standard mathematical tables and formulaes2.bitdownload.ir/ebook/algebra &...

CHAPMAN & HALL/CRCA CRC Press Company

Boca Raton London New York Washington, D.C.

DANIEL ZWILLINGER

31stEDITION

standardMathematicALTABLES andformulae

CRC

2003 by CRC Press LLC

Editor-in-ChiefDaniel ZwillingerRensselaer Polytechnic InstituteTroy, New York

Associate Editors

Steven G. KrantzWashington UniversitySt. Louis, Missouri

Kenneth H. RosenAT&T Bell LaboratoriesHolmdel, New Jersey

Editorial Advisory Board

George E. AndrewsPennsylvania State UniversityUniversity Park, Pennsylvania

Michael F. BridglandCenter for Computing SciencesBowie, Maryland

J. Douglas FairesYoungstown State UniversityYoungstown, Ohio

Gerald B. FollandUniversity of WashingtonSeattle, Washington

Ben FusaroFlorida State UniversityTallahassee, Florida

Alan F. KarrNational Institute Statistical SciencesResearch Triangle Park, North Carolina

Al MardenUniversity of MinnesotaMinneapolis, Minnesota

William H. PressLos Alamos National LabLos Alamos, NM 87545


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 referencematerial available. Prior to the preparation of this 31 st Edition of the CRC StandardMathematical 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 votingpower, heuristic search techniques, quadratic elds, reliability, risk analysis and de-cision rules, a table of solutions to Pells equation, a table of irreducible polynomialsin , 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. Intotal, 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 adozen 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 than3,000 separate items (a de nition , a fact, a table, or a property). The index has alsobeen 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 consistentfrom page to page. Material is presented in a multi-sectional format, with each sec-tion containing a valuable collection of fundamental reference materialtabular andexpository.

In line with the established policy of CRC Press, the Handbook will be kept ascurrent and timely as is possible. Revisions and anticipated uses of newer materialsand tables will be introduced as the need arises. Suggestions for the inclusion of newmaterial in subsequent editions and comments regarding the present edition are wel-comed. The home page for this book, which will include errata, will be maintainedat

The major material in this new edition is as follows:

Chapter 1: Analysis begins with numbers and then combines them into series andproducts. 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 aredetails on polynomials and a separate section on number theory. This chapterincludes many new tables.

Chapter 3: Discrete Mathematics covers traditional discrete topics such as combi-natorics, graph theory, coding theory and information theory, operations re-


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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 tableof 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 thetrigonometric, exponential, and hyperbolic functions, and leading to many ofthe 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 andLaplace 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 (pointestimates, con d ence intervals, hypothesis testing, and ANOVA). Tables of thenormal distribution, and other distributions, are included. Also included in thischapter 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 tonumerically 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 amortgage). 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 electronicresources, 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, JanetTaylor and Kent Taylor Zwillinger.

Daniel Zwillinger

15 October 2002


Contributors

Karen BolingerClarion UniversityClarion, Pennsylvania

Patrick J. DriscollU.S. Military AcademyWest Point, New York

M. Lawrence GlasserClarkson UniversityPotsdam, New York

Jeff GoldbergUniversity of ArizonaTucson, Arizona

Rob GrossBoston CollegeChestnut Hill, Massachusetts

George W. HartSUNY Stony BrookStony Brook, New York

Melvin HausnerCourant Institute (NYU)New York, New York

Victor J. KatzMAAWashington, DC

Silvio LevyMSRIBerkeley, California

Michael MascagniFlorida State UniversityTallahassee, Florida

Ray McLenaghanUniversity of WaterlooWaterloo, Ontario, Canada

John MichaelsSUNY BrockportBrockport, New York

Roger B. NelsenLewis & Clark CollegePortland, Oregon

William C. RinamanLeMoyne CollegeSyracuse, New York

Catherine RobertsCollege of the Holy CrossWorcester, Massachusetts

Joseph J. RushananMITRE CorporationBedford, Massachusetts

Les ServiMIT Lincoln LaboratoryLexington, Massachusetts

Peter SherwoodInteractive Technology, Inc.Newton, Massachusetts

Neil J. A. SloaneAT&T Bell LabsMurray Hill, New Jersey

Cole SmithUniversity of ArizonaTucson, Arizona

Mike SousaVeridianAnn Arbor, Michigan

Gary L. StanekYoungstown State UniversityYoungstown, Ohio

Michael T. StraussHMENewburyport, Massachusetts

Nico M. TemmeCWIAmsterdam, The Netherlands

Ahmed I. ZayedDePaul UniversityChicago, Illinois


Table of Contents

Chapter 1Analysis

Karen Bolinger, M. Lawrence Glasser, Rob Gross, andNeil J. A. Sloane

Chapter 2Algebra

Patrick J. Driscoll, Rob Gross, John Michaels, Roger B.Nelsen, and Brad Wilson

Chapter 3Discrete Mathematics

Jeff Goldberg, Melvin Hausner, Joseph J. Rushanan, LesServi, and Cole Smith

Chapter 4Geometry

George W. Hart, Silvio Levy, and Ray McLenaghan

Chapter 5Continuous Mathematics

Ray McLenaghan and Catherine Roberts

Chapter 6Special Functions

Nico M. Temme and Ahmed I. Zayed

Chapter 7Probability and Statistics

Michael Mascagni, William C. Rinaman, Mike Sousa, andMichael T. Strauss

Chapter 8Scientific Computing

Gary Stanek

Chapter 9Financial Analysis

Daniel Zwillinger

Chapter 10Miscellaneous

Rob Gross, Victor J. Katz, and Michael T. Strauss


Table of Contents

Chapter 1Analysis

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 2Algebra

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 3Discrete Mathematics

3.1 Symbolic logic3.2 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 chaos . . . . . . . . . . . . . . .3.9 Game theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.10 Operations research . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 4Geometry

4.1 Coordinate systems in the plane . . . . . . . . . . . . . . . . . . .4.2 Plane symmetries or isometries . . . . . . . . . . . . . . . . . . .4.3 Other transformations of the plane . . . . . . . . . . . . . . . . .4.4 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


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 5Continuous 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 6Special 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 . . . . . . . . . . . . . . . . . . . . . .


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 ClebschGordan 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 7Probability 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 8Scienti c Computing

8.1 Basic numerical analysis . . . . . . . . . . . . . . . . . . . . .8.2 Numerical linear algebra . . . . . . . . . . . . . . . . . . . . . .


8.3 Numerical integration and differentiation . . . . . . . . . . . . . .8.4 Programming techniques . . . . . . . . . . . . . . . . . . . . . .

Chapter 9Financial Analysis

9.1 Financial formulae . . . . . . . . . . . . . . . . . . . . . . . . .9.2 Financial tables . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 10Miscellaneous

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


List of References

Chapter 1 Analysis

1. J. W. Brown and R. V. Churchill, Complex variables and applications,6th edition, McGrawHill, 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, BocaRaton, FL, 1995.

5. J. P. Lambert, Voting Games, Power Indices, and Presidential Elec-tions, The UMAP Journal, Module 690, 9, No. 3, pages 214267, 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 441448, 2002.

2. I. N. Herstein, Topics in Algebra, 2nd edition, John Wiley & Sons, NewYork, 1975.

3. P. Ribenboim, The book of Prime Number Records, SpringerVerlag,New York, 1988.

4. G. Strang, Linear Algebra and Its Applications, 3rd edition, InternationalThomson Publishing, 1988.

Chapter 3 Discrete Mathematics

1. B. Bollobas, Graph Theory, SpringerVerlag, Berlin, 1979.


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 7494,1990.

4. D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Ma-chine Learning, AddisonWesley, Reading, MA, 1989.

5. J. Gross, Handbook of Graph Theory & Applications, CRC Press, BocaRaton, 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-CorrectingCodes, NorthHolland, 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 10871092, 1953.

9. K. H. Rosen, Handbook of Discrete and Combinatorial Mathematics,CRC Press, Boca Raton, FL, 2000.

10. J. ORourke 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, CRCPress, 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, Greens 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, MarcelDekker, 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.


7. J. A. Schouten, Ricci-Calculus, SpringerVerlag, 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., AcademicPress, 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. Erdelyi, Ed., Tables of Inte-gral Transforms, in 3 volumes, McGrawHill, 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 forthe Special Functions of Mathematical Physics, SpringerVerlag, NewYork, 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 Littles Lawand the FuhrmannCooper Decomposition, Operations Research Let-ters, Volume 9, pages 237247, 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 arehard to nd, Comm. ACM, October 1988, 31, 10, pages 11921201.

6. G. Strang and T. Nguyen, Wavelets and Filter Banks, WellesleyCambridgePress, Wellesley, MA, 1995.

7. D. Zwillinger and S. Kokoska, Standard Probability and Statistics Tablesand 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., TheJohns Hopkins Press, Baltimore, 1989.


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, 2ndedition, McGrawHill, New York, 1978.

5. R. Rubinstein, Simulation and the Monte Carlo Method, Wiley, NewYork, 1981.

Chapter 10 Miscellaneous

1. American Mathematical Society, Mathematical Sciences Professional Di-rectory, Providence, 1995.

2. E. T. Bell, Men of Mathematics, Dover, New York, 1945.

3. C. C. Gillispie, Ed., Dictionary of Scientific Biography, Scribners, NewYork, 19701990.

4. H. S. Tropp, The Origins and History of the Fields Medal, HistoriaMathematica, 3, pages 167181, 1976.

5. E. W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press,Boca Raton, FL, 1999.


List of Figures

2.1 Depiction of right-hand rule

3.1 Hasse diagrams3.2 Three graphs that are isomorphic3.3 Examples of graphs with 6 or 7 vertices3.4 Trees with 7 or fewer vertices3.5 Trees with 8 vertices3.6 Julia sets3.7 The Mandlebrot set3.8 Directed network modeling a flow problem

4.1 Change of coordinates by a rotation4.2 Cartesian coordinates: the 4 quadrants4.3 Polar coordinates4.4 Homogeneous coordinates4.5 Oblique coordinates4.6 A shear with factor

4.7 A perspective transformation4.8 The normal form of a line4.9 Simple polygons4.10 Notation for a triangle4.11 Triangles: isosceles and right4.12 Cevas theorem and Menelauss theorem4.13 Quadrilaterals4.14 Conics: ellipse, parabola, and hyperbola4.15 Conics as a function of eccentricity4.16 Ellipse and components4.17 Hyperbola and components4.18 Arc of a circle4.19 Angles within a circle4.20 The general cubic parabola4.21 Curves: semi-cubic parabola, cissoid of Diocles, witch of Agnesi4.22 The folium of Descartes in two positions, and the strophoid


4.23 Cassinis ovals4.24 The conchoid of Nichomedes4.25 The limacon of Pascal4.26 Cycloid and trochoids4.27 Epicycloids: nephroid, and epicycloid4.28 Hypocycloids: deltoid and astroid4.29 Spirals: Bernoulli, Archimedes, and Cornu4.30 Cartesian coordinates in space4.31 Cylindrical coordinates4.32 Spherical coordinates4.33 Relations between Cartesian, cylindrical, and spherical coordinates4.34 Euler angles4.35 The Platonic solids4.36 Cylinders: oblique and right circular4.37 Right circular cone and frustram4.38 A torus of revolution4.39 The ve nondegenerate real quadrics4.40 Spherical cap, zone, and segment4.41 Right spherical triangle and Napiers rule

5.1 Types of critical points

6.1 Notation for trigonometric functions6.2 Definitions of angles6.3 Sine and cosine6.4 Tangent and cotangent6.5 Different triangles requiring solution6.6 Graphs of and . . . . . . . . . . . . . . . . 5416.7 Cornu spiral6.8 Sine and cosine integrals and 5496.9 Legendre functions6.10 Graphs of the Airy functions and 566

7.1 Approximation to binomial distributions7.2 Conceptual layout of a queue7.3 Sample size code letters for MIL-STD-105 D7.4 Master table for single sampling inspection (normal inspection)7.5 Area of a normal random variable7.6 Illustration of and regions of a normal distribution

8.1 Illustration of Newtons method8.2 Formulae for integration rules with various weight functions8.3 Illustration of the MonteCarlo method


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 . . . . . . . . . . . . . . 228graph union . . . . . . . . . . . . . . . . . .229set 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 . . 399integration symbol . . . . . . . . . . . 399

falling factorial . . . . . . . . . . . . . . . . . .17 logical not . . . . . . . . . . . . . . . . . . . . . . 199

partial differentiation . . . . . . . . . . . . 386 dual code to . . . . . . . . . . . . . . . . 257 partial order . . . . . . . . . . . . . . . . . . . . 204

product symbol . . . . . . . . . . . . . . . . . . 47summation symbol . . . . . . . . . . . . . . 31

empty set . . . . . . . . . . . . . . . . . . . . . . . 202

asymptotic relation . . . . . . . . . . . . 75logical not . . . . . . . . . . . . . . . . . . . 199vertex similarity . . . . . . . . . . . . . .226

logical or . . . . . . . . . . . . . . . . . . . . 199pseudoscalar product . . . . . . . . . 467

graph conjunction . . . . . . . . . . . . 228logical and . . . . . . . . . . . . . . . . . . 199wedge product . . . . . . . . . . . . . . . 395

divergence . . . . . . . . . . . . . . . 493 curl . . . . . . . . . . . . . . . . . . . . .493 Laplacian . . . . . . . . . . . . . . . . 493backward difference . . . . . . . . . . 736gradient . . . . . . . . . . . . . . . . 390, 493linear connection . . . . . . . . . . . . . 484

[ ] graph composition . . . . 228 commutator . . . . . . . . . 155, 467vuw scalar triple product . . . . 136 continued fraction . 96 Christoffel symbol, rst kind

487

Stirling cycle numbers . . . . 212


*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

ClebschGordan

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 . . . . . . . . . 659arithmetic mean . . . . . . . . . . . . . . 659complex conjugate . . . . . . . . . . . . 54set complement . . . . . . . . . . . . . . 203

divisibility . . . . . . . . . . . . . . . . . . . . . . . . 93

determinant of a matrix . . . . . . . 144graph order . . . . . . . . . . . . . . . . . . 226norm . . . . . . . . . . . . . . . . . . . . . . . . 133order of algebraic structure . . . . 160polynomial norm . . . . . . . . . . . . . . 91used in tensor notation . . . . . . . . 487

norm . . . . . . . . . . . . . . . . 133 norm . . . . . . . . . . . . . . . . 133 Frobenius norm . . . . . . . . . 146 in nity norm . . . . . . . . . . . 133norm . . . . . . . . . . . . . . . . . . . . 91, 133

a b vector inner product . . . . . 133group operation . . . . . . . . . . . . . . 161inner product . . . . . . . . . . . . . . . . 132

crystallographic group . . . 309, 311degrees in an angle . . . . . . . . . . . 503function composition . . . . . . . . . . 67temperature degrees . . . . . . . . . . 798translation . . . . . . . . . . . . . . . . . . . 307

2222 crystallographic group . . 310333 crystallographic group . . . 311442 crystallographic group . . . 310632 crystallographic group . . . 311 crystallographic group . . . . . 309binary operation . . . . . . . . . . . . . .160convolution operation . . . . . . . . .579dual of a tensor . . . . . . . . . . . . . . 489group operation . . . . . . . . . . . . . . 161re ection . . . . . . . . . . . . . . . . . . . . 307

a b vector cross product . . . . 135 crystallographic group . . . . . 309 crystallographic group . . . . 309glide-re ection . . . . . . . . . . . . . . .307graph product . . . . . . . . . . . . . . . . 228group operation . . . . . . . . . . . . . . 161product . . . . . . . . . . . . . . . . . . . . . . . 66

Kronecker product . . . . . . . . . . . 159symmetric difference . . . . . . . . . 203

exclusive or . . . . . . . . . . . . . . . . . .645factored graph . . . . . . . . . . . . . . . 224graph edge sum . . . . . . . . . . . . . . 228Kronecker sum . . . . . . . . . . . . . . .160

continuedfraction . . . . . . . . . . . . . . . . . 97

graph join . . . . . . . . . . . . . . . . . . . 228group operation . . . . . . . . . . . . . . 161pseudo-inverse operator . . 149, 151vector addition . . . . . . . . . . . . . . . 132

Greek Letters

maximum vertex degree 223 change in the

argument . . . . . . . . . . . . . . . . 58forward difference . . . . . . . 265, 728Laplacian . . . . . . . . . . . . . . . . . . . .493

gamma function . . . . . . . . 540

Christoffel symbol of secondkind . . . . . . . . . . . . . . . . . . . 487

connection coef cients . . . . . 484

asymptotic function . . . . . . . . . . . 75ohm . . . . . . . . . . . . . . . . . . . . . . . . 792


normal distribution function . . .634 asymptotic function . . . . . . . . . . . . . . 75 graph arboricity . . . . . . . . . . . . . 220

graph independence number225

function, related to zetafunction . . . . . . . . . . . . . . . . . 23

one minus the con dencecoef 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 . . . . . . . . . . . 483designed distance . . . . . . . . . . . . 257Feigenbaums constant . . . . . . . .272

LeviCivita symbol . . . . . . . . .489

power of a test . . . . . . . . . . . . . 661 component of in nitesimal

generator . . . . . . . . . . . . . . . 466

Eulers constantde nition . . . . . . . . . . . . . . . . . . 15in different bases . . . . . . . . . . . 16value . . . . . . . . . . . . . . . . . . . . . . 16

graph genus . . . . . . . . . . . . 224 function, related to zeta

function . . . . . . . . . . . . . . . . . 23 skewness . . . . . . . . . . . . . . . . . 620 excess . . . . . . . . . . . . . . . . . . . . 620

!! connectivity . . . . . . . . . . . . 222! curvature . . . . . . . . . . . . . . . 374! cumulant . . . . . . . . . . . . . . . . . 620

edge connectivity . . . . . . . 223average arrival rate . . . . . . . . . . . 638eigenvalue . . . . . . . . . . 152, 477, 478number of blocks . . . . . . . . . . . . .241

"" Mobius function . . . . . . . . 102" centered moments . . . . . . . . . 620" moments . . . . . . . . . . . . . . . . . 620" MTBF for parallel system . . 655" MTBF for series system . . . 655average service rate . . . . . . . . . . .638mean . . . . . . . . . . . . . . . . . . . . . . . .620

## rectilinear graph crossing

number . . . . . . . . . . . . . . . . 222# graph crossing number . . 222

$ size of the largest clique . . . . . . 221%

% totient function . . . . . 128, 169% characteristic function . . . .620Euler constant . . . . . . . . . . . . . . . . .21golden ratio

de ned . . . . . . . . . . . . . . . . . . . . 16value . . . . . . . . . . . . . . . . . . . . . . 16

incidence mapping . . . . . . . . . . . 219zenith . . . . . . . . . . . . . . . . . . . . . . . 346

prime counting function . . . . 103probability distribution . . . . . 640

constants containing . . . . . . . . . . . 14continued fraction . . . . . . . . . . . . . 97distribution of digits . . . . . . . . . . . 15identities . . . . . . . . . . . . . . . . . . . . . 14number . . . . . . . . . . . . . . . . . . . . . . . 13

in different bases . . . . . . . . . . . 16permutation . . . . . . . . . . . . . . . . . 172sums involving . . . . . . . . . . . . . . . . 24

& logarithmic derivative of thegamma function . . . . . . . . .543

'' spectral radius . . . . . . . . . . 154' radius of curvature . . . . . . . 374' correlation coef cient . . . . . 622server 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


)) Ramanujan function . . . . . . . . . 31) number of divisors . . . . . . 128) torsion . . . . . . . . . . . . . . . . . 374

** graph thickness . . . . . . . . . 227angle in polar coordinates . . . . . 302argument of a complex number . 53azimuth . . . . . . . . . . . . . . . . . . . . . 346

++ component of in nitesimal

generator . . . . . . . . . . . . . . . 466+ quantile of order , . . . . . . . . . 659

- Riemann zeta function . . . . . . . . . 23

Numbers group inverse . . . . . . . . . . . . . . . . 161 matrix inverse . . . . . . . . . . . . . . . . 1380 null vector . . . . . . . . . . . . . . . . . . . . . . 1371

1, group identity . . . . . . . . . . . . . 1611-form . . . . . . . . . . . . . . . . . . . . . . 39510, powers of . . . . . . . . . . 6, 13, 798105 D standard . . . . . . . . . . . . . . .65216, powers of . . . . . . . . . . . . . . . . . 1217 crystallographic groups . . . . 307

2 power set of . . . . . . . . . . . . 203222 crystallographic group . . . 3102, negative powers of . . . . . . . . . . 102, powers of . . . . . . . . . . . . . 6, 10, 272-(,3,1) Steiner triple system . 2492-form . . . . . . . . . . . . . . . . . . . . . . 3962-sphere . . . . . . . . . . . . . . . . . . . . . 4912-switch . . . . . . . . . . . . . . . . . . . . . 22722 crystallographic group . . . . 30922 crystallographic group . . . .3092222 crystallographic group . . . 310230 crystallographic groups,

three-dimensional . . . . . . . 3073

33 crystallographic group . . . . 3113, powers of . . . . . . . . . . . . . . . . . . 293-design (Hadamard matrices) . 2503-form . . . . . . . . . . . . . . . . . . . . . . 3973-sphere . . . . . . . . . . . . . . . . . . . . . 491333 crystallographic group . . . . 311360, degrees in a circle . . . . . . . 503

442 crystallographic group . . . . 3104, powers of . . . . . . . . . . . . . . . . . . 30442 crystallographic group . . . . 310

55, powers of . . . . . . . . . . . . . . . . . . 305-(12,6,1) table . . . . . . . . . . . . . . 2445-design, Mathieu . . . . . . . . . . . . 244

632 crystallographic group . . . . . . . . . 311

Roman LettersA

A interarrival time . . . . . . . . . . . .637 . number of codewords . 259/ skew symmetric part of a

tensor . . . . . . . . . . . . . . . . . . 484A ampere . . . . . . . . . . . . . . . . . . . .792

alternating group on 4 elements

188 radius of circumscribed circle

324 alternating group . . . . . 163, 172

010203004 queue . . . . . . . . . . . 637 Airy function . . . . . . . . . . . 465, 565ALFS additive lagged-Fibonacci

sequence . . . . . . . . . . . . . . . 646AMS American Mathematical Society

801ANOVA analysis of variance . . . . . . . 686AOQ average outgoing quality . . . . . . 652AOQL average outgoing quality limit 652AQL acceptable quality level . . . . . . . 652AR autoregressive model . . . . . . . 718ARMA 5 mixed model . . . . . . . . . 719 graph automorphism group .220

a unit vector . . . . . . . . . . . . . . . . 492 Fourier coef cients . . . . . . . . . 48 proportion of customers . . . .637

6 almost everywhere . . . . . . . . . . . . . 74am amplitude . . . . . . . . . . . . . . . . . . . . . 572arg argument . . . . . . . . . . . . . . . . . . . . . . . 53


BB

1 amount borrowed . . . . . . . . . . 7791 service time . . . . . . . . . . . . . . . 6371, 7 beta function . . . . . . . . . 544 set of blocks . . . . . . . . . . . . . . . 241

1 1 Bell number . . . . . . . . . . . . . .2111 Bernoulli number . . . . . . . . . . 191 a block . . . . . . . . . . . . . . . . . . 2411 Bernoulli polynomial . . . 19

B.C.E (before the common era, B.C.) 810BFS basic feasible solution . . . . . . . . . 283 Airy function . . . . . . . . . . . 465, 565BIBD balanced incomplete block design

245Bq becquerel . . . . . . . . . . . . . . . . . . . . . 792b unit binormal vector . . . . . . . . . . . . . 374

CC

channel capacity . . . . . . . . . . . 255 -combination . . . 206, 215 Fresnel integral . . . . . . . . . 547 combinations with

replacement . . . . . . . . . . . . 206 complex numbers . . . . . . . . 3, 167 complex element vectors 131 integration contour . . . . . 399, 404C coulomb . . . . . . . . . . . . . . . . . . 792C 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

cc cardinality of real numbers . . 2042 number of identical servers . . 6372 speed of light . . . . . . . . . . . . . . .794

cas combination of sin and cos . . . . . .591cd candela . . . . . . . . . . . . . . . . . . . . . . . . 792cm crystallographic group . . . . . . . . . . 309cmm crystallographic group . . . . . . . . 3102 Fourier coef cients . . . . . . . . . . . . . . 508 elliptic function . . . . . . . . . . . 572cof cofactor of matrix . . . . . . 145cond() condition number . . . . . . . . . 148cos trigonometric function . . . . . . . . . 505cosh hyperbolic function . . . . . . . . . . . 524cot trigonometric function . . . . . . . . . . 505coth hyperbolic function . . . . . . . . . . . 524covers trigonometric function . . . . . . .505csc trigonometric function . . . . . . . . . .505csch hyperbolic function . . . . . . . . . . . 524cyc number of cycles . . . . . . . . . . . . . . 172

DD

9 constant service time . . . . . . . 6379 diagonal matrix . . . . . . . . . . . .1389 differentiation operator 456, 466D Roman numeral (500) . . . . . . . . 4

9 9 dihedral group of order 8 . . 1829 dihedral group of order 10 . 1859 dihedral group of order 12 . 1869 region of convergence . . . . . 5959 derangement . . . . . . . . . . . . . 2109 dihedral group . . . . . . . 163, 172

DFT discrete Fourier transform . . . . . 582DLG double loop graph . . . . 230.

.8 distance between vertices223

derivative operator . . . . . . . . . . . 386exterior derivative . . . . . . . . . . . . 397minimum distance . . . . . . . . . . . .256

. . proportion of customers . . . .637.Hu v Hamming distance . . . 256.a projection . . . . . . . . . . . . 395

determinant of matrix . . . . 144 graph diameter . . . . . . . . . . .223div divergence . . . . . . . . . . . . . . . . . . . . 4938 elliptic function . . . . . . . . . . 572


. differential surface area . . . . . . . . . 405

.: differential volume . . . . . . . . . . . . 405

.x fundamental differential . . . . . . . . . 377

EE

; edge set . . . . . . . . . . . . . . . . . . . 219; event . . . . . . . . . . . . . . . . . . . . . 617;8 rst fundamental metric

coef cient . . . . . . . . . . . . . . 377E expectation operator . . . . . . 619

; ; Erlang- service time . . . . . 637; Euler numbers . . . . . . . . . . . . . 20; Euler polynomial . . . . . . . 20; exponential integral . . . . 550; identity group . . . . . . . . . . . . 172; elementary matrix . . . . . . . . 138

Ei exbi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6e

6 algebraic identity . . . . . . . . . . . 1616 charge of electron . . . . . . . . . . 7946 constants containing . . . . . . . . . 156 continued fraction . . . . . . . . . . . 976 de nition . . . . . . . . . . . . . . . . . . . 156 eccentricity . . . . . . . . . . . . . . . . 3256 in different bases . . . . . . . . . . . . 1668 second fundamental metric

coef cient . . . . . . . . . . . . . . 3776

e vector of ones . . . . . . . . . . . . . . 137e unit vector . . . . . . . . . . . . . . . . 1376 permutation symbol . . . 489

ecc eccentricity of a vertex . . . . . . 223erf error function . . . . . . . . . . . . . . . . . . 545erfc complementary error function . . 545exsec trigonometric function . . . . . . . 505

FF

< 8 rst fundamental metriccoef cient . . . . . . . . . . . . . . 377

< Dawsons integral . . . . . . .546< probability distribution

function . . . . . . . . . . . . . . . . 619 Fourier transform . . . . . . . . . . 576< 2 hypergeometric

function . . . . . . . . . . . . . . . . 553< sample distribution function658

F farad . . . . . . . . . . . . . . . . . . . . . . 792

> primitive root . . . . . . . . . . . . . . 195> Warings problem . . . . . . . 100> generating polynomial . . . 256

gd function . . . . . . . . . . . . . . . . . . . . . . . 530 Gudermannian function . . . . . . . 530> covariant metric . . . . . . . . . . . . . . . 486> contravariant metric . . . . . . . . . . . . 486glb greatest lower bound . . . . . . . . . . . . 68

HH

? mean curvature . . . . . . . . . . . . 377? parity check matrix . . . . . . . . 256?p entropy . . . . . . . . . . . . . . 253? Haar wavelet . . . . . . . . . . . 723? Heaviside function . . 77, 408" Hilbert transform . . . . . . . . . . 591H Hermitian conjugate . . . . . . . . 138H henry . . . . . . . . . . . . . . . . . . . . . 792

? ? null hypothesis . . . . . . . . . . . 661? alternative hypothesis . . . . . 661

? Hankel function . . . . . . . . .559

? Hankel function . . . . . . . . .559

? -stage hyperexponentialservice time . . . . . . . . . . . . 637

? harmonic numbers . . . . . . . . . 32? Hermite polynomials . . 532" Hankel transform . . . . . . . . . 589

H.M. harmonic mean . . . . . . . . . . . . . . 660Hz hertz . . . . . . . . . . . . . . . . . . . . . . . . . . 792hav trigonometric function . . . . . 372, 505@ metric coef cients . . . . . . . . . . . . . . 492

II

= rst fundamental form . . . . . . 377= identity matrix . . . . . . . . . . . . . 138=AB mutual information . . 254I Roman numeral (1) . . . . . . . . . . . . 4

ICG inversive congruential generator 646== second fundamental form . . . . . . . 377Im imaginary part of a complex number

53= identity matrix . . . . . . . . . . . . . . . . . 138Inv number of invariant elements . . . 172IVP initial-value problem . . . . . . . . . . 265

ii unit vector . . . . . . . . . . . . . . . . . .494i unit vector . . . . . . . . . . . . . . . . . .135C imaginary unit . . . . . . . . . . . . . . . 53C interest rate . . . . . . . . . . . . . . . . 779

iid independent and identicallydistributed . . . . . . . . . . . . . . 619

inf greatest lower bound . . . . . . . . . . . . .68in mum greatest lower bound . . . . . . . 68

JJ

D Jordan form . . . . . . . . . . . . . . . 154J joule . . . . . . . . . . . . . . . . . . . . . . 792

jj unit vector . . . . . . . . . . . . . . . . . 494j unit vector . . . . . . . . . . . . . . . . . 135

D D Bessel function . . . . . . . . 559D Julia set . . . . . . . . . . . . . . . . . . 273

half order Bessel function

563 zero of Bessel function . . . 563

KK

3 Gaussian curvature . . . . . . . . 3773 system capacity . . . . . . . . . . . 637K Kelvin (degrees) . . . . . . . . . . . 792

3 3 complete graph . . . . . . . . . . .2293 complete bipartite graph 2303 complete multipartite

graph . . . . . . . . . . . . . . . . . . 2303 empty graph . . . . . . . . . . . . . 229

Ki kibi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6k

k curvature vector . . . . . . . . . . . . 374k unit vector . . . . . . . . . . . . . . . . . 494k unit vector . . . . . . . . . . . . . . . . . 135 Boltzmann constant . . . . . . . . .794 dimension of a code . . . . . . . . 258 kernel . . . . . . . . . . . . . . . . 478

k geodesic curvature . . . . . . . . 377k normal curvature vector . . . .377 block size . . . . . . . . . . . . . . . . .241

kg kilogram . . . . . . . . . . . . . . . . . . . . . . 792


LL

average number of customers 638 period . . . . . . . . . . . . . . . . . . . . . . 48* expected loss function . . . 656# Laplace transform . . . . . . . . . . 585L length . . . . . . . . . . . . . . . . . . . . . 796L Roman numeral (50) . . . . . . . . . . 4

norm . . . . . . . . . . . . . . . . . . . . 133 norm . . . . . . . . . . . . . . . . . . . . 133 average number of customers

638 norm . . . . . . . . . . . . . . . . . . . . . .73 Lie group . . . . . . . . . . . . . . . . 466 space of measurable functions

73LCG linear congruential generator . . 644LCL lower control limit . . . . . . . . . . . . 650LCM least common multiple . . . . . . . 101 logarithm . . . . . . . . . . . . . . . . . . 551 dilogarithm . . . . . . . . . . . . . . . . 551LIFO last in, rst out . . . . . . . . . . . . . . 637 polylogarithm . . . . . . . . . . . . . . 551 logarithmic integral . . . . . . . . . . .550LP linear programming . . . . . . . . . . . . 280LTPD lot tolerance percent defective 652

* loss function . . . . . . . . . . . . . . . 656lim limits . . . . . . . . . . . . . . . . . . . . . 70, 385liminf limit inferior . . . . . . . . . . . . . . . . . 70limsup limit superior . . . . . . . . . . . . . . . . 70lm lumen . . . . . . . . . . . . . . . . . . . . . . . . . 792ln logarithmic function . . . . . . . . . . . . .522log logarithmic function . . . . . . . . . . . 522 logarithm to base . . . . . . . . . . . . 522lub least upper bound . . . . . . . . . . . . . . . 68lux lux . . . . . . . . . . . . . . . . . . . . . . . . . . . 792

MM

E Mandelbrot set . . . . . . . . . . . . 273E exponential service time . . . 637E number of codewords . . . . . . 258EF measure of a polynomial 93$ Mellin transform . . . . . . . . . . 612M mass . . . . . . . . . . . . . . . . . . . . . 796M Roman numeral (1000) . . . . . . . 4

MA5 moving average . . . . . . . . . . . . 719M.D. mean deviation . . . . . . . . . . . . . . 660

MFLG multiplicative lagged-Fibonaccigenerator . . . . . . . . . . . . . . . 646

E00! queue . . . . . . . . . . . . . . . . . . . . 639E00202 queue . . . . . . . . . . . . . . . . . . 639E00 queue . . . . . . . . . . . . . . . . . . . 639Mi mebi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6MLE maximum likelihood estimator 662E0E0! queue . . . . . . . . . . . . . . . . . . . 638E0E02 queue . . . . . . . . . . . . . . . . . . . .639E Mobius ladder graph . . . . . . . . . . 229MOLS mutually orthogonal Latin

squares . . . . . . . . . . . . . . . . .251MOM method of moments . . . . . . . . . 662MTBF mean time between failures . . 655m

mortgage amount . . . . . . . . . . 779 number in the source . . . . . . . 637m meter . . . . . . . . . . . . . . . . . . . . . 792

mid midrange . . . . . . . . . . . . . . . . . . . . . 660mod modular arithmetic . . . . . . . . . . . . . 94mol mole . . . . . . . . . . . . . . . . . . . . . . . . . 792

NN

G number of zeros . . . . . . . . . . . . 58G null space . . . . . . . . . . . . . 149G" ( normal random variable

619N unit normal vector . . . . . . . . . .378% normal vector . . . . . . . . . . . . . 377 natural numbers . . . . . . . . . . . . . . 3N newton . . . . . . . . . . . . . . . . . . . . 792

G number of monic irreduciblepolynomials . . . . . . . . . . . . 261

nn principal normal unit vector . 374n unit normal vector . . . . . . . . . . 135 code length . . . . . . . . . . . . . . . . 258 number of time periods . . . . . 779 order of a plane . . . . . . . . . . . . 248

OH asymptotic function . . . . . . . . . . . . . . 75H matrix group . . . . . . . . . . . . . . . . 171H odd graph . . . . . . . . . . . . . . . . . . . . . 229I asymptotic function . . . . . . . . . . . . . . . 75


PP

F number of poles . . . . . . . . . . . . 58F principal . . . . . . . . . . . . . . . . . . 779F 1 conditional probability

617F ; probability of event ; . . 617F # auxiliary function . . . . . 561F -permutation . . . . . . . 215F -permutation . . . . . . . . 206F Markov transition function

640F& ' Riemann F function . . . . . 465

F F chromatic polynomial . . 221F path (type of graph) . . . . . . . 229F Lagrange interpolating

polynomial . . . . . . . . . . . . . 733F Legendre function . . . . . 465F Legendre polynomials . . 534F Jacobi polynomials . 533

F Legendre function . . . . . .554F associated Legendre

functions . . . . . . . . . . . . . . . 557Pa pascal . . . . . . . . . . . . . . . . . . . . . . . . . 792Per period of a sequence . . . . . . . 644Pi pebi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6PID principal ideal domain . . . . . . . . . 165F -step Markov transition

matrix . . . . . . . . . . . . . . . . . 641F permutations with replacement

206PRI priority service . . . . . . . . . . . . . . . .637PRNG pseudorandom number generator

644p

, partitions . . . . . . . . . . . . . . . 210," product of prime numbers . 106

p1 crystallographic group . . . . . . 309, 311p2 crystallographic group . . . . . . . . . . 310p3 crystallographic group . . . . . . . . . . 311p31m crystallographic group . . . . . . . 311p3m1 crystallographic group . . . . . . . 311p4 crystallographic group . . . . . . . . . . 310p4g crystallographic group . . . . . . . . . 310p4m crystallographic group . . . . . . . . 310p6 crystallographic group . . . . . . . . . . 311p6m crystallographic group . . . . . . . . 311per permanent . . . . . . . . . . . . . . . . . . . . 145pg crystallographic group . . . . . . . . . . 309

pgg crystallographic group . . . . . . . . . 309pm crystallographic group . . . . . . . . . .309pmg crystallographic group . . . . . . . . 309pmm crystallographic group . . . . . . . . 310,

p joint probability distribution254

, discrete probability . . . . . . . . 619, partitions . . . . . . . . . . . . . . 207, restricted partitions . . . . 210, proportion of time . . . . . . . . . 638

QQ

J quaternion group . . . . . . . . . . 182J# auxiliary function . . . . . 561 rational numbers . . . . . . . . 3, 167

J J cube (type of graph) . . . . . . 229J Legendre function . . . . . 465J Legendre function . . . . . 554

J associated Legendre functions557

7 nome . . . . . . . . . . . . . . . . . . . . . . . . . . . 574

RR

K Ricci tensor . . . . . . . . . . . 485, 488K Riemann tensor . . . . . . . . . . . . 488K curvature tensor . . . . . . . . . . . 485K radius (circumscribed circle) 319,

513K range . . . . . . . . . . . . . . . . . . . . . 650K rate of a code . . . . . . . . . . . . . . 255K range space . . . . . . . . . . . . 149K* . risk function . . . . . . . . .657K reliability function . . . . . . 655 continuity in . . . . . . . . . . . . . . . . 71 convergence in . . . . . . . . . . . . . .70 real numbers . . . . . . . . . . . . 3, 167

K K reliability of a component . . 653K reliability of parallel system 653K reliability of series system . 653K radius of the earth . . . . . . . . 372

real element vectors . . . . . . . . . .131 real matrices . . . . . . . . 137Re real part of a complex number . . . .53R.M.S. root mean square . . . . . . . . . . . 660


RSS random service . . . . . . . . . . . . . . . 637r

distance in polar coordinates . 302 modulus of a complex number 53 radius (inscribed circle) . 318, 512 shearing factor . . . . . . . . . . . . . 352* regret function . . . . . . . . 658

radius of graph . . . . . . . . . . 226rad radian . . . . . . . . . . . . . . . . . . . . . . . . 792 replication number . . . . . . . . . . . . . . 241 Rademacher functions . . . . . . . 722

SS

sample space . . . . . . . . . . . . . . 617 torsion tensor . . . . . . . . . . . . . . 485 Fresnel integral . . . . . . . . . 547 symmetric group . . . . . . . . . . 163 Stirling number second

kind . . . . . . . . . . . . . . . . . . . 213(/ symmetric part of a tensor

484S siemen . . . . . . . . . . . . . . . . . . . . 792

symmetric group . . . . . . . . . . 180 area of inscribed polygon . . 324 star (type of graph) . . . . . . . . 229 symmetric group . . . . . . . . . . 172 surface area of a sphere . 368

SA simulated annealing . . . . . . . . . . . . 291SI Systeme Internationale dUnites . . 792# sine integral . . . . . . . . . . . . . . . . . 549 matrix group . . . . . . . . . . . . 171 matrix group . . . . . . . . . . . . 171H matrix group . . . . . . . . . . . . . . . 172H matrix group . . . . . . . . . . . . . . .172SPRT sequential probability ratio test 681SRS shift-register sequence . . . . . . . . 645STS Steiner triple system . . . . . . . . . . 249L matrix group . . . . . . . . . . . . . . .172SVD singular value decomposition . . 156s

Stirling number rst kind213

arc length parameter . . . . . . . . 373

sample standard deviation . . . 660

semi-perimeter . . . . . . . . . . . . . 512s second . . . . . . . . . . . . . . . . . . . . . 792

area of circumscribed polygon

324

elementary symmetric functions

84sec trigonometric function . . . . . . . . . .505sech hyperbolic function . . . . . . . . . . . 524sgn signum function . . . . . . . . . . . .77, 144sin trigonometric function . . . . . . . . . . 505sinh hyperbolic function . . . . . . . . . . . 524$8 elliptic function . . . . . . . . . . . 572sr steradian . . . . . . . . . . . . . . . . . . . . . . . 792sup least upper bound . . . . . . . . . . . . . . . 68supremum least upper bound . . . . . . . . 68

TT

T transpose . . . . . . . . . . . . . . . . . . 131T tesla . . . . . . . . . . . . . . . . . . . . . . 792T time interval . . . . . . . . . . . . . . . 796transpose . . . . . . . . . . . . . . . . . . . . 138

/ / Chebyshev polynomials 534/ isomorphism class of trees

241Ti tebi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6TN Toeplitz network . . . . . . . . . 230 trace of matrix . . . . . . . . . . . . 150- design nomenclature . . . . . 241

critical value . . . . . . . . . . . . . . 695 ! transition probabilities . . . . 255

tan trigonometric function . . . . . . . . . . 505tanh hyperbolic function . . . . . . . . . . . 524t unit tangent vector . . . . . . . . . . . . . . . 374

UU

L universe . . . . . . . . . . . . . . . . . . 201L matrix group . . . . . . . . . . . 172L uniform random variable

619L Chebyshev polynomials 535

UCL upper control limit . . . . . . . . . . . 650UFD unique factorization domain . . . 165UMVU type of estimator . . . . . . . . . . .663URL Uniform Resource Locators . . . 8038 traf c intensity . . . . . . . . . . . . . . . . . . 6388 unit step function . . . . . . . . . . . . 5958 distance . . . . . . . . . . . . . . . . . . . . . . . 492


VV

: Klein four group . . . . . . . . . . . 179: vertex set . . . . . . . . . . . . . . . . . 219V Roman numeral (5) . . . . . . . . . . . 4V volt . . . . . . . . . . . . . . . . . . . . . . . 792

% vector operation . . . . . . . . . . . . . . .158: volume of a sphere . . . . . . . . . . 368vers trigonometric function . . . . . . . . . 505

WW

M average time . . . . . . . . . . . . . . 638M 8 Wronskian . . . . . . . . . . 462W watt . . . . . . . . . . . . . . . . . . . . . . 792

M M root of unity . . . . . . . . . . . . .582M average time . . . . . . . . . . . . .638M wheel (type of graph) . . . . . 229M Walsh functions . . . . . . .722

Wb weber . . . . . . . . . . . . . . . . . . . . . . . . 792

XX

A in nitesimal generator . . . . . 466A set of points . . . . . . . . . . . . . . . 241X Roman numeral (10) . . . . . . . . . .4

A rst prolongation . . . . . . . . . . . . . 466A second prolongation . . . . . . . . . . 466 C

th order statistic . . . . . . . . . . . . . . 659 rectangular coordinates . . . . . . . . . .492

YB Bessel function . . . . . . . . . . . . . 559

" homogeneous solution . . 456 half order Bessel function

563 particular solution . . . . . . 456 zero of Bessel function . . . 563

ZZ

4 queue discipline . . . . . . . . . . . 6374 center of a graph . . . . . . . 2214 instantaneous hazard rate .655 integers . . . . . . . . . . . . . . . . . 3, 167) 4-transform . . . . . . . . . . . . . . . 594

4 4 4 semidirect group product

187 integers modulo . . . . . . . . 167 a group . . . . . . . . . . . . . . . . . . 163 integers modulo , . . . . . . . . . 167

complex number . . . . . . . . . . . . . . . . . . 53 critical value . . . . . . . . . . . . . . . . . . . 695


Chapter Analysis

1.1 CONSTANTS 31.1.1 Types of numbers 31.1.2 Roman numerals 41.1.3 Arrow notation 41.1.4 Representation of numbers 51.1.5 Binary prefixes 61.1.6 Decimal multiples and prefixes 61.1.7 Decimal equivalents of common fractions 71.1.8 Hexadecimal addition and subtraction table 81.1.9 Hexadecimal multiplication table 81.1.10 Hexadecimaldecimal fraction conversion table 9

1.2 SPECIAL NUMBERS 101.2.1 Powers of 2 101.2.2 Powers of 16 in decimal scale 121.2.3 Powers of 10 in hexadecimal scale 131.2.4 Special constants 131.2.5 Constants in different bases 161.2.6 Factorials 171.2.7 Bernoulli polynomials and numbers 191.2.8 Euler polynomials and numbers 201.2.9 Fibonacci numbers 211.2.10 Powers of integers 211.2.11 Sums of powers of integers 221.2.12 Negative integer powers 231.2.13 de Bruijn sequences 241.2.14 Integer sequences 25

1.3 SERIES AND PRODUCTS 311.3.1 Definitions 311.3.2 General properties 321.3.3 Convergence tests 331.3.4 Types of series 341.3.5 Summation formulae 401.3.6 Improving convergence: Shanks transformation 401.3.7 Summability methods 411.3.8 Operations with power series 411.3.9 Miscellaneous sums and series 411.3.10 Infinite series 421.3.11 Infinite products 47

1-58488-291-3/02/$0.00+$1.50c 2003 CRC Press, Inc.


1.3.12 Infinite products and infinite series 47

1.4 FOURIER SERIES 481.4.1 Special cases 491.4.2 Alternate forms 501.4.3 Useful series 501.4.4 Expansions of basic periodic functions 51

1.5 COMPLEX ANALYSIS 531.5.1 Definitions 531.5.2 Operations on complex numbers 541.5.3 Functions of a complex variable 541.5.4 CauchyRiemann equations 551.5.5 Cauchy integral theorem 551.5.6 Cauchy integral formula 551.5.7 Taylor series expansions 551.5.8 Laurent series expansions 561.5.9 Zeros and singularities 561.5.10 Residues 571.5.11 The argument principle 581.5.12 Transformations and mappings 58

1.6 INTERVAL ANALYSIS 651.6.1 Interval arithmetic rules 651.6.2 Interval arithmetic properties 65

1.7 REAL ANALYSIS 661.7.1 Relations 661.7.2 Functions (mappings) 661.7.3 Sets of real numbers 671.7.4 Topological space 691.7.5 Metric space 691.7.6 Convergence in with metric 701.7.7 Continuity in with metric 711.7.8 Banach space 721.7.9 Hilbert space 741.7.10 Asymptotic relationships 75

1.8 GENERALIZED FUNCTIONS 761.8.1 Delta function 761.8.2 Other generalized functions 77


1.1 CONSTANTS

1.1.1 TYPES OF NUMBERS

1.1.1.1 Natural numbersThe set of natural numbers, , is customarily denoted by . Many authorsdo not consider to be a natural number.

1.1.1.2 IntegersThe set of integers, , is customarily denoted by . The positiveintegers are .

1.1.1.3 Rational numbersThe set of rational numbers, , is customarily denoted by .Two fractions and

are equal if and only if .

Addition of fractions is de ned by

. Multiplication of fractions

is de ned by .

1.1.1.4 Real numbersThe set of real numbers is customarily denoted by . Real numbers are de ned tobe converging sequences of rational numbers or as decimals that might or might notrepeat.

Real numbers are often divided into two subsets. One subset, the algebraicnumbers, are real numbers which solve a polynomial equation in one variable withinteger coef cien ts. For example;

is an algebraic number because it solves the

polynomial equation ; and all rational numbers are algebraic. Real num-bers that are not algebraic numbers are called transcendental numbers. Examples oftranscendental numbers include and .

1.1.1.5 Complex numbersThe set of complex numbers is customarily denoted by . They are numbers of theform , where , and and are real numbers. See page 53.

Operation computation resultaddition multiplication

reciprocal

complex conjugate Properties include: and .


1.1.2 ROMAN NUMERALS

The major symbols in Roman numerals are I , V , X , L , C ,D , and M . The rules for constructing Roman numerals are:

1. A symbol following one of equal or greater value adds its value. (For example,II , XI , and DV .)

2. A symbol following one of lesser value has the lesser value subtracted fromthe larger value. An I is only allowed to precede a V or an X, an X is onlyallowed to precede an L or a C, and a C is only allowed to precede a D oran M. (For example IV , IX , and XL .)

3. When a symbol stands between two of greater value, its value is subtractedfrom the second and the result is added to the rst (for example, XIV , CIX , DXL ).

4. When two ways exist for representing a number, the one in which the symbolof larger value occurs earlier in the string is preferred. (For example, 14 isrepresented as XIV, not as VIX.)

Decimal number 1 2 3 4 5 6 7 8 9Roman numeral I II III IV V VI VII VIII IX

10 14 50 200 400 500 600 999 1000X XIV L CC CD D DC CMXCIX M

1950 1960 1970 1980 1990MCML MCMLX MCMLXX MCMLXXX MCMXC

1995 1999 2000 2001 2004 2010MCMXCV MCMXCIX MM MMI MMIV MMX

1.1.3 ARROW NOTATION

Arrow notation is a way to represent large numbers in which evaluation proceedsfrom the right:

(1.1.1)

For example, , , and .


1.1.4 REPRESENTATION OF NUMBERS

Numerals as usually written have radix or base 10, so that the numeral represents the number . However,other bases can be used, particularly bases 2, 8, and 16. When a number is written inbase 2, the number is said to be in binary notation. The names of other bases are:

2 binary3 ternary4 quaternary5 quinary6 senary7 septenary8 octal

9 nonary10 decimal11 undenary12 duodecimal16 hexadecimal20 vigesimal60 sexagesimal

When writing a number in base , the digits used range from to . If

, then the digit A stands for , B for , etc. When a base other than 10 isused, it is indicated by a subscript:

A

(1.1.2)

To convert a number from base 10 to base , divide the number by , and theremainder will be the last digit. Then divide the quotient by , using the remainderas the previous digit. Continue dividing the quotient by until a quotient of isarrived at.

EXAMPLE To convert 573 to base 12, divide 573 by 12, yielding a quotient of 47 and aremainder of 9; hence, 9 is the last digit. Divide 47 by 12, yielding a quotient of 3 anda remainder of 11 (which we represent with a B). Divide 3 by 12 yielding a quotientof 0 and a remainder of 3. Therefore, B .

In general, to convert from base to base , it is simplest to convert to base 10as an intermediate step. However, it is simple to convert from base to base . Forexample, to convert to base 16, group the digits in fours (because 16is ), yielding , and then convert each group of 4 to base 16 directly,yielding BD.


1.1.5 BINARY PREFIXES

A byte is 8 bits. A kibibyte is bytes. Other pre x es for power of 2 are:

Factor Pre x Symbol

kibi Ki mebi Mi gibi Gi tebi Ti pebi Pi exbi Ei

1.1.6 DECIMAL MULTIPLES AND PREFIXES

The pre x names and symbols below are taken from Conference Generale des Poidset Mesures, 1991. The common names are for the U.S.

Factor Pre x Symbol Common name

googolplex

googol yotta Y heptillion zetta Z hexillion exa E quintillion peta P quadrillion tera T trillion giga G billion mega M million kilo k thousand hecto H hundred deka da ten deci d tenth centi c hundreth milli m thousandth micro (Greek mu) millionth nano n billionth pico p trillionth femto f quadrillionth atto a quintillionth zepto z hexillionth yocto y heptillionth


1.1.7 DECIMAL EQUIVALENTS OF COMMON FRACTIONS1/64 0.015625

1/32 2/64 0.031253/64 0.046875

1/16 2/32 4/64 0.06255/64 0.078125

3/32 6/64 0.093757/64 0.109375

1/8 4/32 8/64 0.1259/64 0.140625

5/32 10/64 0.1562511/64 0.171875

3/16 6/32 12/64 0.187513/64 0.203125

7/32 14/64 0.2187515/64 0.234375

1/4 8/32 16/64 0.2517/64 0.265625

9/32 18/64 0.2812519/64 0.296875

5/16 10/32 20/64 0.312521/64 0.328125

11/32 22/64 0.3437523/64 0.359375

3/8 12/32 24/64 0.37525/64 0.390625

13/32 26/64 0.4062527/64 0.421875

7/16 14/32 28/64 0.437529/64 0.453125

15/32 30/64 0.4687531/64 0.484375

1/2 16/32 32/64 0.5

33/64 0.51562517/32 34/64 0.53125

35/64 0.5468759/16 18/32 36/64 0.5625

37/64 0.57812519/32 38/64 0.59375

39/64 0.6093755/8 20/32 40/64 0.625

41/64 0.64062521/32 42/64 0.65625

43/64 0.67187511/16 22/32 44/64 0.6875

45/64 0.70312523/32 46/64 0.71875

47/64 0.7343753/4 24/32 48/64 0.75

49/64 0.76562525/32 50/64 0.78125

51/64 0.79687513/16 26/32 52/64 0.8125

53/64 0.82812527/32 54/64 0.84375

55/64 0.8593757/8 28/32 56/64 0.875

57/64 0.89062529/32 58/64 0.90625

59/64 0.92187515/16 30/32 60/64 0.9375

61/64 0.95312531/32 62/64 0.96875

63/64 0.9843751/1 32/32 64/64 1


1.1.8 HEXADECIMAL ADDITION AND SUBTRACTION TABLE

A , B , C , D , E , F .Example: ; hence and .Example: E ; hence E and E .

1 2 3 4 5 6 7 8 9 A B C D E F1 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F 102 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F 10 113 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F 10 11 124 05 06 07 08 09 0A 0B 0C 0D 0E 0F 10 11 12 135 06 07 08 09 0A 0B 0C 0D 0E 0F 10 11 12 13 146 07 08 09 0A 0B 0C 0D 0E 0F 10 11 12 13 14 157 08 09 0A 0B 0C 0D 0E 0F 10 11 12 13 14 15 168 09 0A 0B 0C 0D 0E 0F 10 11 12 13 14 15 16 179 0A 0B 0C 0D 0E 0F 10 11 12 13 14 15 16 17 18A 0B 0C 0D 0E 0F 10 11 12 13 14 15 16 17 18 19B 0C 0D 0E 0F 10 11 12 13 14 15 16 17 18 19 1AC 0D 0E 0F 10 11 12 13 14 15 16 17 18 19 1A 1BD 0E 0F 10 11 12 13 14 15 16 17 18 19 1A 1B 1CE 0F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1DF 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E

1.1.9 HEXADECIMAL MULTIPLICATION TABLE

Example: .Example: F E.

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


1.1.10 HEXADECIMALDECIMAL FRACTION CONVERSIONTABLE

The values below are correct to all digits shown.

Hex Decimal Hex Decimal Hex Decimal Hex Decimal

.00 0 .40 0.250000 .80 0.500000 .C0 0.750000

.01 0.003906 .41 0.253906 .81 0.503906 .C1 0.753906

.02 0.007812 .42 0.257812 .82 0.507812 .C2 0.757812

.03 0.011718 .43 0.261718 .83 0.511718 .C3 0.761718

.04 0.015625 .44 0.265625 .84 0.515625 .C4 0.765625

.05 0.019531 .45 0.269531 .85 0.519531 .C5 0.769531

.06 0.023437 .46 0.273437 .86 0.523437 .C6 0.773437

.07 0.027343 .47 0.277343 .87 0.527343 .C7 0.777343

.08 0.031250 .48 0.281250 .88 0.531250 .C8 0.781250

.09 0.035156 .49 0.285156 .89 0.535156 .C9 0.785156

.0A 0.039062 .4A 0.289062 .8A 0.539062 .CA 0.789062

.0B 0.042968 .4B 0.292968 .8B 0.542968 .CB 0.792968

.0C 0.046875 .4C 0.296875 .8C 0.546875 .CC 0.796875

.0D 0.050781 .4D 0.300781 .8D 0.550781 .CD 0.800781

.0E 0.054687 .4E 0.304687 .8E 0.554687 .CE 0.804687

.0F 0.058593 .4F 0.308593 .8F 0.558593 .CF 0.808593

.10 0.062500 .50 0.312500 .90 0.562500 .D0 0.812500

.11 0.066406 .51 0.316406 .91 0.566406 .D1 0.816406

.12 0.070312 .52 0.320312 .92 0.570312 .D2 0.820312

.13 0.074218 .53 0.324218 .93 0.574218 .D3 0.824218

.14 0.078125 .54 0.328125 .94 0.578125 .D4 0.828125

.15 0.082031 .55 0.332031 .95 0.582031 .D5 0.832031

.16 0.085937 .56 0.335937 .96 0.585937 .D6 0.835937

.17 0.089843 .57 0.339843 .97 0.589843 .D7 0.839843

.18 0.093750 .58 0.343750 .98 0.593750 .D8 0.843750

.19 0.097656 .59 0.347656 .99 0.597656 .D9 0.847656

.1A 0.101562 .5A 0.351562 .9A 0.601562 .DA 0.851562

.1B 0.105468 .5B 0.355468 .9B 0.605468 .DB 0.855468

.1C 0.109375 .5C 0.359375 .9C 0.609375 .DC 0.859375

.1D 0.113281 .5D 0.363281 .9D 0.613281 .DD 0.863281

.1E 0.117187 .5E 0.367187 .9E 0.617187 .DE 0.867187

.1F 0.121093 .5F 0.371093 .9F 0.621093 .DF 0.871093

.20 0.125000 .60 0.375000 .A0 0.625000 .E0 0.875000

.21 0.128906 .61 0.378906 .A1 0.628906 .E1 0.878906

.22 0.132812 .62 0.382812 .A2 0.632812 .E2 0.882812

.23 0.136718 .63 0.386718 .A3 0.636718 .E3 0.886718

.24 0.140625 .64 0.390625 .A4 0.640625 .E4 0.890625

.25 0.144531 .65 0.394531 .A5 0.644531 .E5 0.894531

.26 0.148437 .66 0.398437 .A6 0.648437 .E6 0.898437

.27 0.152343 .67 0.402343 .A7 0.652343 .E7 0.902343


Hex Decimal Hex Decimal Hex Decimal Hex Decimal

.28 0.156250 .68 0.406250 .A8 0.656250 .E8 0.906250

.29 0.160156 .69 0.410156 .A9 0.660156 .E9 0.910156

.2A 0.164062 .6A 0.414062 .AA 0.664062 .EA 0.914062

.2B 0.167968 .6B 0.417968 .AB 0.667968 .EB 0.917968

.2C 0.171875 .6C 0.421875 .AC 0.671875 .EC 0.921875

.2D 0.175781 .6D 0.425781 .AD 0.675781 .ED 0.925781

.2E 0.179687 .6E 0.429687 .AE 0.679687 .EE 0.929687

.2F 0.183593 .6F 0.433593 .AF 0.683593 .EF 0.933593

.30 0.187500 .70 0.437500 .B0 0.687500 .F0 0.937500

.31 0.191406 .71 0.441406 .B1 0.691406 .F1 0.941406

.32 0.195312 .72 0.445312 .B2 0.695312 .F2 0.945312

.33 0.199218 .73 0.449218 .B3 0.699218 .F3 0.949218

.34 0.203125 .74 0.453125 .B4 0.703125 .F4 0.953125

.35 0.207031 .75 0.457031 .B5 0.707031 .F5 0.957031

.36 0.210937 .76 0.460937 .B6 0.710937 .F6 0.960937

.37 0.214843 .77 0.464843 .B7 0.714843 .F7 0.964843

.38 0.218750 .78 0.468750 .B8 0.718750 .F8 0.968750

.39 0.222656 .79 0.472656 .B9 0.722656 .F9 0.972656

.3A 0.226562 .7A 0.476562 .BA 0.726562 .FA 0.976562

.3B 0.230468 .7B 0.480468 .BB 0.730468 .FB 0.980468

.3C 0.234375 .7C 0.484375 .BC 0.734375 .FC 0.984375

.3D 0.238281 .7D 0.488281 .BD 0.738281 .FD 0.988281

.3E 0.242187 .7E 0.492187 .BE 0.742187 .FE 0.992187

.3F 0.246093 .7F 0.496093 .BF 0.746093 .FF 0.996093

1.2 SPECIAL NUMBERS

1.2.1 POWERS OF 2

1 2 0.52 4 0.253 8 0.1254 16 0.06255 32 0.031256 64 0.0156257 128 0.00781258 256 0.003906259 512 0.00195312510 1024 0.0009765625


11 2048 0.0004882812512 4096 0.00024414062513 8192 0.000122070312514 16384 0.0000610351562515 32768 0.00003051757812516 65536 0.000015258789062517 131072 0.0000076293945312518 262144 0.00000381469726562519 524288 0.000001907348632812520 1048576 0.0000009536743164062521 2097152 0.00000047683715820312522 4194304 0.000000238418579101562523 8388608 0.0000001192092895507812524 16777216 0.00000005960464477539062525 33554432 0.000000029802322387695312526 67108864 0.0000000149011611938476562527 134217728 0.00000000745058059692382812528 268435456 0.000000003725290298461914062529 536870912 0.0000000018626451492309570312530 1073741824 0.00000000093132257461547851562531 2147483648 0.000000000465661287307739257812532 4294967296 0.0000000002328306436538696289062533 8589934592 0.00000000011641532182693481445312534 17179869184 0.000000000058207660913467407226562535 34359738368 0.0000000000291038304567337036132812536 68719476736 0.00000000001455191522836685180664062537 137438953472 0.000000000007275957614183425903320312538 274877906944 0.0000000000036379788070917129516601562539 549755813888 0.00000000000181898940354585647583007812540 1099511627776 0.0000000000009094947017729282379150390625

41 2199023255552 42 439804651110443 8796093022208 44 1759218604441645 35184372088832 46 7036874417766447 140737488355328 48 28147497671065649 562949953421312 50 1125899906842624

51 2251799813685248 52 450359962737049653 9007199254740992 54 1801439850948198455 36028797018963968 56 7205759403792793657 144115188075855872 58 28823037615171174459 576460752303423488 60 1152921504606846976

61 2305843009213693952 62 461168601842738790463 9223372036854775808 64 18446744073709551616


65 36893488147419103232 66 7378697629483820646467 147573952589676412928 68 29514790517935282585669 590295810358705651712 70 1180591620717411303424

71 2361183241434822606848 72 472236648286964521369673 9444732965739290427392 74 1888946593147858085478475 37778931862957161709568 76 7555786372591432341913677 151115727451828646838272 78 30223145490365729367654479 604462909807314587353088 80 1208925819614629174706176

81 2417851639229258349412352 82 483570327845851669882470483 9671406556917033397649408 84 1934281311383406679529881685 38685626227668133590597632 86 7737125245533626718119526487 154742504910672534362390528 88 30948500982134506872478105689 618970019642690137449562112 90 1237940039285380274899124224

1.2.2 POWERS OF 16 IN DECIMAL SCALE

0 1 11 16 0.06252 256 0.003906253 4096 0.0002441406254 65536 0.00001525878906255 1048576 0.000000953674316406256 16777216 0.0000000596046447753906257 268435456 0.00000000372529029846191406258 4294967296 0.000000000232830643653869628906259 68719476736 0.000000000014551915228366851806640625

10 1099511627776 0.000000000000909494701772928237915039062511 17592186044416 12 281474976710656 13 4503599627370496

14 72057594037927936 15 1152921504606846976

16 18446744073709551616

17 295147905179352825856 18 4722366482869645213696 19 75557863725914323419136 20 1208925819614629174706176


1.2.3 POWERS OF 10 IN HEXADECIMAL SCALE

0 1 11 A 0.19999999999999999999. . . 2 64 0.028F5C28F5C28F5C28F5. . .3 3E8 0.004189374BC6A7EF9DB2. . .4 2710 0.00068DB8BAC710CB295E. . .5 186A0 0.0000A7C5AC471B478423. . .6 F4240 0.000010C6F7A0B5ED8D36. . .7 989680 0.000001AD7F29ABCAF485. . .8 5F5E100 0.0000002AF31DC4611873. . .9 3B9ACA00 0.000000044B82FA09B5A5. . .

10 2540BE400 0.000000006DF37F675EF6. . .11 174876E800 0.000000000AFEBFF0BCB2. . .12 E8D4A51000 0.000000000119799812DE. . .13 9184E72A000 0.00000000001C25C26849. . .14 5AF3107A4000 0.000000000002D09370D4. . .15 38D7EA4C68000 0.000000000000480EBE7B. . .16 2386F26FC10000 0.0000000000000734ACA5. . .

1.2.4 SPECIAL CONSTANTS

1.2.4.1 The constant The transcendental number is de ned as the ratio of the circumference of a circleto the diameter. It is also the ratio of the area of a circle to the square of the radius() and appears in several formulae in geometry and trigonometry (see Section 6.1)

circumference of a circle volume of a sphere

area of a circle surface area of a sphere

One method of computing is to use the in nite series for the function andone of the identities

(1.2.1)


There are many other identities involving . See Section 1.4.3. For example:

square roots

square roots

(1.2.2)

To 200 decimal places:

3. 14159 26535 89793 23846 26433 83279 50288 41971 69399 3751058209 74944 59230 78164 06286 20899 86280 34825 34211 7067982148 08651 32823 06647 09384 46095 50582 23172 53594 0812848111 74502 84102 70193 85211 05559 64462 29489 54930 38196


0.15707 96326 79489 66192 31321 69163 97514 42098 58469 96876 0.20943 95102 39319 54923 08428 92218 63352 56131 44626 62501 0.26179 93877 99149 43653 85536 15273 29190 70164 30783 28126 0.28559 93321 44526 65804 20584 89389 04571 67451 97218 12501 0.31415 92653 58979 32384 62643 38327 95028 84197 16939 93751 0.34906 58503 98865 91538 47381 53697 72254 26885 74377 70835 0.39269 90816 98724 15480 78304 22909 93786 05246 46174 92189 0.44879 89505 12827 60549 46633 40468 50041 20281 67057 05359 0.52359 87755 98298 87307 71072 30546 58381 40328 61566 56252 0.62831 85307 17958 64769 25286 76655 90057 68394 33879 87502 0.78539 81633 97448 30961 56608 45819 87572 10492 92349 84378 1.04719 75511 96597 74615 42144 61093 16762 80657 23133 12504 1.57079 63267 94896 61923 13216 91639 75144 20985 84699 68755 2.09439 51023 93195 49230 84289 22186 33525 61314 46266 25007 4.71238 89803 84689 85769 39650 74919 25432 62957 54099 06266


7.85398 16339 74483 09615 66084 58198 75721 04929 23498 43776

1.77245 38509 05516 02729 81674 83341 14518 27975 49456 12239

In 1999 was computed to decimal digits. Thefrequency distribution of the digits for , up to 200,000,000,000 decimal places,is:

digit 0: 20000030841 digit 5: 19999917053digit 1: 19999914711 digit 6: 19999881515digit 2: 20000136978 digit 7: 19999967594digit 3: 20000069393 digit 8: 20000291044digit 4: 19999921691 digit 9: 19999869180

1.2.4.2 The constant The transcendental number is the base of natural logarithms. It is given by

(1.2.3)


2. 71828 18284 59045 23536 02874 71352 66249 77572 47093 6999595749 66967 62772 40766 30353 54759 45713 82178 52516 6427427466 39193 20030 59921 81741 35966 29043 57290 03342 9526059563 07381 32328 62794 34907 63233 82988 07531 95251 01901


0.33978 52285 57380 65442 00359 33919 08281 22196 55886 71249 0.38832 59754 94149 31933 71839 24478 95178 53938 92441 95714 0.45304 69714 09840 87256 00479 11892 11041 62928 74515 61666 0.54365 63656 91809 04707 20574 94270 53249 95514 49418 73999 0.67957 04571 14761 30884 00718 67838 16562 44393 11773 42499 0.90609 39428 19681 74512 00958 23784 22083 25857 49031 23332 1.35914 09142 29522 61768 01437 35676 33124 88786 23546 84998 1.81218 78856 39363 49024 01916 47568 44166 51714 98062 46664 23.14069 26327 79269 00572 90863 67948 54738 02661 06242 60021 22.45915 77183 61045 47342 71522 04543 73502 75893 15133 99669

The function is de ned by

(see page 521). The numbers and are

related by the formula (1.2.4)

1.2.4.3 The constant Eulers constant is de ned by

(1.2.5)


It is not known whether is rational or irrational. To 200 decimal places:

0. 57721 56649 01532 86060 65120 90082 40243 10421 59335 9399235988 05767 23488 48677 26777 66467 09369 47063 29174 6749514631 44724 98070 82480 96050 40144 86542 83622 41739 9764492353 62535 00333 74293 73377 37673 94279 25952 58247 09492

1.2.4.4 The constant The golden ratio, , is de ned as the positive root of t

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