CRC Standard Mathematical Tables and Formulas, 33rd Edition

873 Pages·2012·9.13 MB·English

by  Daniel Zwillinger (ed.)

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CRC
STANDARD
MATHEMATICAL
TABLES AND
FORMULAS
33RD EDITION

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Advances in Applied Mathematics

Series Editor: Daniel Zwillinger

Published Titles

Advanced Engineering Mathematics with MATLAB, Fourth Edition

Dean G. Duffy

CRC Standard Curves and Surfaces with Mathematica®, Third Edition

David H. von Seggern

CRC Standard Mathematical Tables and Formulas, 33rd Edition

Dan Zwillinger

Dynamical Systems for Biological Modeling: An Introduction

Fred Brauer and Christopher Kribs

Fast Solvers for Mesh-Based Computations Maciej Paszy´nski

Green’s Functions with Applications, Second Edition Dean G. Duffy

Handbook of Peridynamic Modeling Floriin Bobaru, John T. Foster,

Philippe H. Geubelle, and Stewart A. Silling

Introduction to Financial Mathematics Kevin J. Hastings

Linear and Complex Analysis for Applications John P. D’Angelo

Linear and Integer Optimization: Theory and Practice, Third Edition

Gerard Sierksma and Yori Zwols

Markov Processes James R. Kirkwood

Pocket Book of Integrals and Mathematical Formulas, 5th Edition

Ronald J. Tallarida

Stochastic Partial Differential Equations, Second Edition Pao-Liu Chow

Quadratic Programming with Computer Programs Michael J. Best

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Advances in Applied Mathematics

Edited by
Dan Zwillinger

CRC
STANDARD
MATHEMATICAL
TABLES AND
FORMULAS
33RD EDITION

“K29784_FM” — 2017/12/6 — 17:41 — page 8 — #8

CRC Press
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Table of Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi

Chapter 1
Numbers and Elementary Mathematics . . . . . . . . . . . . . . . . . . .
1

1.1
Proofs without words . . . . . . . . . . . . . . . . . . . . . . . .
3

1.2
Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5

1.3
Special numbers . . . . . . . . . . . . . . . . . . . . . . . . . .
13

1.4
Interval analysis
. . . . . . . . . . . . . . . . . . . . . . . . . .
24

1.5
Fractal Arithmetic
. . . . . . . . . . . . . . . . . . . . . . . . .
25

1.6
Max-Plus Algebra
. . . . . . . . . . . . . . . . . . . . . . . . .
26

1.7
Coupled-analogues of Functions . . . . . . . . . . . . . . . . . .
27

1.8
Number theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
28

1.9
Series and products . . . . . . . . . . . . . . . . . . . . . . . . .
47

Chapter 2
Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67

2.1
Elementary algebra . . . . . . . . . . . . . . . . . . . . . . . . .
69

2.2
Polynomials
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
73

2.3
Vector algebra
. . . . . . . . . . . . . . . . . . . . . . . . . . .
78

2.4
Linear and matrix algebra
. . . . . . . . . . . . . . . . . . . . .
83

2.5
Abstract algebra
. . . . . . . . . . . . . . . . . . . . . . . . . .
106

Chapter 3
Discrete Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

3.1
Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
135

3.2
Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . .
140

3.3
Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
151

3.4
Combinatorial design theory . . . . . . . . . . . . . . . . . . . .
172

3.5
Difference equations . . . . . . . . . . . . . . . . . . . . . . . .
184

Chapter 4
Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

4.1
Euclidean geometry
. . . . . . . . . . . . . . . . . . . . . . . .
193

4.2
Grades and Degrees
. . . . . . . . . . . . . . . . . . . . . . . .
193

4.3
Coordinate systems in the plane . . . . . . . . . . . . . . . . . .
194

4.4
Plane symmetries or isometries
. . . . . . . . . . . . . . . . . .
200

4.5
Other transformations of the plane . . . . . . . . . . . . . . . . .
207

4.6
Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
209

4.7
Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
211

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vi
Table of Contents

4.8
Surfaces of revolution: the torus . . . . . . . . . . . . . . . . . .
219

4.9
Quadrics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
219

4.10
Spherical geometry and trigonometry . . . . . . . . . . . . . . .
224

4.11
Conics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
229

4.12
Special plane curves . . . . . . . . . . . . . . . . . . . . . . . .
240

4.13
Coordinate systems in space . . . . . . . . . . . . . . . . . . . .
249

4.14
Space symmetries or isometries . . . . . . . . . . . . . . . . . .
252

4.15
Other transformations of space . . . . . . . . . . . . . . . . . . .
255

4.16
Direction angles and direction cosines . . . . . . . . . . . . . . .
257

4.17
Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
257

4.18
Lines in space
. . . . . . . . . . . . . . . . . . . . . . . . . . .
259

4.19
Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
261

4.20
Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
265

4.21
Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
265

4.22
Differential geometry . . . . . . . . . . . . . . . . . . . . . . . .
267

Chapter 5
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

5.1
Differential calculus . . . . . . . . . . . . . . . . . . . . . . . .
277

5.2
Differential forms
. . . . . . . . . . . . . . . . . . . . . . . . .
288

5.3
Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
291

5.4
Table of indefinite integrals
. . . . . . . . . . . . . . . . . . . .
305

5.5
Table of definite integrals
. . . . . . . . . . . . . . . . . . . . .
343

5.6
Ordinary differential equations . . . . . . . . . . . . . . . . . . .
350

5.7
Partial differential equations . . . . . . . . . . . . . . . . . . . .
362

5.8
Integral equations . . . . . . . . . . . . . . . . . . . . . . . . . .
375

5.9
Tensor analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .
378

5.10
Orthogonal coordinate systems . . . . . . . . . . . . . . . . . . .
388

5.11
Real analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
393

5.12
Generalized functions
. . . . . . . . . . . . . . . . . . . . . . .
403

5.13
Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
405

5.14
Significant Mathematical Equations . . . . . . . . . . . . . . . .
417

Chapter 6
Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

6.1
Ceiling and floor functions . . . . . . . . . . . . . . . . . . . . .
421

6.2
Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . .
421

6.3
Exponential function . . . . . . . . . . . . . . . . . . . . . . . .
422

6.4
Logarithmic functions . . . . . . . . . . . . . . . . . . . . . . .
422

6.5
Trigonometric functions . . . . . . . . . . . . . . . . . . . . . .
424

6.6
Circular functions and planar triangles . . . . . . . . . . . . . . .
433

6.7
Tables of trigonometric functions
. . . . . . . . . . . . . . . . .
437

6.8
Angle conversion . . . . . . . . . . . . . . . . . . . . . . . . . .
440

6.9
Inverse circular functions . . . . . . . . . . . . . . . . . . . . . .
441

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vii

6.10
Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . .
443

6.11
Inverse hyperbolic functions . . . . . . . . . . . . . . . . . . . .
447

6.12
Gudermannian function
. . . . . . . . . . . . . . . . . . . . . .
449

6.13
Orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . .
451

6.14
Clebsch–Gordan coefficients . . . . . . . . . . . . . . . . . . . .
458

6.15
Bessel functions
. . . . . . . . . . . . . . . . . . . . . . . . . .
460

6.16
Beta function . . . . . . . . . . . . . . . . . . . . . . . . . . . .
469

6.17
Elliptic integrals
. . . . . . . . . . . . . . . . . . . . . . . . . .
470

6.18
Jacobian elliptic functions . . . . . . . . . . . . . . . . . . . . .
473

6.19
Error functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
475

6.20
Fresnel integrals
. . . . . . . . . . . . . . . . . . . . . . . . . .
476

6.21
Gamma function . . . . . . . . . . . . . . . . . . . . . . . . . .
478

6.22
Hypergeometric functions . . . . . . . . . . . . . . . . . . . . .
481

6.23
Lambert Function . . . . . . . . . . . . . . . . . . . . . . . . . .
483

6.24
Legendre functions . . . . . . . . . . . . . . . . . . . . . . . . .
484

6.25
Polylogarithms . . . . . . . . . . . . . . . . . . . . . . . . . . .
488

6.26
Prolate Spheroidal Wave Functions
. . . . . . . . . . . . . . . .
489

6.27
Sine, cosine, and exponential integrals . . . . . . . . . . . . . . .
490

6.28
Weierstrass Elliptic Function . . . . . . . . . . . . . . . . . . . .
492

6.29
Integral transforms: List . . . . . . . . . . . . . . . . . . . . . .
493

6.30
Integral transforms: Preliminaries . . . . . . . . . . . . . . . . .
494

6.31
Fourier integral transform
. . . . . . . . . . . . . . . . . . . . .
494

6.32
Discrete Fourier transform (DFT) . . . . . . . . . . . . . . . . .
500

6.33
Fast Fourier transform (FFT) . . . . . . . . . . . . . . . . . . . .
502

6.34
Multidimensional Fourier transforms
. . . . . . . . . . . . . . .
502

6.35
Hankel transform . . . . . . . . . . . . . . . . . . . . . . . . . .
503

6.36
Hartley transform . . . . . . . . . . . . . . . . . . . . . . . . . .
504

6.37
Hilbert transform . . . . . . . . . . . . . . . . . . . . . . . . . .
505

6.38
Laplace transform
. . . . . . . . . . . . . . . . . . . . . . . . .
508

6.39
Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . .
512

6.40
Z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . .
512

6.41
Tables of transforms . . . . . . . . . . . . . . . . . . . . . . . .
517

Chapter 7
Probability and Statistics
. . . . . . . . . . . . . . . . . . . . . . . . . . 533

7.1
Probability theory
. . . . . . . . . . . . . . . . . . . . . . . . .
535

7.2
Classical probability problems . . . . . . . . . . . . . . . . . . .
545

7.3
Probability distributions . . . . . . . . . . . . . . . . . . . . . .
553

7.4
Queuing theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
562

7.5
Markov chains . . . . . . . . . . . . . . . . . . . . . . . . . . .
565

7.6
Random number generation . . . . . . . . . . . . . . . . . . . .
568

7.7
Random matrices . . . . . . . . . . . . . . . . . . . . . . . . . .
574

7.8
Control charts and reliability . . . . . . . . . . . . . . . . . . . .
575

7.9
Statistics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
580

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viii
Table of Contents

7.10
Confidence intervals . . . . . . . . . . . . . . . . . . . . . . . .
588

7.11
Tests of hypotheses . . . . . . . . . . . . . . . . . . . . . . . . .
595

7.12
Linear regression . . . . . . . . . . . . . . . . . . . . . . . . . .
609

7.13
Analysis of variance (ANOVA)
. . . . . . . . . . . . . . . . . .
613

7.14
Sample size . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
620

7.15
Contingency tables . . . . . . . . . . . . . . . . . . . . . . . . .
623

7.16
Acceptance sampling . . . . . . . . . . . . . . . . . . . . . . . .
626

7.17
Probability tables . . . . . . . . . . . . . . . . . . . . . . . . . .
628

Chapter 8
Scientific Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645

8.1
Basic numerical analysis . . . . . . . . . . . . . . . . . . . . . .
646

8.2
Numerical linear algebra . . . . . . . . . . . . . . . . . . . . . .
659

8.3
Numerical integration and differentiation . . . . . . . . . . . . .
668

8.4
Programming techniques . . . . . . . . . . . . . . . . . . . . . .
688

Chapter 9
Mathematical Formulas from the Sciences
. . . . . . . . . . . . . . . . . 689

9.1
Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
691

9.2
Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
692

9.3
Atmospheric physics . . . . . . . . . . . . . . . . . . . . . . . .
694

9.4
Atomic Physics . . . . . . . . . . . . . . . . . . . . . . . . . . .
695

9.5
Basic mechanics . . . . . . . . . . . . . . . . . . . . . . . . . .
696

9.6
Beam dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . .
698

9.7
Biological Models . . . . . . . . . . . . . . . . . . . . . . . . .
699

9.8
Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
700

9.9
Classical mechanics
. . . . . . . . . . . . . . . . . . . . . . . .
701

9.10
Coordinate systems – Astronomical . . . . . . . . . . . . . . . .
702

9.11
Coordinate systems – Terrestrial . . . . . . . . . . . . . . . . . .
703

9.12
Earthquake engineering
. . . . . . . . . . . . . . . . . . . . . .
704

9.13
Economics (Macro)
. . . . . . . . . . . . . . . . . . . . . . . .
705

9.14
Electromagnetic Transmission . . . . . . . . . . . . . . . . . . .
707

9.15
Electrostatics and magnetism
. . . . . . . . . . . . . . . . . . .
708

9.16
Electromagnetic Field Equations . . . . . . . . . . . . . . . . . .
709

9.17
Electronic circuits
. . . . . . . . . . . . . . . . . . . . . . . . .
710

9.18
Epidemiology . . . . . . . . . . . . . . . . . . . . . . . . . . . .
711

9.19
Fluid mechanics
. . . . . . . . . . . . . . . . . . . . . . . . . .
712

9.20
Human body . . . . . . . . . . . . . . . . . . . . . . . . . . . .
713

9.21
Modeling physical systems . . . . . . . . . . . . . . . . . . . . .
714

9.22
Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
715

9.23
Population genetics . . . . . . . . . . . . . . . . . . . . . . . . .
716

9.24
Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . .
717

9.25
Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
719

9.26
Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
720

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ix

9.27
Relativistic mechanics . . . . . . . . . . . . . . . . . . . . . . .
721

9.28
Solid mechanics
. . . . . . . . . . . . . . . . . . . . . . . . . .
722

9.29
Statistical mechanics . . . . . . . . . . . . . . . . . . . . . . . .
723

9.30
Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . .
724

Chapter 10
Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725

10.1
Calendar computations . . . . . . . . . . . . . . . . . . . . . . .
727

10.2
Cellular automata . . . . . . . . . . . . . . . . . . . . . . . . . .
728

10.3
Communication theory . . . . . . . . . . . . . . . . . . . . . . .
729

10.4
Control theory
. . . . . . . . . . . . . . . . . . . . . . . . . . .
734

10.5
Computer languages . . . . . . . . . . . . . . . . . . . . . . . .
736

10.6
Compressive Sensing . . . . . . . . . . . . . . . . . . . . . . . .
737

10.7
Constrained Least Squares . . . . . . . . . . . . . . . . . . . . .
738

10.8
Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . .
739

10.9
Discrete dynamical systems and chaos . . . . . . . . . . . . . . .
740

10.10
Elliptic curves
. . . . . . . . . . . . . . . . . . . . . . . . . . .
743

10.11
Financial formulas . . . . . . . . . . . . . . . . . . . . . . . . .
746

10.12
Game theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .
754

10.13
Knot theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
757

10.14
Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
759

10.15
Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
761

10.16
Moments of inertia . . . . . . . . . . . . . . . . . . . . . . . . .
766

10.17
Music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
767

10.18
Operations research
. . . . . . . . . . . . . . . . . . . . . . . .
769

10.19
Proof Methods . . . . . . . . . . . . . . . . . . . . . . . . . . .
781

10.20
Recreational mathematics
. . . . . . . . . . . . . . . . . . . . .
782

10.21
Risk analysis and decision rules . . . . . . . . . . . . . . . . . .
783

10.22
Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . .
785

10.23
Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
794

10.24
Voting power . . . . . . . . . . . . . . . . . . . . . . . . . . . .
801

10.25
Greek alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . .
803

10.26
Braille code . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
803

10.27
Morse code . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
803

10.28
Bar Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
804

List of References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809

List of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811

Index
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819

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Preface

It has long been the established policy of CRC Press to publish, in handbook form,
the most up-to-date, authoritative, logically arranged, and readily usable reference
material available.
Just as pocket calculators have replaced tables of square roots and trig functions;
the internet has made printed tabulation of many tables and formulas unnecessary.
As the content and capabilities of the internet continue to grow, the content of this
book also evolves. For this edition of Standard Mathematical Tables and Formulae
the content was reconsidered and reviewed. The criteria for inclusion in this edition
includes:

• information that is immediately useful as a reference (e.g., interpretation of
powers of 10);

• information that is useful and not commonly known (e.g., proof methods);

• information that is more complete or concise than that which can be easily
found on the internet (e.g., table of conformal mappings);

• information difficult to find on the internet due to the challenges of entering an
appropriate query (e.g., integral tables).

Applying these criteria, practitioners from mathematics, engineering, and the sci-
ences have made changes in several sections and have added new material.

• The “Mathematical Formulas from the Sciences” chapter now includes topics
from biology, chemistry, and radar.

• Material has been augmented in many areas, including: acceptance sampling,
card games, lattices, and set operations.

• New material has been added on the following topics: continuous wavelet trans-
form, contour integration, coupled analogues, financial options, fractal arith-
metic, generating functions, linear temporal logic, matrix pseudospectra, max
plus algebra, proof methods, and two dimensional integrals.

• Descriptions of new functions have been added: Lambert, prolate spheroidal,
and Weierstrass.

Of course, the same successful format which has characterized earlier editions of the
Handbook has been retained. Material is presented in a multi-sectional format, with
each section containing a valuable collection of fundamental reference material—
tabular and expository.

xi

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xii
Preface

In line with the established policy of CRC Press, the Handbook will be updated
in as current and timely manner as is possible. Suggestions for the inclusion of new
material in subsequent editions and comments regarding the present edition are wel-
comed. The home page for this book, which will include errata, will be maintained
at http://www.mathtable.com/smtf.
This new edition of the Handbook will continue to support the needs of practi-
tioners of mathematics in the mathematical and scientific fields, as it has for almost
90 years. Even as the internet becomes more powerful, it is this editor’s opinion that
the new edition will continue to be a valued reference.

MATLAB R
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