Functions

861

(3)

21.16.1 Hypergeometric Functions

861

(1)

21.16.2 Confluent Hypergeometric Functions

862

(1)

21.16.3 Coulomb Wave Functions

863

(1)

21.17 Elliptic Integrals and Elliptic Functions

864

(6)

21.17.1 Overview of Elliptic Integrals and Elliptic Functions

864

(1)

21.17.2 Elliptic Integrals

864

(3)

21.17.3 Jacobi Elliptic Functions and Theta Functions

867

(3)

21.18 Parabolic Cylinder Functions

870

(2)

21.18.1 Parabolic Cylinder Functions

870

(2)

21.19 Miscellaneous Functions for Science and Engineering

872

(7)

21.19.1 Debye Functions

872

(1)

21.19.2 Sievert Integral

873

(1)

21.19.3 Abramowitz Function

873

(1)

21.19.4 Spence Integral

874

(1)

21.19.5 Clausen Integral

875

(1)

21.19.6 Voigt (Plasma Dispersion) Funtion

876

(1)

21.19.7 Angular Momentum Coupling Coefficients

876

(3)

APPENDIX: File Names for PC-Based Systems

879

(4)

INDEXES

883

Index of Function Notations

883

(3)

Index of Programs and Dependencies

886

(3)

Index of Subjects and Authors

889

WILLIAM J. THOMPSON, PhD, DSc, is Professor of Physics at the University of North Carolina, Chapel Hill. The author of three previous Wiley titles, Computing in Applied Science, Computing for Scientists and Engineers, and Angular Momentum, he has published more than 120 papers in theoretical physics, applied mathematics, and statistics.


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