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 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 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 [...]... pages 11921201 6 G Strang and T Nguyen, Wavelets and Filter Banks, WellesleyCambridge 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 Scientic 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... 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 Products, edited by A Jeffrey and D Zwillinger, 6th edition, Academic Press, Orlando, Florida, 2000 3 N H Ibragimov, Ed., CRC Handbook of Lie... 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 lyi, Ed., Tables of Intee gral Transforms, in 3 volumes, McGrawHill, New York, 1954 2 I S Gradshteyn and M Ryzhik, Tables of Integrals, Series, and Products, edited by A Jeffrey and D Zwillinger, 6th edition, Academic Press, Orlando, Florida, 2000... 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 Ordinary Differential Equations, CRC Press, Boca Raton, FL, 1995 â 2003 by CRC Press LLC 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... Search, Optimization, and Machine Learning, AddisonWesley, 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, NorthHolland, Amsterdam, 1977 8 N Metropolis, A W Rosenbluth, M N Rosenbluth, A H Teller and E Teller, Equation... cap, zone, and segment Right spherical triangle and Napiers rule 5.1 Types of critical points 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 Notation for trigonometric functions Definitions of angles Sine and cosine Tangent and cotangent Different triangles requiring solution Graphs of ĩà and ẵ ĩà Cornu spiral Sine and cosine integrals ậ ĩà and ĩà Legendre functions Graphs of the Airy functions ĩà and 7.1 7.2... 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 Computational 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,... T Vetterling, and B P Flannery, Numerical Recipes in C++: The Art of Scientic Computing, 2nd edition, Cambridge University Press, New York, 2002 4 A Ralston and P Rabinowitz, A First Course in Numerical Analysis, 2nd edition, McGrawHill, 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... Triangles: isosceles and right Cevas theorem and Menelauss theorem Quadrilaterals Conics: ellipse, parabola, and hyperbola Conics as a function of eccentricity Ellipse and components Hyperbola and components Arc of a circle Angles within a circle The general cubic parabola Curves: semi-cubic parabola, cissoid of Diocles, witch of Agnesi The folium of Descartes in two positions, and the strophoid â 2003... of Pascal á Cycloid and trochoids Epicycloids: nephroid, and epicycloid Hypocycloids: deltoid and astroid Spirals: Bernoulli, Archimedes, and Cornu Cartesian coordinates in space Cylindrical coordinates Spherical coordinates Relations between Cartesian, cylindrical, and spherical coordinates Euler angles The Platonic solids Cylinders: oblique and right circular Right circular cone and frustram A torus . 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,. 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. 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