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
¾
, 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
http://www.mathtable.com/.
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
