Atlas for Computing Mathematical Functions

by Thompson, William J.

• ISBN13: 9780471181712
• ISBN10: 0471181714
• Edition: CD
• Format: Hardcover
• Copyright: 1997-06-01
• Publisher: Wiley-Interscience

Summary

An invaluable reference and learning guide to more than 150 special functions of applied mathematics, statistics, physics, chemistry, computer science, and engineering. This book 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 Fortran-90 source programs on the CD-ROM, which is both Windows and Macintosh compatible.

Table of Contents

Preface	xiii
INTRODUCTION	1	(12)
The Atlas of Functions	1	(2)
What This Atlas Contains	1	(1)
How to Use the Atlas	2	(1)
About the Production of the Atlas	2	(1)
The Computer Interface	3	(10)
What the CD-ROM Contains	3	(1)
How to Locate a Function	3	(1)
Exploring Functions with Mathematica	4	(3)
The Fortran Functions: No Assembly Required	7	(1)
The Fortran-90 Standard	8	(1)
File Names for PC-Based Systems	9	(1)
Reliability of Programs: Disclaimer	9	(1)
References on the Computer Interface	9	(4)
PART I. THE FUNCTIONS	13	(578)
1 Introduction to the Functions	13	(2)
How the Function Descriptions Are Organized	14	(1)
2 A Visual Tour of the Atlas	15	(8)
3 Computing Strategies	23	(10)
3.1 General Computing Strategies	23	(2)
3.2 Iteration and Recursion	25	(3)
3.3 Continued Fractions and Rational Approximations	28	(1)
3.4 Using Asymptotic Expansions	29	(1)
3.5 Euler-Maclaurin Summation Formula	30	(1)
3.6 Accuracy and Precision of the Functions	31	(1)
3.7 Mathematical Constants Used in the Atlas	32	(1)
References on Computing Strategies	32	(1)
4 Elementary Transcendental Functions	33	(24)
4.1 Exponential and Logarithmic Functions	33	(5)
4.1.1 Exponentials	34	(2)
4.1.2 Logarithms	36	(2)
4.2 Circular and Inverse Circular Functions	38	(9)
4.2.1 Circular Functions	38	(4)
4.2.2 Inverse Circular Functions	42	(5)
4.3 Hyperbolic and Inverse Hyperbolic Functions	47	(9)
4.3.1 Hyperbolic Functions	47	(4)
4.3.2 Inverse Hyperbolic Functions	51	(5)
References on Elementary Transcendental Functions	56	(1)
5 Exponential Integrals and Related Functions	57	(20)
5.1 Exponential and Logarithmic Integrals	57	(13)
5.1.1 Exponential Integral of the First Kind	57	(5)
5.1.2 Exponential Integral of the Second Kind	62	(5)
5.1.3 Logarithmic Integral	67	(3)
5.2 Cosine and Sine Integrals	70	(6)
References on Exponential Integrals and Related Functions	76	(1)
6 Gamma and Beta Functions	77	(30)
6.1 Gamma Function and Beta Function	77	(7)
6.1.1 Gamma Function	77	(5)
6.1.2 Beta Function	82	(2)
6.2 Psi (Digamma) and Polygamma Functions	84	(11)
6.2.1 Psi Function	85	(4)
6.2.2 Polygamma Functions	89	(6)
6.3 Incomplete Gamma and Beta Functions	95	(9)
6.3.1 Incomplete Gamma Function	96	(4)
6.3.2 Incomplete Beta Function	100	(4)
References on Gamma and Beta Functions	104	(3)
7 Combinatorial Functions	107	(24)
7.1 Factorials and Rising Factorials	107	(6)
7.1.1 Factorial Function	108	(3)
7.1.2 Rising Factorial Function	111	(2)
7.2 Binomial and Multinomial Coefficients	113	(6)
7.2.1 Binomial Coefficients	113	(3)
7.2.2 Multinomial Coefficients	116	(3)
7.3 Stirling Numbers of First and Second Kinds	119	(5)
7.3.1 Stirling Numbers of the First Kind	119	(3)
7.3.2 Stirling Numbers of the Second Kind	122	(2)
7.4 Fibonacci and Lucas Polynomials	124	(5)
7.4.1 Fibonacci Polynomials and Fibonacci Numbers	124	(3)
7.4.2 Lucas Polynomials and Lucas Numbers	127	(2)
References on Combinatorial Functions	129	(2)
8 Number Theory Functions	131	(26)
8.1 Bernoulli Numbers and Bernoulli Polynomials	131	(6)
8.1.1 Bernoulli Numbers	132	(3)
8.1.2 Bernoulli Polynomials	135	(2)
8.2 Euler Numbers and Euler Polynomials	137	(5)
8.2.1 Euler Numbers	137	(3)
8.2.2 Euler Polynomials	140	(2)
8.3 Riemann Zeta Function	142	(3)
8.4 Other Sums of Reciprocal Powers	145	(4)
8.5 Polylogarithms	149	(6)
References on Number Theory Functions	155	(2)
9 Probability Distributions	157	(64)
9.1 Overview of Probability Distribution Functions	158	(1)
9.2 Discrete Probability Distributions	158	(15)
9.2.1 Binomial Distribution	160	(2)
9.2.2 Negative Binomial (Pascal) Distribution	162	(2)
9.2.3 Geometric Distribution	164	(2)
9.2.4 Hypergeometric Distribution	166	(3)
9.2.5 Logarithmic Series Distribution	169	(2)
9.2.6 Poisson Distribution	171	(2)
9.3 Normal Probability Distributions	173	(24)
9.3.1 Gauss (Normal) Probability Function	175	(2)
9.3.2 Bivariate Normal Probability Function	177	(3)
9.3.3 Chi-Square Probability Functions	180	(6)
9.3.4 F-(Variance-Ratio) Distribution Functions	186	(4)
9.3.5 Student's t-Distribution Functions	190	(4)
9.3.6 Lognormal Distribution	194	(3)
9.4 Other Continuous Probability Distributions	197	(22)
9.4.1 Cauchy (Lorentz) Distribution	198	(3)
9.4.2 Exponential Distribution	201	(2)
9.4.3 Pareto Distribution	203	(3)
9.4.4 Weibull Distribution	206	(3)
9.4.5 Logistic Distribution	209	(2)
9.4.6 Laplace Distribution	211	(2)
9.4.7 Kolmogorov-Smirnov Distribution	213	(3)
9.4.8 Beta Distribution	216	(3)
References on Probability Distribution Functions	219	(2)
10 Error Function, Fresnel and Dawson Integrals	221	(16)
10.1 Error Function	221	(3)
10.2 Fresnel Integrals	224	(8)
10.3 Dawson Integral	232	(4)
References on Error Functions, Fresnel and Dawson Integrals	236	(1)
11 Orthogonal Polynomials	237	(30)
11.1 Overview of Orthogonal Polynomials	237	(5)
11.2 Chebyshev Polynomials	242	(7)
11.2.1 Chebyshev Polynomials of the First Kind	243	(2)
11.2.2 Chebyshev Polynomials of the Second Kind	245	(4)
11.3 Gegenbauer (Ultraspherical) Polynomials	249	(4)
11.4 Hermite Polynomials	253	(2)
11.5 Laguerre Polynomials	255	(3)
11.6 Legendre Polynomials	258	(3)
11.7 Jacobi Polynomials	261	(4)
References on Orthogonal Polynomials	265	(2)
12 Legendre Functions	267	(38)
12.1 Overview of Legendre Functions	267	(9)
12.1.1 Visualizing Legendre Functions of the First Kind	268	(3)
12.1.2 Visualizing Legendre Functions of the Second Kind	271	(4)
12.1.3 Legendre Functions and Coordinate Systems	275	(1)
12.2 Spherical Legendre Functions	276	(13)
12.2.1 Spherical Polar Coordinates	276	(1)
12.2.2 Legendre Functions of the First Kind for Integer m and n	277	(5)
12.2.3 Legendre Functions of the Second Kind for Integer m and n	282	(7)
12.3 Toroidal Legendre Functions	289	(9)
12.3.1 Toroidal Coordinates	289	(2)
12.3.2 Toroidal Functions of the First Kind	291	(3)
12.3.3 Toroidal Functions of the Second Kind	294	(4)
12.4 Conical Legendre Functions	298	(4)
12.4.1 Laplace Equation on a Cone	298	(1)
12.4.2 Conical Functions	299	(3)
References on Legendre Functions	302	(3)
13 Spheroidal Wave Functions	305	(38)
13.1 Overview of Spheroidal Wave Functions	305	(14)
13.1.1 Spheroidal Coordinates	306	(1)
13.1.2 Scalar Wave Equation in Spheroidal Coordinates	307	(1)
13.1.3 Eigenvalues for Spheroidal Equations	308	(10)
13.1.4 Auxiliary Functions for Eigenvalues	318	(1)
13.2 Spheroidal Angular Functions	319	(15)
13.2.1 Expansion Coefficients for Angular Functions	319	(10)
13.2.2 Spheroidal Angular Functions	329	(5)
13.3 Spheroidal Radial Functions	334	(8)
13.3.1 Expansion Coefficients for Radial Functions	334	(4)
13.3.2 Spheroidal Radial Functions	338	(4)
References on Spheroidal Wave Functions	342	(1)
14 Bessel Functions	343	(90)
14.1 Overview of Bessel Functions	343	(5)
14.2 Bessel Functions of Integer Order	348	(25)
14.2.1 Regular Cylindrical Bessel Function	349	(6)
14.2.2 Irregular Cylindrical Bessel Function	355	(4)
14.2.3 Regular Hyperbolic Bessel Function	359	(7)
14.2.4 Irregular Hyperbolic Bessel Function	366	(7)
14.3 Kelvin Functions	373	(19)
14.3.1 Regular Kelvin Functions	373	(10)
14.3.2 Irregular Kelvin Functions	383	(9)
14.4 Bessel Functions of Half-Integer Order	392	(22)
14.4.1 Regular Spherical Bessel Function	392	(7)
14.4.2 Irregular Spherical Bessel Function	399	(4)
14.4.3 Regular Modified Spherical Bessel Function	403	(7)
14.4.4 Irregular Modified Spherical Bessel Function	410	(4)
14.5 Airy Functions	414	(18)
14.5.1 Airy Functions	414	(9)
14.5.2 Derivatives of Airy Functions	423	(9)
References on Bessel Functions	432	(1)
15 Struve, Anger, and Weber Functions	433	(26)
15.1 Struve Functions	433	(13)
15.1.1 Struve Function	433	(7)
15.1.2 Modified Struve Function	440	(6)
15.2 Anger and Weber Functions	446	(10)
15.2.1 Overview of Anger and Weber Functions	446	(1)
15.2.2 Anger Function	447	(6)
15.2.3 Weber Function	453	(3)
References on Struve, Anger, and Weber Functions	456	(3)
16 Hypergeometric Functions and Coulomb Wave Functions	459	(34)
16.1 Hypergeometric Functions	459	(4)
16.2 Confluent Hypergeometric Functions	463	(13)
16.2.1 Regular Function	463	(6)
16.2.2 Irregular Function	469	(7)
16.3 Coulomb Wave Functions	476	(15)
16.3.1 Regular Functions and Derivatives	476	(9)
16.3.2 Irregular Functions and Derivatives	485	(6)
References on Hypergeometric Functions and Coulomb Wave Functions	491	(2)
17 Elliptic Integrals and Elliptic Functions	493	(44)
17.1 Overview of Elliptic Integrals and Elliptic Functions	493	(1)
17.2 Elliptic Integrals	494	(21)
17.2.1 Elliptic Integrals of the First Kind	494	(6)
17.2.2 Elliptic Integrals of the Second Kind	500	(4)
17.2.3 Jacobi Zeta Function	504	(4)
17.2.4 Heuman Lambda Function	508	(3)
17.2.5 Elliptic Integrals of the Third Kind	511	(4)
17.3 Jacobi Elliptic Functions and Theta Functions	515	(19)
17.3.1 Jacobi Elliptic Functions	515	(8)
17.3.2 Theta Functions	523	(6)
17.3.3 Logarithmic Derivatives of Theta Functions	529	(5)
References on Elliptic Integrals and Elliptic Functions	534	(3)
18 Parabolic Cylinder Functions	537	(12)
18.1 Parabolic Cylinder Coordinates	537	(1)
18.2 Parabolic Cylinder Functions	538	(10)
18.2.1 Parabolic Cylinder Functions U	538	(6)
18.2.2 Parabolic Cylinder Functions V	544	(4)
References on Parabolic Cylinder Functions	548	(1)
19 Miscellaneous Functions for Science and Engineering	549	(42)
19.1 Debye Functions	549	(3)
19.2 Sievert Integral	552	(3)
19.3 Abramowitz Function	555	(5)
19.4 Spence Integral	560	(3)
19.5 Clausen Integral	563	(5)
19.6 Voigt (Plasma Dispersion) Function	568	(6)
19.7 Angular Momentum Coupling Coefficients	574	(13)
19.7.1 3-j Coefficients	576	(4)
19.7.2 6-j Coefficients	580	(4)
19.7.3 9-j Coefficients	584	(3)
References on Miscellaneous Functions for Science and Engineering	587	(4)
PART II. THE COMPUTER INTERFACE	591	(274)
20 The Mathematica Notebooks	591	(204)
20.1 Introduction to the Notebooks	591	(1)
20.2 Exploring with the Notebook Cells	592	(1)
20.3 The Annotated Notebooks	592	(1)
20.4 Elementary Transcendental Functions	593	(7)
20.5 Exponential Integrals and Related Functions	600	(5)
20.6 Gamma and Beta Functions	605	(11)
20.7 Combinatorial Functions	616	(9)
20.8 Number Theory Functions	625	(6)
20.9 Probability Distributions	631	(30)
20.10 Error Function, Fresnel and Dawson Integrals	661	(4)
20.11 Orthogonal Polynomials	665	(9)
20.12 Legendre Functions	674	(16)
20.13 Spheroidal Wave Functions	690	(16)
20.14 Bessel Functions	706	(39)
20.15 Struve, Anger, and Weber Functions	745	(9)
20.16 Hypergeometric Functions and Coulomb Wave Functions	754	(9)
20.17 Elliptic Integrals and Elliptic Functions	763	(17)
20.18 Parabolic Cylinder Functions	780	(6)
20.19 Miscellaneous Functions for Science and Engineering	786	(9)
21 The Fortran Driver Programs	795	(70)
21.1 Introduction to the Fortran Driver Programs	795	(1)
21.2 How the Fortran Drivers Are Organized	795	(1)
21.3 Annotations to the Fortran Driver Programs	796	(1)
21.4 Elementary Transcendental Functions	796	(4)
21.4.1 Exponential and Logarithmic Functions	796	(1)
21.4.2 Circular and Inverse Circular Functions	797	(1)
21.4.3 Hyperbolic and Inverse Hyperbolic Functions	798	(2)
21.5 Exponential Integrals and Related Functions	800	(2)
21.5.1 Exponential and Logarithmic Integrals	800	(1)
21.5.2 Cosine and Sine Integrals	801	(1)
21.6 Gamma and Beta Functions	802	(3)
21.6.1 Gamma Function and Beta Function	802	(1)
21.6.2 Psi (Digamma) and Poygamma Functions	803	(1)
21.6.3 Incomplete Gamma and Beta Functions	804	(1)
21.7 Combinatorial Functions	805	(4)
21.7.1 Factorials and Rising Factorials	805	(1)
21.7.2 Binomial and Multinomial Coefficients	806	(2)
21.7.3 Stirling Numbers of the First and Second Kinds	808	(1)
21.7.4 Fibonacci and Lucas Polynomials	808	(1)
21.8 Number Theory Functions	809	(4)
21.8.1 Bernoulli Numbers and Bernoulli Polynomials	810	(1)
21.8.2 Euler Numbers and Euler Polynomials	810	(1)
21.8.3 Riemann Zeta Function	811	(1)
21.8.4 Other Sums of Reciprocal Powers	811	(1)
21.8.5 Polylogarithms	812	(1)
21.9 Probability Distributions	813	(11)
21.9.1 Organization of the PDFs	813	(1)
21.9.2 Discrete Probability Distributions	813	(3)
21.9.3 Normal Probability Distributions	816	(3)
21.9.4 Other Continuous Probability Distributions	819	(5)
21.10 Error Function, Fresnel and Dawson Integrals	824	(2)
21.10.1 Error Function	824	(1)
21.10.2 Fresnel Integrals	825	(1)
21.10.3 Dawson Integral	825	(1)
21.11 Orthogonal Polynomials	826	(4)
21.11.1 Orthogonal Polynomial Functions	826	(1)
21.11.2 Chebyshev Polynomials	826	(1)
21.11.3 Gegenbauer (Ultraspherical) Polynomials	827	(1)
21.11.4 Hermite Polynomials	828	(1)
21.11.5 Laguerre Polynomials	828	(1)
21.11.6 Legendre Polynomials	829	(1)
21.11.7 Jacobi Polynomials	829	(1)
21.12 Legendre Functions	830	(3)
21.12.1 Overview of Legendre Functions	830	(1)
21.12.2 Spherical Legendre Functions	830	(2)
21.12.3 Toroidal Legendre Functions	832	(1)
21.12.4 Conical Legendre Functions	832	(1)
21.13 Spheroidal Wave Functions	833	(3)
21.13.1 Overview of Spheroidal Wave Functions	833	(1)
21.13.2 Spheroidal Angular Functions	834	(1)
21.13.3 Spheroidal Radial Functions	835	(1)
21.14 Bessel Functions	836	(10)
21.14.1 Overview of Bessel Functions	837	(1)
21.14.2 Bessel Functions of Integer Order	837	(3)
21.14.3 Kelvin Functions	840	(2)
21.14.4 Bessel Functions of Half-Integer Order	842	(3)
21.14.5 Airy Functions	845	(1)
21.15 Struve, Anger, and Weber Functions	846	(3)
21.15.1 Struve Functions	846	(2)
21.15.2 Anger and Weber Functions	848	(1)
21.16 Hypergeometric Functions and Coulomb Wave Functions	849	(3)
21.16.1 Hypergeometric Functions	849	(1)
21.16.2 Confluent Hypergeometric Functions	850	(1)
21.16.3 Coulomb Wave Functions	851	(1)
21.17 Elliptic Integrals and Elliptic Functions	852	(5)
21.17.1 Overview of Elliptic Integrals and Elliptic Functions	852	(1)
21.17.2 Elliptic Integrals	853	(2)
21.17.3 Jacobi Elliptic Functions and Theta Functions	855	(2)
21.18 Parabolic Cylinder Functions	857	(2)
21.18.1 Parabolic Cylinder Functions	857	(2)
21.19 Miscellaneous Functions for Science and Engineering	859	(6)
21.19.1 Debye Functions	859	(1)
21.19.2 Sievert Integral	860	(1)
21.19.3 Abramowitz Function	860	(1)
21.19.4 Spence Integral	861	(1)
21.19.5 Clausen Integral	862	(1)
21.19.6 Voigt (Plasma Dispersion) Function	862	(1)
21.19.7 Angular Momentum Coupling Coefficients	863	(2)
APPENDIX: File Names for PC-Based Systems	865	(4)
INDEXES	869
Index of Function Notations	869	(3)
Index of Programs and Dependencies	872	(3)
Index of Subjects and Authors	875