standard mathematical tables and formulae 31st edition - zwillinger

CHAPMAN & HALL/CRC
A CRC Press Company
Boca Raton London New York Washington, D.C.
DANIEL ZWILLINGER
31
st
EDITION
standard
MathematicAL
TABLES
and
formulae
CRC
© 2003 by CRC Press LLC
Editor-in-Chief
Daniel Zwillinger
Rensselaer Polytechnic Institute
Troy, New York
Associate Editors
Steven G. Krantz
Washington University
St. Louis, Missouri
Kenneth H. Rosen
AT&T Bell Laboratories
Holmdel, New Jersey
Editorial Advisory Board
George E. Andrews
Pennsylvania State University
University Park, Pennsylvania
Michael F. Bridgland
Center for Computing Sciences

Bowie, Maryland
J. Douglas Faires
Youngstown State University
Youngstown, Ohio
Gerald B. Folland
University of Washington
Seattle, Washington
Ben Fusaro
Florida State University
Tallahassee, Florida
Alan F. Karr
National Institute Statistical Sciences
Research Triangle Park, North Carolina
Al Marden
University of Minnesota
Minneapolis, Minnesota
William H. Press
Los Alamos National Lab
Los Alamos, NM 87545
© 2003 by CRC Press LLC
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. Prior to the preparation of this 31
st
Edition of the CRC Standard
Mathematical Tables and Formulae, the content of such a book was reconsidered.
The previous edition was carefully analyzed, and input was obtained from practi-
tioners in the many branches of mathematics, engineering, and the physical sciences.
The consensus was that numerous small additions were required in several sections,

and several new areas needed to be added.
Some of the new materials included in this edition are: game theory and voting
power, heuristic search techniques, quadratic elds, reliability, risk analysis and de-
cision rules, a table of solutions to Pell’s equation, a table of irreducible polynomials
in
, a longer table of prime numbers, an interpretation of powers of 10, a col-
lection of “proofs without words”, and representations of groups of small order. In
total, there are more than 30 completely new sections, more than 50 new and mod-
i ed entries in the sections, more than 90 distinguished examples, and more than a
dozen new tables and gures. This brings the total number of sections, sub-sections,
and sub-sub-sections to more than 1,000. Within those sections are now more than
3,000 separate items (a de nition , a fact, a table, or a property). The index has also
been extensively re-worked and expanded to make nding results faster and easier;
there are now more than 6,500 index references (with 75 cross-references of terms)
and more than 750 notation references.
The same successful format which has characterized earlier editions of the Hand-
book is retained, while its presentation has been updated and made more consistent
from page to page. Material is presented in a multi-sectional format, with each sec-
tion containing a valuable collection of fundamental reference material—tabular and
expository.
In line with the established policy of CRC Press, the Handbook will be kept as
current and timely as is possible. Revisions and anticipated uses of newer materials
and tables will be introduced as the need arises. 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
The major material in this new edition is as follows:
Chapter 1: Analysis begins with numbers and then combines them into series and
products. Series lead naturally into Fourier series. Numbers also lead to func-
tions which results in coverage of real analysis, complex analysis, and gener-

alized functions.
Chapter 2: Algebra covers the different types of algebra studied: elementary al-
gebra, vector algebra, linear algebra, and abstract algebra. Also included are
details on polynomials and a separate section on number theory. This chapter
includes many new tables.
Chapter 3: Discrete Mathematics covers traditional discrete topics such as combi-
natorics, graph theory, coding theory and information theory, operations re-
© 2003 by CRC Press LLC
/>search, and game theory. Also included in this chapter are logic, set theory,
and chaos.
Chapter 4: Geometry covers all aspects of geometry: points, lines, planes, sur-
faces, polyhedra, coordinate systems, and differential geometry.
Chapter 5: Continuous Mathematics covers calculus material: differentiation, in-
tegration, differential and integral equations, and tensor analysis. A large table
of integrals is included. This chapter also includes differential forms and or-
thogonal coordinate systems.
Chapter 6: Special Functions contains a sequence of functions starting with the
trigonometric, exponential, and hyperbolic functions, and leading to many of
the common functions encountered in applications: orthogonal polynomials,
gamma and beta functions, hypergeometric functions, Bessel and elliptic func-
tions, and several others. This chapter also contains sections on Fourier and
Laplace transforms, and includes tables of these transforms.
Chapter 7: Probability and Statistics begins with basic probability information (de n -
ing several common distributions) and leads to common statistical needs (point
estimates, con d ence intervals, hypothesis testing, and ANOVA). Tables of the
normal distribution, and other distributions, are included. Also included in this
chapter are queuing theory, Markov chains, and random number generation.
Chapter 8: Scientific Computing explores numerical solutions of linear and non-
linear algebraic systems, numerical algorithms for linear algebra, and how to
numerically solve ordinary and partial differential equations.

Chapter 9: Financial Analysis contains the formulae needed to determine the re-
turn on an investment and how to determine an annuity (i.e., the cost of a
mortgage). Numerical tables covering common values are included.
Chapter 10: Miscellaneous contains details on physical units (de nition s and con-
versions), formulae for date computations, lists of mathematical and electronic
resources, and biographies of famous mathematicians.
It has been exciting updating this edition and making it as useful as possible.
But it would not have been possible without the loving support of my family, Janet
Taylor and Kent Taylor Zwillinger.
Daniel Zwillinger
15 October 2002
© 2003 by CRC Press LLC
Contributors
Karen Bolinger
Clarion University
Clarion, Pennsylvania
Patrick J. Driscoll
U.S. Military Academy
West Point, New York
M. Lawrence Glasser
Clarkson University
Potsdam, New York
Jeff Goldberg
University of Arizona
Tucson, Arizona
Rob Gross
Boston College
Chestnut Hill, Massachusetts
George W. Hart
SUNY Stony Brook

Stony Brook, New York
Melvin Hausner
Courant Institute (NYU)
New York, New York
Victor J. Katz
MAA
Washington, DC
Silvio Levy
MSRI
Berkeley, California
Michael Mascagni
Florida State University
Tallahassee, Florida
Ray McLenaghan
University of Waterloo
Waterloo, Ontario, Canada
John Michaels
SUNY Brockport
Brockport, New York
Roger B. Nelsen
Lewis & Clark College
Portland, Oregon
William C. Rinaman
LeMoyne College
Syracuse, New York
Catherine Roberts
College of the Holy Cross
Worcester, Massachusetts
Joseph J. Rushanan
MITRE Corporation

Bedford, Massachusetts
Les Servi
MIT Lincoln Laboratory
Lexington, Massachusetts
Peter Sherwood
Interactive Technology, Inc.
Newton, Massachusetts
Neil J. A. Sloane
AT&T Bell Labs
Murray Hill, New Jersey
Cole Smith
University of Arizona
Tucson, Arizona
Mike Sousa
Veridian
Ann Arbor, Michigan
Gary L. Stanek
Youngstown State University
Youngstown, Ohio
Michael T. Strauss
HME
Newburyport, Massachusetts
Nico M. Temme
CWI
Amsterdam, The Netherlands
Ahmed I. Zayed
DePaul University
Chicago, Illinois
© 2003 by CRC Press LLC
Table of Contents

Chapter 1
Analysis
Karen Bolinger, M. Lawrence Glasser, Rob Gross, and
Neil J. A. Sloane
Chapter 2
Algebra
Patrick J. Driscoll, Rob Gross, John Michaels, Roger B.
Nelsen, and Brad Wilson
Chapter 3
Discrete Mathematics
Jeff Goldberg, Melvin Hausner, Joseph J. Rushanan, Les
Servi, and Cole Smith
Chapter 4
Geometry
George W. Hart, Silvio Levy, and Ray McLenaghan
Chapter 5
Continuous Mathematics
Ray McLenaghan and Catherine Roberts
Chapter 6
Special Functions
Nico M. Temme and Ahmed I. Zayed
Chapter 7
Probability and Statistics
Michael Mascagni, William C. Rinaman, Mike Sousa, and
Michael T. Strauss
Chapter 8
Scientific Computing
Gary Stanek
Chapter 9
Financial Analysis

Daniel Zwillinger
Chapter 10
Miscellaneous
Rob Gross, Victor J. Katz, and Michael T. Strauss
© 2003 by CRC Press LLC
Table of Contents
Chapter 1
Analysis
1.1 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Special numbers . . . . . . .
1.3 Series and products . . . . .
1.4 Fourier series . . . . . . . .
1.5 Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Interval analysis . . .
1.7 Real analysis . . . . .
1.8 Generalized functions
Chapter 2
Algebra
2.1 Proofs without words
2.2 Elementary algebra . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Polynomials . . . . .
2.4 Number theory . . . . . . . . . . . . . . . . . . .
2.5 Vector algebra . . . . . . . .
2.6 Linear and matrix algebra . .
2.7 Abstract algebra . . . . . . . . . . .
Chapter 3
Discrete Mathematics
3.1 Symbolic logic
3.2 Set theory . . . . . .
3.3 Combinatorics . . . . . . . .

3.4 Graphs . . . . . . . . . . . . . . . .
3.5 Combinatorial design theory . . . . . . . . . . . . . . . . . . . .
3.6 Communication theory . . . . . . . . . . . . . . . . . . . . . . .
3.7 Difference equations .
3.8 Discrete dynamical systems and chao
s . . . . . .
3.9 Game theory . . . . .
3.10 Operations research .
Chapter 4
Geometry
4.1 Coordinate systems in the plane . . . . . . . . . . . . . . . . . . .
4.2 Plane symmetries or isometries . . . . . . . . . . . . . . . . . . .
4.3 Other transformations of the plane . . . . . . . . . . . . . . . . .
4.4 Lines . .
© 2003 by CRC Press LLC
4.5 Polygons
4.6 Conics . . . . . . . .
4.7 Special plane curves .
4.8 Coordinate systems in space . . . . . . . . . . . . . . . . . . . .
4.9 Space symmetries or isometries . . .
4.10 Other transformations of space . . . . . . . . . . . . . . . . . . .
4.11 Direction angles and direction cosines . . . . . . . . . . . . . .
4.12 Planes .
4.13 Lines in space . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.14 Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.15 Cylinders . . . . . .
4.16 Cones .
4.17 Surfaces of revolution: the torus . . . . . . . . . . . . . . . . . .
4.18 Quadrics
4.19 Spherical geometry & trigonometry . . . . . . . . . . . . . . . . .

4.20 Differential geometry . . . .
4.21 Angle conversion . . . . . .
4.22 Knots up to eight crossings . . . . . . . . . . . . . . . . . . . .
Chapter 5
Continuous Mathematics
5.1 Differential calculus . . . . .
5.2 Differential forms . .
5.3 Integration . . . . . .
5.4 Table of inde n ite integrals . . . . .
5.5 Table of de nite integrals . . . . . . . . . . . . . . . . .
5.6 Ordinary differential equations . . .
5.7 Partial differential equations . . . . .
5.8 Eigenvalues . . . . . . . . .
5.9 Integral equations . .
5.10 Tensor analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.11 Orthogonal coordinate systems . . .
5.12 Control theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 6
Special Functions
6.1 Trigonometric or circular functions . . . . . . . . . . . .
6.2 Circular functions and planar triangles . . . . . . . . . . . . . .
6.3 Inverse circular functions . . . . . .
6.4 Ceiling and oor functions . . . . . . . . . . . . . . . . . . . .
6.5 Exponential function . . . .
6.6 Logarithmic functions . . . . . . . .
6.7 Hyperbolic functions . . . .
6.8 Inverse hyperbolic functions . . . .
6.9 Gudermannian function . . . . . . . . . . . . . . . . . . . . . . .
6.10 Orthogonal polynomials . . .
© 2003 by CRC Press LLC

6.11 Gamma function . . . . . . .
6.12 Beta function . . . .
6.13 Error functions . . . . . . . . . . . .
6.14 Fresnel integrals
6.15 Sine, cosine, and exponential integrals . . . . . . . . . . . . . .
6.16 Polylogarithms . . . .
6.17 Hypergeometric functions . . . . . .
6.18 Legendre functions . . . . .
6.19 Bessel functions . . .
6.20 Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.21 Jacobian elliptic functions . .
6.22 Clebsch–Gordan coef cients . . . . . . . . . . . . . . . . . . . .
6.23 Integral transforms: Preliminaries . . . . . . . . . . . . . . . . . .
6.24 Fourier transform . . . . . .
6.25 Discrete Fourier transform (DFT) . .
6.26 Fast Fourier transform (FFT) . . . . . . . . . . .
6.27 Multidimensional Fourier transform
6.28 Laplace transform . .
6.29 Hankel transform . .
6.30 Hartley transform . . . . . .
6.31 Hilbert transform . . . . . .
6.32
-Transform . . . . .
6.33 Tables of transforms .
Chapter 7
Probability and Statistics
7.1 Probability theory . . . . . .
7.2 Classical probability problems . . . . . . . . . . . . . . . . . .
7.3 Probability distributions . . .
7.4 Queuing theory . . .

7.5 Markov chains . . . .
7.6 Random number generation .
7.7 Control charts and reliability . . . . . . . . . . . . . . . . . . .
7.8 Risk analysis and decision rules . . .
7.9 Statistics . . . . . . .
7.10 Con de nce intervals . . . . .
7.11 Tests of hypotheses .
7.12 Linear regression . . . . . .
7.13 Analysis of variance (ANOVA) . . . . . . . . . . . . . . . . . . .
7.14 Probability tables . . . . . .
7.15 Signal processing . . . . . .
Chapter 8
Scienti c Computing
8.1 Basic numerical analysis . .
8.2 Numerical linear algebra . .
© 2003 by CRC Press LLC
8.3 Numerical integration and differentiation . . . . . . . . .
8.4 Programming techniques . . . . . .
Chapter 9
Financial Analysis
9.1 Financial formulae . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Financial tables . . . . . . . . . . . . . . . . . . . . . .
Chapter 10
Miscellaneous
10.1 Units . .
10.2 Interpretations of powers of 10 . . .
10.3 Calendar computations . . .
10.4 AMS classi cation scheme .
10.5 Fields medals . . . . . . . .
10.6 Greek alphabet . . . .

10.7 Computer languages . . . . . . . . . . . . . . . . . . . . . . . .
10.8 Professional mathematical organizations . . . . . . . . . . . . . .
10.9 Electronic mathematical resources . . . . . . . .
10.10 Biographies of mathematicians . . . . . . . . . . . . . .
List of references
List of Figures
List of notation
835
© 2003 by CRC Press LLC
List of References
Chapter 1 Analysis
1. J. W. Brown and R. V. Churchill, Complex variables and applications,
6th edition, McGraw–Hill, New York, 1996.
2. L. B. W. Jolley, Summation of Series, Dover Publications, New York,
1961.
3. S. G. Krantz, Real Analysis and Foundations, CRC Press, Boca Raton,
FL, 1991.
4. S. G. Krantz, The Elements of Advanced Mathematics, CRC Press, Boca
Raton, FL, 1995.
5. J. P. Lambert, “Voting Games, Power Indices, and Presidential Elec-
tions”, The UMAP Journal, Module 690, 9, No. 3, pages 214–267, 1988.
6. L. D. Servi, “Nested Square Roots of 2”, American Mathematical Monthly,
to appear in 2003.
7. N. J. A. Sloane and S. Plouffe, Encyclopedia of Integer Sequences, Aca-
demic Press, New York, 1995.
Chapter 2 Algebra
1. C. Caldwell and Y. Gallot, “On the primality of
and
”, Mathematics of Computation, 71:237, pages 441–448, 2002.
2. I. N. Herstein, Topics in Algebra, 2nd edition, John Wiley & Sons, New

York, 1975.
3. P. Ribenboim, The book of Prime Number Records, Springer–Verlag,
New York, 1988.
4. G. Strang, Linear Algebra and Its Applications, 3rd edition, International
Thomson Publishing, 1988.
Chapter 3 Discrete Mathematics
1. B. Bollob´as, Graph Theory, Springer–Verlag, Berlin, 1979.
© 2003 by CRC Press LLC
2. C. J. Colbourn and J. H. Dinitz, Handbook of Combinatorial Designs,
CRC Press, Boca Raton, FL, 1996.
3. F. Glover, “Tabu Search: A Tutorial”, Interfaces, 20(4), pages 74–94,
1990.
4. D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Ma-
chine Learning, Addison–Wesley, Reading, MA, 1989.
5. J. Gross, Handbook of Graph Theory & Applications, CRC Press, Boca
Raton, FL, 1999.
6. D. Luce and H. Raiffa, Games and Decision Theory, Wiley, 1957.
7. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting
Codes, North–Holland, Amsterdam, 1977.
8. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E.
Teller, “Equation of State Calculations by Fast Computing Machines”, J.
Chem. Phys., V 21, No. 6, pages 1087–1092, 1953.
9. K. H. Rosen, Handbook of Discrete and Combinatorial Mathematics,
CRC Press, Boca Raton, FL, 2000.
10. J. O’Rourke and J. E. Goodman, Handbook of Discrete and Computa-
tional Geometry, CRC Press, Boca Raton, FL, 1997.
Chapter 4 Geometry
1. A. Gray, Modern Differential Geometry of Curves and Surfaces, CRC
Press, Boca Raton, FL, 1993.
2. C. Livingston, Knot Theory, The Mathematical Association of America,

Washington, D.C., 1993.
3. D. J. Struik, Lectures in Classical Differential Geometry, 2nd edition,
Dover, New York, 1988.
Chapter 5 Continuous Mathematics
1. A. G. Butkovskiy, Green’s Functions and Transfer Functions Handbook,
Halstead Press, John Wiley & Sons, New York, 1982.
2. I. S. Gradshteyn and M. Ryzhik, Tables of Integrals, Series, and Prod-
ucts, edited by A. Jeffrey and D. Zwillinger, 6th edition, Academic Press,
Orlando, Florida, 2000.
3. N. H. Ibragimov, Ed., CRC Handbook of Lie Group Analysis of Differen-
tial Equations, Volume 1, CRC Press, Boca Raton, FL, 1994.
4. A. J. Jerri, Introduction to Integral Equations with Applications, Marcel
Dekker, New York, 1985.
5. P. Moon and D. E. Spencer, Field Theory Handbook, Springer-Verlag,
Berlin, 1961.
6. A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solution for Ordi-
nary Differential Equations, CRC Press, Boca Raton, FL, 1995.
© 2003 by CRC Press LLC
7. J. A. Schouten, Ricci-Calculus, Springer–Verlag, Berlin, 1954.
8. J. L. Synge and A. Schild, Tensor Calculus, University of Toronto Press,
Toronto, 1949.
9. D. Zwillinger, Handbook of Differential Equations, 3rd ed., Academic
Press, New York, 1997.
10. D. Zwillinger, Handbook of Integration, A. K. Peters, Boston, 1992.
Chapter 6 Special Functions
1. Staff of the Bateman Manuscript Project, A. Erd´elyi, Ed., Tables of Inte-
gral Transforms, in 3 volumes, McGraw–Hill, New York, 1954.
2. I. S. Gradshteyn and M. Ryzhik, Tables of Integrals, Series, and Prod-
ucts, edited by A. Jeffrey and D. Zwillinger, 6th edition, Academic Press,
Orlando, Florida, 2000.

3. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for
the Special Functions of Mathematical Physics, Springer–Verlag, New
York, 1966.
4. N. I. A. Vilenkin, Special Functions and the Theory of Group Represen-
tations, American Mathematical Society, Providence, RI, 1968.
Chapter 7 Probability and Statistics
1. I. Daubechies, Ten Lectures on Wavelets, SIAM Press, Philadelphia, 1992.
2. W. Feller, An Introduction to Probability Theory and Its Applications,
Volume 1, John Wiley & Sons, New York, 1968.
3. J. Keilson and L. D. Servi, “The Distributional Form of Little’s Law
and the Fuhrmann–Cooper Decomposition”, Operations Research Let-
ters, Volume 9, pages 237–247, 1990.
4. Military Standard 105 D, U.S. Government Printing Of ce, Washington,
D.C., 1963.
5. S. K. Park and K. W. Miller, “Random number generators: good ones are
hard to nd”, Comm. ACM, October 1988, 31, 10, pages 1192–1201.
6. G. Strang and T. Nguyen, Wavelets and Filter Banks, Wellesley–Cambridge
Press, Wellesley, MA, 1995.
7. D. Zwillinger and S. Kokoska, Standard Probability and Statistics Tables
and Formulae, Chapman & Hall/CRC, Boca Raton, Florida, 2000.
Chapter 8 Scientific Computing
1. R. L. Burden and J. D. Faires, Numerical Analysis, 7th edition, Brooks/Cole,
Paci c Grove, CA, 2001.
2. G. H. Golub and C. F. Van Loan, Matrix Computations, 2nd ed., The
Johns Hopkins Press, Baltimore, 1989.
© 2003 by CRC Press LLC
3. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Nu-
merical Recipes in C++: The Art of Scientific Computing, 2nd edition,
Cambridge University Press, New York, 2002.
4. A. Ralston and P. Rabinowitz, A First Course in Numerical Analysis, 2nd

edition, McGraw–Hill, New York, 1978.
5. R. Rubinstein, Simulation and the Monte Carlo Method, Wiley, New
York, 1981.
Chapter 10 Miscellaneous
1. American Mathematical Society, Mathematical Sciences ProfessionalDi-
rectory, Providence, 1995.
2. E. T. Bell, Men of Mathematics, Dover, New York, 1945.
3. C. C. Gillispie, Ed., Dictionary of Scientific Biography, Scribners, New
York, 1970–1990.
4. H. S. Tropp, “The Origins and History of the Fields Medal”, Historia
Mathematica, 3, pages 167–181, 1976.
5. E. W. Weisstein, CRC Concise Encyclopediaof Mathematics, CRC Press,
Boca Raton, FL, 1999.
© 2003 by CRC Press LLC
List of Figures
2.1 Depiction of right-hand rule
3.1 Hasse diagrams
3.2 Three graphs that are isomorphic
3.3 Examples of graphs with 6 or 7 vertices
3.4 Trees with 7 or fewer vertices
3.5 Trees with 8 vertices
3.6 Julia sets
3.7 The Mandlebrot set
3.8 Directed network modeling a flow problem
4.1 Change of coordinates by a rotation
4.2 Cartesian coordinates: the 4 quadrants
4.3 Polar coordinates
4.4 Homogeneous coordinates
4.5 Oblique coordinates
4.6 A shear with factor

4.7 A perspective transformation
4.8 The normal form of a line
4.9 Simple polygons
4.10 Notation for a triangle
4.11 Triangles: isosceles and right
4.12 Ceva’s theorem and Menelaus’s theorem
4.13 Quadrilaterals
4.14 Conics: ellipse, parabola, and hyperbola
4.15 Conics as a function of eccentricity
4.16 Ellipse and components
4.17 Hyperbola and components
4.18 Arc of a circle
4.19 Angles within a circle
4.20 The general cubic parabola
4.21 Curves: semi-cubic parabola, cissoid of Diocles, witch of Agnesi
4.22 The folium of Descartes in two positions, and the strophoid


© 2003 by CRC Press LLC
4.23 Cassini’s ovals
4.24 The conchoid of Nichomedes
4.25 The limac¸on of Pascal
4.26 Cycloid and trochoids
4.27 Epicycloids: nephroid, and epicycloid
4.28 Hypocycloids: deltoid and astroid
4.29 Spirals: Bernoulli, Archimedes, and Cornu
4.30 Cartesian coordinates in space
4.31 Cylindrical coordinates
4.32 Spherical coordinates
4.33 Relations between Cartesian, cylindrical, and spherical coordinates

4.34 Euler angles
4.35 The Platonic solids
4.36 Cylinders: oblique and right circular
4.37 Right circular cone and frustram
4.38 A torus of revolution
4.39 The ve nondegenerate real quadrics
4.40 Spherical cap, zone, and segment
4.41 Right spherical triangle and Napier’s rule
5.1 Types of critical points
6.1 Notation for trigonometric functions
6.2 Definitions of angles
6.3 Sine and cosine
6.4 Tangent and cotangent
6.5 Different triangles requiring solution
6.6 Graphs of
and 541
6.7 Cornu spiral
6.8 Sine and cosine integrals
and 549
6.9 Legendre functions
6.10 Graphs of the Airy functions
and 566
7.1 Approximation to binomial distributions
7.2 Conceptual layout of a queue
7.3 Sample size code letters for MIL-STD-105 D
7.4 Master table for single sampling inspection (normal inspection)
7.5 Area of a normal random variable
7.6 Illustration of
and regions of a normal distribution
8.1 Illustration of Newton’s method

8.2 Formulae for integration rules with various weight functions
8.3 Illustration of the Monte–Carlo method
© 2003 by CRC Press LLC
List of Notation
Symbols
! factorial 17
!! double factorial 17
tensor differentiation . . . 484
tensor differentiation . . . 484
cyclic subgroup generated by . 162
set complement 203
derivative, rst 386
derivative, second . 386
ceiling function 520
oor function 520
Stirling subset numbers . . 213
aleph null 204
universal quanti er . 201
arrow notation . . 4
if and only if 199
implies 199
logical implication 199
set intersection 203
graph edge sum 228
graph union . . 229
set union 203
group isomorphism 170, 225
congruence 94
existential quanti er . 201
Plank constant over 794

in nity 68
de nite integral 399
integral around closed path . .399
integration symbol 399
falling factorial 17
logical not 199
partial differentiation 386
dual code to 257
partial order 204
product symbol 47
summation symbol 31
empty set . . . 202
asymptotic relation 75
logical not . 199
vertex similarity . . . 226
logical or . 199
pseudoscalar product . 467
graph conjunction . . 228
logical and 199
wedge product . 395
divergence 493
curl 493
Laplacian 493
backward difference 736
gradient . . . 390, 493
linear connection . . . 484
[]
graph composition 228
commutator . 155, 467
vuw scalar triple product 136

continued fraction . 96
Christoffel symbol, rst kind
487
Stirling cycle numbers . . . . 212


© 2003 by CRC Press LLC
*Page numbers listed do not match PDF page numbers due to deletion of blank pages.
()
poset notation 205
shifted factorial 17
type of tensor 483
design nomenclature . 245
point in three-dimensional
space 345
homogeneous
coordinates . 303
homogeneous
coordinates . 348
Clebsch–Gordan
coef cient . . 574
binomial coef cient 208
multinomial
coef cient . . 209
Jacobi symbol 94
Legendre symbol . . 94
fourth derivative 386
th
derivative 386
fth derivative 386

trimmed mean 659
arithmetic mean . . 659
complex conjugate 54
set complement 203
divisibility 93
determinant of a matrix . . 144
graph order 226
norm 133
order of algebraic structure 160
polynomial norm 91
used in tensor notation 487
norm 133
norm 133
Frobenius norm . 146
in nity norm 133
norm 91,133
a b vector inner product . . . . . 133
group operation . 161
inner product 132
crystallographic group . . . 309, 311
degrees in an angle 503
function composition 67
temperature degrees 798
translation 307
2222 crystallographic group . . 310
333 crystallographic group . . . 311
442 crystallographic group . . . 310
632 crystallographic group . . . 311
crystallographic group 309
binary operation 160

convolution operation 579
dual of a tensor 489
group operation . 161
re ection 307
a b vector cross product 135
crystallographic group 309
crystallographic group 309
glide-re ection . . . 307
graph product . . 228
group operation . 161
product . . . 66
Kronecker product 159
symmetric difference . . 203
exclusive or . . 645
factored graph 224
graph edge sum . 228
Kronecker sum 160
continued
fraction . 97
graph join 228
group operation . 161
pseudo-inverse operator . . 149, 151
vector addition 132
Greek Letters
maximum vertex degree 223
change in the
argument 58
forward difference 265, 728
Laplacian . . 493
gamma function 540

Christoffel symbol of second
kind 487
connection coef cients 484
asymptotic function . . . 75
ohm 792
© 2003 by CRC Press LLC
normal distribution function . . .634
asymptotic function . 75
graph arboricity 220
graph independence number
225
function, related to zeta
function . . . 23
one minus the con dence
coef cient . . 666
probability of type I error . . . . . 661
probability of type II error . . 661
function, related to zeta
function . . . 23
chromatic index . 221
chromatic number . 221
-distribution . . 703
critical value . . . 696
chi-square distributed 619
minimum vertex degree . .223
delta function 76
Kronecker delta 483
designed distance 257
Feigenbaum’s constant 272
Levi–Civita symbol 489

power of a test 661
component of in nitesimal
generator . 466
Euler’s constant
de nition . 15
in different bases . . . 16
value . 16
graph genus . 224
function, related to zeta
function . . . 23
skewness 620
excess . . 620
connectivity 222
curvature . . . 374
cumulant 620
edge connectivity 223
average arrival rate 638
eigenvalue . . 152, 477, 478
number of blocks . . . 241
M¨obius function 102
centered moments . 620
moments 620
MTBF for parallel system . . 655
MTBF for series system . . . 655
average service rate . . 638
mean . 620
rectilinear graph crossing
number 222
graph crossing number . . 222
size of the largest clique 221

totient function . . . . . 128, 169
characteristic function 620
Euler constant 21
golden ratio
de ned . 16
value . . 16
incidence mapping . 219
zenith 346
prime counting function 103
probability distribution 640
constants containing 14
continued fraction 97
distribution of digits 15
identities . 14
number . . . 13
in different bases . . . 16
permutation 172
sums involving 24
logarithmic derivative of the
gamma function . . . 543
spectral radius 154
radius of curvature 374
correlation coef cient 622
server utilization 638
standard deviation 620
sum of divisors 128
variance . . . 620
singular value of a matrix . . 152
sum of
th

powers of divisors
128
variance . . . 622
covariance . 622
© 2003 by CRC Press LLC
Ramanujan function 31
number of divisors 128
torsion . . 374
graph thickness 227
angle in polar coordinates 302
argument of a complex number . 53
azimuth 346
component of in nitesimal
generator . 466
quantile of order 659
Riemann zeta function . . . 23
Numbers
group inverse . . . 161
matrix inverse 138
0 null vector . . 137
1
1, group identity 161
1-form 395
10, powers of . . 6,13,798
105 D standard . . . 652
16, powers of . . . 12
17 crystallographic groups . . . . 307
2
power set of 203
2

22 crystallographic group . . . 310
2, negative powers of . . . 10
2, powers of 6,10,27
2-(
,3,1) Steiner triple system . 249
2-form 396
2-sphere 491
2-switch . 227
22
crystallographic group 309
22
crystallographic group 309
2222 crystallographic group . . . 310
230 crystallographic groups,
three-dimensional . . . 307
3
3
3 crystallographic group 311
3, powers of . . 29
3-design (Hadamard matrices) . 250
3-form 397
3-sphere 491
333 crystallographic group 311
360, degrees in a circle 503
4
4
2 crystallographic group 310
4, powers of . . 30
442 crystallographic group 310
5

5, powers of . . 30
5-(12,6,1) table 244
5-design, Mathieu . . 244
632 crystallographic group . . 311
Roman Letters
A
A
interarrival time 637
number of codewords . 259
skew symmetric part of a
tensor . 484
A ampere . 792
alternating group on 4 elements
188
radius of circumscribed circle
324
alternating group 163, 172
queue . . 637
Airy function 465, 565
ALFS additive lagged-Fibonacci
sequence 646
AMS American Mathematical Society
801
ANOVA analysis of variance 686
AOQ average outgoing quality . . . . . . 652
AOQL average outgoing quality limit 652
AQL acceptable quality level 652
AR
autoregressive model . . 718
ARMA

mixed model 719
graph automorphism group . 220
a unit vector . 492
Fourier coef cients . . . 48
proportion of customers 637
almost everywhere . . 74
am amplitude 572
arg argument . 53
© 2003 by CRC Press LLC
B
B
amount borrowed . 779
service time 637
beta function 544
set of blocks 241
Bell number 211
Bernoulli number . 19
a block . . . 241
Bernoulli polynomial . . . 19
B.C.E (before the common era, B.C.) 810
BFS basic feasible solution 283
Airy function 465, 565
BIBD balanced incomplete block design
245
Bq becquerel 792
b unit binormal vector 374
C
C
channel capacity . . 255
-combination . . . 206, 215

Fresnel integral 547
combinations with
replacement 206
complex numbers . . 3,167
complex element vectors 131
integration contour . . . . . 399, 404
C coulomb 792
C Roman numeral (100) . . 4
cyclic group of order 2 178
direct group product
181
cyclic group of order 3 178
direct group product . 184
cyclic group of order 4 178
direct group product . 181
cyclic group of order 5 179
cyclic group of order 6 179
cyclic group of order 7 180
cyclic group of order 8 180
cyclic group of order 9 184
Catalan numbers . . . 212
cycle graph . 229
cyclic group 172
cyclic group of order 10 . . 185
C.E. (common era, A.D.) 810
cosine integral 549
c
c cardinality of real numbers . . 204
number of identical servers . . 637
speed of light 794

cas combination of sin and cos 591
cd candela 792
cm crystallographic group 309
cmm crystallographic group . . . 310
Fourier coef cients 50
elliptic function . . . 572
cof
cofactor of matrix 145
cond(
) condition number 148
cos trigonometric function . 505
cosh hyperbolic function . 524
cot trigonometric function 505
coth hyperbolic function 524
covers trigonometric function 505
csc trigonometric function 505
csch hyperbolic function 524
cyc number of cycles 172
D
D
constant service time . 637
diagonal matrix 138
differentiation operator 456, 466
D Roman numeral (500) 4
dihedral group of order 8 . . 182
dihedral group of order 10 . 185
dihedral group of order 12 . 186
region of convergence . 595
derangement . . 210
dihedral group . 163, 172

DFT discrete Fourier transform 582
DLG
double loop graph 230
distance between vertices
223
derivative operator 386
exterior derivative . . . 397
minimum distance 256
proportion of customers 637
H
u v Hamming distance . . . 256
a projection 395
determinant of matrix 144
graph diameter 223
div divergence 493
elliptic function 572
© 2003 by CRC Press LLC
differential surface area 405
differential volume . . 405
x fundamental differential 377
E
E
edge set . . . 219
event 617
rst fundamental metric
coef cient . . 377
E
expectation operator 619
Erlang- service time . 637
Euler numbers . . . 20

Euler polynomial . 20
exponential integral 550
identity group 172
elementary matrix 138
Eiexbi 6
e
algebraic identity . . . 161
charge of electron 794
constants containing 15
continued fraction 97
de nition . . . 15
eccentricity . 325
in different bases 16
second fundamental metric
coef cient . . 377
e vector of ones 137
e
unit vector 137
permutation symbol . . . 489
ecc
eccentricity of a vertex . 223
erf error function 545
erfc complementary error function . . 545
exsec trigonometric function 505
F
F
rst fundamental metric
coef cient . . 377
Dawson’s integral . 546
probability distribution

function . 619
Fourier transform 576
hypergeometric
function . 553
sample distribution function
658
F farad 792
Fibonacci numbers 21
critical value 696
Galois eld . 169
Fourier cosine transform . . . 582
Fourier sine transform 582
discrete Fourier transform . 582
hypergeometric function . . . 36
FCFS rst come, rst served 637
FFT fast Fourier transform . . . 584
FIFO rst in, rst out . . 637
f
sample density function . .658
second fundamental metric
coef cient . . . 377
probability density function
619
density function 467
G
G
Green’s function . . . 471
general service time distribution
637
generating matrix . . 256

graph 219
gravitational constant . . . . . . 794
primitive root 195
Waring’s problem 100
generating function . . . . . 620
rst fundamental metric
coef cient . . . 377
Green’s function 463
generating function . . . . 268
G Catalan constant 23
isomorphism classes . . . . 241
GCD greatest common divisor 101
Galois eld 169
general interarrival time 637
Gi gibi . 6
matrix group 171
matrix group 171
G.M. geometric mean . . . 659
Gy gray 792
graph girth . 224
g
determinant of the metric tensor
489
metric tensor . . 487
© 2003 by CRC Press LLC
primitive root 195
Waring’s problem . . . 100
generating polynomial . . . 256
gd function 530
Gudermannian function 530

covariant metric 486
contravariant metric 486
glb greatest lower bound 68
H
H
mean curvature 377
parity check matrix 256
p entropy 253
Haar wavelet 723
Heaviside function . . 77, 408
Hilbert transform 591
H
Hermitian conjugate . . . 138
H henry 792
null hypothesis . . 661
alternative hypothesis 661
Hankel function . . . 559
Hankel function . . . 559
-stage hyperexponential
service time 637
harmonic numbers 32
Hermite polynomials . . 532
Hankel transform 589
H.M. harmonic mean 660
Hz hertz 792
hav trigonometric function 372, 505
metric coef cients . . 492
I
I
rst fundamental form . . 377

identity matrix 138
mutual information . . 254
I Roman numeral (1) . 4
ICG inversive congruential generator 646
second fundamental form . 377
Im imaginary part of a complex number
53
identity matrix 138
Inv number of invariant elements . . . 172
IVP initial-value problem . . 265
i
i unit vector 494
i unit vector 135
imaginary unit . . 53
interest rate 779
iid independent and identically
distributed 619
inf greatest lower bound 68
in mum greatest lower bound 68
J
J
Jordan form 154
J joule 792
j
j unit vector . . 494
j unit vector . . 135
Bessel function 559
Julia set 273
half order Bessel function
563

zero of Bessel function . . . 563
K
K
Gaussian curvature . 377
system capacity 637
K Kelvin (degrees) 792
complete graph . . . 229
complete bipartite graph 230
complete multipartite
graph 230
empty graph 229
Ki kibi . 6
k
k curvature vector 374
k unit vector 494
k unit vector 135
Boltzmann constant 794
dimension of a code 258
kernel 478
k geodesic curvature 377
k
normal curvature vector 377
block size . . 241
kg kilogram 792
© 2003 by CRC Press LLC
L
L
average number of customers 638
period 48
expected loss function . . . 656

Laplace transform 585
L length 796
L Roman numeral (50) 4
norm 133
norm 133
average number of customers
638
norm . 73
Lie group 466
space of measurable functions
73
LCG linear congruential generator . . 644
LCL lower control limit 650
LCM least common multiple . . 101
logarithm 551
dilogarithm . . . 551
LIFO last in, rst out 637
polylogarithm 551
logarithmic integral . 550
LP linear programming . 280
LTPD lot tolerance percent defective 652
loss function 656
lim limits 70,385
liminf limit inferior 70
limsup limit superior 70
lm lumen 792
ln logarithmic function . . 522
log logarithmic function 522
logarithm to base 522
lub least upper bound . . . 68

luxlux 792
M
M
Mandelbrot set 273
exponential service time . . . 637
number of codewords 258
measure of a polynomial 93
Mellin transform 612
M mass 796
M Roman numeral (1000) . 4
MA
moving average 719
M.D. mean deviation 660
MFLG multiplicative lagged-Fibonacci
generator . . 646
queue . 639
queue . . 639
queue 639
Mi mebi . . . 6
MLE maximum likelihood estimator 662
queue . . . 638
queue . . . 639
M¨obius ladder graph . . 229
MOLS mutually orthogonal Latin
squares . . 251
MOM method of moments . . . 662
MTBF mean time between failures . .655
m
mortgage amount 779
number in the source . 637

m meter 792
mid midrange 660
mod modular arithmetic 94
mol mole 792
N
N
number of zeros 58
null space . 149
normal random variable
619
N unit normal vector . 378
normal vector . . 377
natural numbers 3
N newton . . 792
number of monic irreducible
polynomials . . 261
n
n principal normal unit vector . 374
n unit normal vector 135
code length 258
number of time periods 779
order of a plane . . . 248
O
asymptotic function 75
matrix group . . . 171
odd graph 229
asymptotic function 75
© 2003 by CRC Press LLC
P
P

number of poles 58
principal 779
conditional probability
617
probability of event . . 617
auxiliary function 561
-permutation . . 215
-permutation . 206
Markov transition function
640
Riemann function . . . . .465
chromatic polynomial . .221
path (type of graph) . . . 229
Lagrange interpolating
polynomial . 733
Legendre function 465
Legendre polynomials . . 534
Jacobi polynomials . 533
Legendre function 554
associated Legendre
functions . 557
Pa pascal . 792
Per
period of a sequence 644
Pi pebi . 6
PID principal ideal domain . . . 165
-step Markov transition
matrix 641
permutations with replacement
206

PRI priority service . . 637
PRNG pseudorandom number generator
644
p
partitions 210
product of prime numbers . 106
p1 crystallographic group . 309, 311
p2 crystallographic group . 310
p3 crystallographic group . 311
p31m crystallographic group . . 311
p3m1 crystallographic group . . 311
p4 crystallographic group . 310
p4g crystallographic group . . 310
p4m crystallographic group . 310
p6 crystallographic group . 311
p6m crystallographic group . 311
per permanent 145
pg crystallographic group . 309
pgg crystallographic group . . 309
pm crystallographic group 309
pmg crystallographic group . 309
pmm crystallographic group 310
p joint probability distribution
254
discrete probability . . . 619
partitions . . . 207
restricted partitions . . . . 210
proportion of time 638
Q
Q

quaternion group . . 182
auxiliary function . . . . . 561
rational numbers 3,167
cube (type of graph) 229
Legendre function . . . . . 465
Legendre function . 554
associated Legendre functions
557
nome 574
R
R
Ricci tensor . . 485, 488
Riemann tensor 488
curvature tensor . . 485
radius (circumscribed circle) 319,
513
range . 650
rate of a code 255
range space 149
risk function . . . 657
reliability function 655
continuity in 71
convergence in 70
real numbers . . . 3, 167
reliability of a component . . 653
reliability of parallel system 653
reliability of series system . 653
radius of the earth . . . 372
real element vectors 131
real matrices . 137

Re real part of a complex number 53
R.M.S. root mean square . . . 660
© 2003 by CRC Press LLC