Standard Mathematical Tables and Formulae

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st

31

EDITION

CRC

standard MathematicAL TABLES and formulae DANIEL ZWILLINGER

CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C.

© 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

Ben Fusaro Florida State University Tallahassee, Florida

Michael F. Bridgland Center for Computing Sciences Bowie, Maryland

Alan F. Karr National Institute Statistical Sciences Research Triangle Park, North Carolina

J. Douglas Faires Youngstown State University Youngstown, Ohio

Al Marden University of Minnesota Minneapolis, Minnesota

Gerald B. Folland University of Washington Seattle, Washington

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 practitioners 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 decision rules, a table of solutions to Pell’s equation, a table of irreducible polynomials in ¾

, a longer table of prime numbers, an interpretation of powers of 10, a collection 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 modi 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 Handbook 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 section 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 welcomed. The home page for this book, which will include errata, will be maintained at http://www.mathtable.com/.




   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 functions which results in coverage of real analysis, complex analysis, and generalized functions. Chapter 2: Algebra covers the different types of algebra studied: elementary algebra, 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 combinatorics, graph theory, coding theory and information theory, operations re-

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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, surfaces, polyhedra, coordinate systems, and differential geometry. Chapter 5: Continuous Mathematics covers calculus material: differentiation, integration, differential and integral equations, and tensor analysis. A large table of integrals is included. This chapter also includes differential forms and orthogonal 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 functions, 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 nonlinear 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 return 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 conversions), 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

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Contributors Karen Bolinger Clarion University Clarion, Pennsylvania

William C. Rinaman LeMoyne College Syracuse, New York

Patrick J. Driscoll U.S. Military Academy West Point, New York

Catherine Roberts College of the Holy Cross Worcester, Massachusetts

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

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

John Michaels SUNY Brockport Brockport, New York

Nico M. Temme CWI Amsterdam, The Netherlands

Roger B. Nelsen Lewis & Clark College Portland, Oregon

Ahmed I. Zayed DePaul University Chicago, Illinois

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Table of Contents Chapter 1 Analysis



Chapter 2 Algebra





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



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





Nico M. Temme and Ahmed I. Zayed

Chapter 7 Probability and Statistics



Gary Stanek

Chapter 9 Financial Analysis







Daniel Zwillinger

Chapter 10 Miscellaneous





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

Chapter 8 Scientific Computing





Ray McLenaghan and Catherine Roberts

Chapter 6 Special Functions







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

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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 . . . . .



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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 chaos 3.9 Game theory . . . . . . . . . . . . . . 3.10 Operations research . . . . . . . . . .

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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 . . . . . . . . . . . . . . . .

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4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22

Polygons . . . . . . . . . . . . . . . . Conics . . . . . . . . . . . . . . . . . Special plane curves . . . . . . . . . . Coordinate systems in space . . . . . Space symmetries or isometries . . . . Other transformations of space . . . . Direction angles and direction cosines Planes . . . . . . . . . . . . . . . . . Lines in space . . . . . . . . . . . . . Polyhedra . . . . . . . . . . . . . . . Cylinders . . . . . . . . . . . . . . . Cones . . . . . . . . . . . . . . . . . Surfaces of revolution: the torus . . . Quadrics . . . . . . . . . . . . . . . . Spherical geometry & trigonometry . . Differential geometry . . . . . . . . . Angle conversion . . . . . . . . . . . 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 . . . . . . . . .

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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 . . . . . . . .

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6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22 6.23 6.24 6.25 6.26 6.27 6.28 6.29 6.30 6.31 6.32 6.33

Gamma function . . . . . . . . . . . . Beta function . . . . . . . . . . . . . Error functions . . . . . . . . . . . . . Fresnel integrals . . . . . . . . . . . . Sine, cosine, and exponential integrals Polylogarithms . . . . . . . . . . . . . Hypergeometric functions . . . . . . . Legendre functions . . . . . . . . . . Bessel functions . . . . . . . . . . . . Elliptic integrals . . . . . . . . . . . . Jacobian elliptic functions . . . . . . . Clebsch–Gordan coef cients . . . . . Integral transforms: Preliminaries . . . Fourier transform . . . . . . . . . . . Discrete Fourier transform (DFT) . . . Fast Fourier transform (FFT) . . . . . Multidimensional Fourier transform . Laplace transform . . . . . . . . . . . Hankel transform . . . . . . . . . . . Hartley transform . . . . . . . . . . . Hilbert transform . . . . . . . . . . . -Transform . . . . . . . . . . . . . . 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 . . . . . . . .

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Chapter 8 Scienti c Computing                               8.1 Basic numerical analysis . . . . . . . . . . . . . . . . . . . . . 8.2 Numerical linear algebra . . . . . . . . . . . . . . . . . . . . . .

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8.3 8.4

Numerical integration and differentiation . . . . . . . . . . . . . . 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 

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List of Figures



List of notation



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List of References Chapter 1

Analysis

1. J. W. Brown and R. V. Churchill, Complex variables and applications, 6th edition, McGraw–Hill, New York, 1996. 2. L. B. W. Jolley, Summation of Series, Dover Publications, New York, 1961. 3. S. G. Krantz, Real Analysis and Foundations, CRC Press, Boca Raton, FL, 1991. 4. S. G. Krantz, The Elements of Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. 5. J. P. Lambert, “Voting Games, Power Indices, and Presidential Elections”, The UMAP Journal, Module 690, 9, No. 3, pages 214–267, 1988. 6. L. D. Servi, “Nested Square Roots of 2”, American Mathematical Monthly, to appear in 2003. 7. N. J. A. Sloane and S. Plouffe, Encyclopedia of Integer Sequences, Academic Press, New York, 1995. Chapter 2

Algebra

1. C. Caldwell and Y. Gallot, “On the primality of 
and      




”, Mathematics of Computation, 71:237, pages 441–448, 2002. 2. I. N. Herstein, Topics in Algebra, 2nd edition, John Wiley & Sons, New York, 1975. 3. P. Ribenboim, The book of Prime Number Records, Springer–Verlag, New York, 1988. 4. G. Strang, Linear Algebra and Its Applications, 3rd edition, International Thomson Publishing, 1988. Chapter 3

Discrete Mathematics

1. B. Bollob´as, Graph Theory, Springer–Verlag, Berlin, 1979.

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2. C. J. Colbourn and J. H. Dinitz, Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 1996. 3. F. Glover, “Tabu Search: A Tutorial”, Interfaces, 20(4), pages 74–94, 1990. 4. D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison–Wesley, Reading, MA, 1989. 5. J. Gross, Handbook of Graph Theory & Applications, CRC Press, Boca Raton, FL, 1999. 6. D. Luce and H. Raiffa, Games and Decision Theory, Wiley, 1957. 7. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North–Holland, Amsterdam, 1977. 8. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, “Equation of State Calculations by Fast Computing Machines”, J. Chem. Phys., V 21, No. 6, pages 1087–1092, 1953. 9. K. H. Rosen, Handbook of Discrete and Combinatorial Mathematics, CRC Press, Boca Raton, FL, 2000. 10. J. O’Rourke and J. E. Goodman, Handbook of Discrete and 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, Washington, D.C., 1993. 3. D. J. Struik, Lectures in Classical Differential Geometry, 2nd edition, Dover, New York, 1988. Chapter 5

Continuous Mathematics

1. A. G. Butkovskiy, Green’s Functions and Transfer Functions Handbook, Halstead Press, John Wiley & Sons, New York, 1982. 2. I. S. Gradshteyn and M. Ryzhik, Tables of Integrals, Series, and Products, 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 Differential Equations, Volume 1, CRC Press, Boca Raton, FL, 1994. 4. A. J. Jerri, Introduction to Integral Equations with Applications, Marcel Dekker, New York, 1985. 5. P. Moon and D. E. Spencer, Field Theory Handbook, Springer-Verlag, Berlin, 1961. 6. A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solution for Ordinary Differential Equations, CRC Press, Boca Raton, FL, 1995.

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7. J. A. Schouten, Ricci-Calculus, Springer–Verlag, Berlin, 1954. 8. J. L. Synge and A. Schild, Tensor Calculus, University of Toronto Press, Toronto, 1949. 9. D. Zwillinger, Handbook of Differential Equations, 3rd ed., Academic Press, New York, 1997. 10. D. Zwillinger, Handbook of Integration, A. K. Peters, Boston, 1992. Chapter 6

Special Functions

1. Staff of the Bateman Manuscript Project, A. Erd´elyi, Ed., Tables of Integral Transforms, in 3 volumes, McGraw–Hill, 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. 3. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer–Verlag, New York, 1966. 4. N. I. A. Vilenkin, Special Functions and the Theory of Group Representations, American Mathematical Society, Providence, RI, 1968. Chapter 7

Probability and Statistics

1. I. Daubechies, Ten Lectures on Wavelets, SIAM Press, Philadelphia, 1992. 2. W. Feller, An Introduction to Probability Theory and Its Applications, Volume 1, John Wiley & Sons, New York, 1968. 3. J. Keilson and L. D. Servi, “The Distributional Form of Little’s Law and the Fuhrmann–Cooper Decomposition”, Operations Research Letters, Volume 9, pages 237–247, 1990. 4. Military Standard 105 D, U.S. Government Printing Of ce, Washington, D.C., 1963. 5. S. K. Park and K. W. Miller, “Random number generators: good ones are hard to nd”, Comm. ACM, October 1988, 31, 10, pages 1192–1201. 6. G. Strang and T. Nguyen, Wavelets and Filter Banks, Wellesley–Cambridge Press, Wellesley, MA, 1995. 7. D. Zwillinger and S. Kokoska, Standard Probability and Statistics Tables and Formulae, Chapman & Hall/CRC, Boca Raton, Florida, 2000. Chapter 8

Scientific Computing

1. R. L. Burden and J. D. Faires, Numerical Analysis, 7th edition, Brooks/Cole, Paci c Grove, CA, 2001. 2. G. H. Golub and C. F. Van Loan, Matrix Computations, 2nd ed., The Johns Hopkins Press, Baltimore, 1989.

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3. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++: The Art of Scientific Computing, 2nd edition, Cambridge University Press, New York, 2002. 4. A. Ralston and P. Rabinowitz, A First Course in Numerical Analysis, 2nd edition, McGraw–Hill, New York, 1978. 5. R. Rubinstein, Simulation and the Monte Carlo Method, Wiley, New York, 1981. Chapter 10

Miscellaneous

1. American Mathematical Society, Mathematical Sciences Professional Directory, Providence, 1995. 2. E. T. Bell, Men of Mathematics, Dover, New York, 1945. 3. C. C. Gillispie, Ed., Dictionary of Scientific Biography, Scribners, New York, 1970–1990. 4. H. S. Tropp, “The Origins and History of the Fields Medal”, Historia Mathematica, 3, pages 167–181, 1976. 5. E. W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, Boca Raton, FL, 1999.

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List of Figures 2.1

Depiction of right-hand rule

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

Hasse diagrams Three graphs that are isomorphic Examples of graphs with 6 or 7 vertices Trees with 7 or fewer vertices Trees with 8 vertices Julia sets The Mandlebrot set Directed network modeling a flow problem

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22

Change of coordinates by a rotation Cartesian coordinates: the 4 quadrants Polar coordinates Homogeneous coordinates Oblique coordinates A shear with factor   ½¾ A perspective transformation The normal form of a line Simple polygons Notation for a triangle Triangles: isosceles and right Ceva’s theorem and Menelaus’s 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

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4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 4.33 4.34 4.35 4.36 4.37 4.38 4.39 4.40 4.41

Cassini’s ovals The conchoid of Nichomedes The limac¸on of Pascal 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 of revolution The ve nondegenerate real quadrics Spherical cap, zone, and segment Right spherical triangle and Napier’s 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  and
 Cornu spiral Sine and cosine integrals  and   Legendre functions Graphs of the Airy functions  and 

7.1 7.2 7.3 7.4 7.5 7.6

Approximation to binomial distributions Conceptual layout of a queue Sample size code letters for MIL-STD-105 D Master table for single sampling inspection (normal inspection) Area of a normal random variable Illustration of  and  regions of a normal distribution

8.1 8.2 8.3

Illustration of Newton’s method Formulae for integration rules with various weight functions Illustration of the Monte–Carlo method

© 2003 by CRC Press LLC

List of Notation *Page numbers listed do not match PDF page numbers due to deletion of blank pages.

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









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

 



 partial order . . . . . . . . . . . . . . . . . . . . 204  product symbol . . . . . . . . . . . . . . . . . . 47 summation symbol . . . . . . . . . . . . . . 31  empty set . . . . . . . . . . . . . . . . . . . . . . . 202 



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

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

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

divergence . . . . . . . . . . . . . . . 493



graph edge sum . . . . . . . . . . . . . . 228 graph union . . . . . . . . . . . . . . . . . .229 set union . . . . . . . . . . . . . . . . . . . . 203 
group isomorphism . . . . . . . . . 170, 225  congruence . . . . . . . . . . . . . . . . . . . . . . 94  existential quanti er . . . . . . . . . . . . . 201 Plank constant over  . . . . . . . . . . . 794  in nity . . . . . . . . . . . . . . . . . . . . . . . . . 68 

 curl . . . . . . . . . . . . . . . . . . . . . 493
 Laplacian . . . . . . . . . . . . . . . . 493

  de nite integral . . . . . . . . . . . 399

integral around closed path . . 399 integration symbol . . . . . . . . . . . 399
falling factorial . . . . . . . . . . . . . . . . . . 17  logical not . . . . . . . . . . . . . . . . . . . . . . 199

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backward difference . . . . . . . . . . 736 gradient . . . . . . . . . . . . . . . . 390, 493 linear connection . . . . . . . . . . . . . 484 [] graph composition . . . . 228 commutator . . . . . . . . . 155, 467 vuw scalar triple product . . . . 136 
  
 continued fraction . 96   Christoffel symbol, rst kind   487
Stirling cycle numbers . . . . 212  
 



()



poset notation . . . . . . . . . 205 
 shifted factorial . . . . . . . . . . 17   type of tensor . . . . . . . . 483   design nomenclature . 245    point in three-dimensional space . . . . . . . . . . . . . . . . . . 345      homogeneous coordinates . . . . . . . . . . . . . 303        homogeneous . . . . . . . . . . . . . 348  coordinates 

















Clebsch–Gordan

binary operation . . . . . . . . . . . . . .160 convolution operation . . . . . . . . . 579 dual of a tensor . . . . . . . . . . . . . . 489 group operation . . . . . . . . . . . . . . 161 re ection . . . . . . . . . . . . . . . . . . . . 307 

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

 

 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 ½  ¾ 



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

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

 

 trimmed mean . . . . . . . . . 659 arithmetic mean . . . . . . . . . . . . . . 659 complex conjugate . . . . . . . . . . . . 54 set complement . . . . . . . . . . . . . . 203  divisibility . . . . . . . . . . . . . . . . . . . . . . . . 93 



determinant of a matrix . . . . . . . 144 graph order . . . . . . . . . . . . . . . . . . 226 norm . . . . . . . . . . . . . . . . . . . . . . . . 133 order of algebraic structure . . . . 160 polynomial norm . . . . . . . . . . . . . . 91 used in tensor notation . . . . . . . . 487 norm . . . . . . . . . . . . . . . . 133 norm . . . . . . . . . . . . . . . . 133   Frobenius norm . . . . . . . . . 146   in nity norm . . . . . . . . . . . 133 norm . . . . . . . . . . . . . . . . . . . . 91, 133  



 





Æ

a  b vector inner product . . . . . 133 group operation . . . . . . . . . . . . . . 161 inner product . . . . . . . . . . . . . . . . 132 crystallographic group . . . 309, 311 degrees in an angle . . . . . . . . . . . 503 function composition . . . . . . . . . . 67 temperature degrees . . . . . . . . . . 798 translation . . . . . . . . . . . . . . . . . . . 307

© 2003 by CRC Press LLC



Greek Letters

maximum vertex degree 223

   change in the argument . . . . . . . . . . . . . . . . 58 forward difference . . . . . . . 265, 728 Laplacian . . . . . . . . . . . . . . . . . . . .493 







½ ¾ ¿ ½  ¾  ¿     continued fraction . . . . . . . . . . . . . . . . . 97 graph join . . . . . . . . . . . . . . . . . . . 228 group operation . . . . . . . . . . . . . . 161 pseudo-inverse operator . . 149, 151 vector addition . . . . . . . . . . . . . . . 132





 

2222 crystallographic group . . 310 333 crystallographic group . . . 311 442 crystallographic group . . . 310 632 crystallographic group . . . 311 crystallographic group . . . . . 309

gamma function . . . . . . . . 540  
Christoffel symbol of second kind . . . . . . . . . . . . . . . . . . . 487 connection coef cients . . . . . 484





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

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

"

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



"




"



graph independence number 225   function, related to zeta function . . . . . . . . . . . . . . . . . 23 one minus the con dence coef cient . . . . . . . . . . . . . . 666 probability of type I error . . . . . 661 



probability of type II error . . 661 function, related to zeta function . . . . . . . . . . . . . . . . . 23



 

#

rectilinear graph crossing number . . . . . . . . . . . . . . . . 222 #  graph crossing number . . 222 $ size of the largest clique . . . . . . 221 # 

%

totient function . . . . . 128, 169 characteristic function . . . . 620 Euler constant . . . . . . . . . . . . . . . . . 21 golden ratio de ned . . . . . . . . . . . . . . . . . . . . 16 value . . . . . . . . . . . . . . . . . . . . . . 16 incidence mapping . . . . . . . . . . . 219 zenith . . . . . . . . . . . . . . . . . . . . . . . 346 %






%

chromatic index . . . . . . . 221 chromatic number . . . . . . 221
 -distribution . . . . . . . . . . . . . . . 703
 critical value . . . . . . . . . . . . . 696
 chi-square distributed . . . . . 619   

Æ

minimum vertex degree . .223 delta function . . . . . . . . . . . . 76 Æ Kronecker delta . . . . . . . . . . . 483 designed distance . . . . . . . . . . . . 257 Feigenbaum’s constant . . . . . . . . 272 
Levi–Civita symbol . . . . . . . . . 489 Æ



Æ

½





 

power of a test . . . . . . . . . . . . . 661 component of in nitesimal generator . . . . . . . . . . . . . . . 466

 

Euler’s constant de nition . . . . . . . . . . . . . . . . . . 15 in different bases . . . . . . . . . . . 16 value . . . . . . . . . . . . . . . . . . . . . . 16  graph genus . . . . . . . . . . . . 224   function, related to zeta function . . . . . . . . . . . . . . . . . 23  skewness . . . . . . . . . . . . . . . . . 620
excess . . . . . . . . . . . . . . . . . . . . 620 !



prime counting function . . . . 103 probability distribution . . . . . 640 constants containing . . . . . . . . . . . 14 continued fraction . . . . . . . . . . . . . 97 distribution of digits . . . . . . . . . . . 15 identities . . . . . . . . . . . . . . . . . . . . . 14 number . . . . . . . . . . . . . . . . . . . . . . . 13 in different bases . . . . . . . . . . . 16 permutation . . . . . . . . . . . . . . . . . 172 sums involving . . . . . . . . . . . . . . . . 24 & logarithmic derivative of the gamma function . . . . . . . . . 543 '

spectral radius . . . . . . . . . . 154 radius of curvature . . . . . . . 374 ' correlation coef cient . . . . . 622 server utilization . . . . . . . . . . . . . 638 '

' 

(

standard deviation . . . . . . . . . . 620 sum of divisors . . . . . . . . . 128
( variance . . . . . . . . . . . . . . . . . . 620 ( singular value of a matrix . . 152 th (
 sum of powers of divisors 128 ( variance . . . . . . . . . . . . . . . . . 622 ( covariance . . . . . . . . . . . . . . . 622 (

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



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



© 2003 by CRC Press LLC

(

4

)

4 2 crystallographic group . . . . 310 4, powers of . . . . . . . . . . . . . . . . . . 30 442 crystallographic group . . . . 310

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

5

*

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

5, powers of . . . . . . . . . . . . . . . . . . 30 5-(12,6,1) table . . . . . . . . . . . . . . 244 5-design, Mathieu . . . . . . . . . . . . 244 632 crystallographic group . . . . . . . . . 311

+

Roman Letters A

component of in nitesimal generator . . . . . . . . . . . . . . . 466 + quantile of order , . . . . . . . . . 659 -   Riemann zeta function . . . . . . . . . 23 + 

Numbers

A interarrival time . . . . . . . . . . . .637 number of codewords . 259 /  skew symmetric part of a tensor . . . . . . . . . . . . . . . . . . 484 A ampere . . . . . . . . . . . . . . . . . . . . 792 

 group inverse . . . . . . . . . . . . . . . . 161  matrix inverse . . . . . . . . . . . . . . . . 138

 .

 

0 null vector . . . . . . . . . . . . . . . . . . . . . . 137 1 1, group identity . . . . . . . . . . . . . 161 1-form . . . . . . . . . . . . . . . . . . . . . . 395 10, powers of . . . . . . . . . . 6, 13, 798 105 D standard . . . . . . . . . . . . . . . 652 16, powers of . . . . . . . . . . . . . . . . . 12 17 crystallographic groups . . . . 307 2  power set of  . . . . . . . . . . . . 203  2 22 crystallographic group . . . 310 2, negative powers of . . . . . . . . . . 10 2, powers of . . . . . . . . . . . . . 6, 10, 27 2-( ,3,1) Steiner triple system . 249 2-form . . . . . . . . . . . . . . . . . . . . . . 396 2-sphere . . . . . . . . . . . . . . . . . . . . . 491 2-switch . . . . . . . . . . . . . . . . . . . . . 227 22 crystallographic group . . . . 309 22 crystallographic group . . . .309 2222 crystallographic group . . . 310 230 crystallographic groups, three-dimensional . . . . . . . 307 3 3 3 crystallographic group . . . . 311 3, powers of . . . . . . . . . . . . . . . . . . 29 3-design (Hadamard matrices) . 250 3-form . . . . . . . . . . . . . . . . . . . . . . 397 3-sphere . . . . . . . . . . . . . . . . . . . . . 491 333 crystallographic group . . . . 311 360, degrees in a circle . . . . . . . 503



© 2003 by CRC Press LLC

 

alternating group on 4 elements 188 
radius of circumscribed circle 324  alternating group . . . . . 163, 172 010203004 queue . . . . . . . . . . . 637  Airy function . . . . . . . . . . . 465, 565 ALFS additive lagged-Fibonacci sequence . . . . . . . . . . . . . . . 646 AMS American Mathematical Society 801 ANOVA analysis of variance . . . . . . . 686 AOQ average outgoing quality . . . . . . 652 AOQL average outgoing quality limit 652 AQL acceptable quality level . . . . . . . 652 AR  autoregressive model . . . . . . . 718 ARMA 5 mixed model . . . . . . . . . 719  graph automorphism group . 220 


 

a unit vector . . . . . . . . . . . . . . . . 492
Fourier coef cients . . . . . . . . . 48
proportion of customers . . . .637
6 almost everywhere . . . . . . . . . . . . . 74 am amplitude . . . . . . . . . . . . . . . . . . . . . 572 arg argument . . . . . . . . . . . . . . . . . . . . . . . 53

B B amount borrowed . . . . . . . . . . 779 service time . . . . . . . . . . . . . . . 637 1, 7 beta function . . . . . . . . . 544  set of blocks . . . . . . . . . . . . . . . 241 1 1

1 

Bell number . . . . . . . . . . . . . .211 Bernoulli number . . . . . . . . . . 19 1 a block . . . . . . . . . . . . . . . . . . 241 1  Bernoulli polynomial . . . 19 B.C.E (before the common era, B.C.) 810 BFS basic feasible solution . . . . . . . . . 283  Airy function . . . . . . . . . . . 465, 565 BIBD balanced incomplete block design 245 Bq becquerel . . . . . . . . . . . . . . . . . . . . . 792 b unit binormal vector . . . . . . . . . . . . . 374 1 1

C

c c cardinality of real numbers . . 204 number of identical servers . . 637 2 speed of light . . . . . . . . . . . . . . .794 cas combination of sin and cos . . . . . .591 cd candela . . . . . . . . . . . . . . . . . . . . . . . . 792 cm crystallographic group . . . . . . . . . . 309 cmm crystallographic group . . . . . . . . 310 2 Fourier coef cients . . . . . . . . . . . . . . 50 8  elliptic function . . . . . . . . . . . 572 cof  cofactor of matrix  . . . . . . 145 cond() condition number . . . . . . . . . 148 cos trigonometric function . . . . . . . . . 505 cosh hyperbolic function . . . . . . . . . . . 524 cot trigonometric function . . . . . . . . . . 505 coth hyperbolic function . . . . . . . . . . . 524 covers trigonometric function . . . . . . . 505 csc trigonometric function . . . . . . . . . .505 csch hyperbolic function . . . . . . . . . . . 524 cyc number of cycles . . . . . . . . . . . . . . 172 2

D

C channel capacity . . . . . . . . . . . 255    -combination . . . 206, 215   Fresnel integral . . . . . . . . . 547     combinations with replacement . . . . . . . . . . . . 206 complex numbers . . . . . . . . 3, 167 complex  element vectors 131  integration contour . . . . . 399, 404 C coulomb . . . . . . . . . . . . . . . . . . 792 C Roman numeral (100) . . . . . . . . . 4 

 

cyclic group of order 2 . . . . 178 direct group product 181  cyclic group of order 3 . . . . 178    direct group product . 184  cyclic group of order 4 . . . . 178   
direct group product . 181  cyclic group of order 5 . . . . 179  cyclic group of order 6 . . . . 179  cyclic group of order 7 . . . . 180  cyclic group of order 8 . . . . 180  cyclic group of order 9 . . . . 184  Catalan numbers . . . . . . . . . . 212  cycle graph . . . . . . . . . . . . . . 229  cyclic group . . . . . . . . . . . . . . 172  cyclic group of order 10 . . 185 C.E. (common era, A.D.) . . . . . . . . . . .810  cosine integral . . . . . . . . . . . . . . 549 



 
 


© 2003 by CRC Press LLC

D constant service time . . . . . . . 637 diagonal matrix . . . . . . . . . . . . 138 9 differentiation operator 456, 466 D Roman numeral (500) . . . . . . . . 4 9 9

9 

dihedral group of order 8 . . 182 dihedral group of order 10 . 185 9 dihedral group of order 12 . 186 9 region of convergence . . . . . 595 9 derangement . . . . . . . . . . . . . 210 9 dihedral group . . . . . . . 163, 172 DFT discrete Fourier transform . . . . . 582 DLG
 double loop graph . . . . 230 9

9

.

distance between vertices 223 derivative operator . . . . . . . . . . . 386 exterior derivative . . . . . . . . . . . . 397 minimum distance . . . . . . . . . . . . 256 .8 

. 

proportion of customers . . . .637 u v Hamming distance . . . 256 .
a projection . . . . . . . . . . . . 395  determinant of matrix  . . . . 144   graph diameter . . . . . . . . . . .223 div divergence . . . . . . . . . . . . . . . . . . . . 493 8  elliptic function . . . . . . . . . . 572 .

.H 

differential surface area . . . . . . . . . 405 differential volume . . . . . . . . . . . . 405 .x fundamental differential . . . . . . . . . 377

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

. .:

E E edge set . . . . . . . . . . . . . . . . . . . 219 event . . . . . . . . . . . . . . . . . . . . . 617 ;8  rst fundamental metric coef cient . . . . . . . . . . . . . . 377 E   expectation operator . . . . . . 619 ; ;

; 


Erlang- service time . . . . . 637 Euler numbers . . . . . . . . . . . . . 20 ;  Euler polynomial . . . . . . . 20 ;  exponential integral . . . . 550 ; identity group . . . . . . . . . . . . 172 ; elementary matrix . . . . . . . . 138 Ei exbi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 e 6 algebraic identity . . . . . . . . . . . 161 6 charge of electron . . . . . . . . . . 794 6 constants containing . . . . . . . . . 15 6 continued fraction . . . . . . . . . . . 97 6 de nition . . . . . . . . . . . . . . . . . . . 15 6 eccentricity . . . . . . . . . . . . . . . . 325 6 in different bases . . . . . . . . . . . . 16 68  second fundamental metric coef cient . . . . . . . . . . . . . . 377 ; ;

6 

e vector of ones . . . . . . . . . . . . . . 137 e unit vector . . . . . . . . . . . . . . . . 137 6½ 
permutation symbol . . . 489 ecc eccentricity of a vertex . . . . . . 223 erf error function . . . . . . . . . . . . . . . . . . 545 erfc complementary error function . . 545 exsec trigonometric function . . . . . . . 505



F F rst fundamental metric coef cient . . . . . . . . . . . . . . 377 <  Dawson’s integral . . . . . . .546 <  probability distribution function . . . . . . . . . . . . . . . . 619 Fourier transform . . . . . . . . . . 576 < 
 2  hypergeometric function . . . . . . . . . . . . . . . . 553  sample distribution function < 658 < 8 

© 2003 by CRC Press LLC

 contravariant metric . . . . . . . . . . . . 486 glb greatest lower bound . . . . . . . . . . . . 68 >

> 

i i unit vector . . . . . . . . . . . . . . . . . .494 i unit vector . . . . . . . . . . . . . . . . . .135 C imaginary unit . . . . . . . . . . . . . . . 53 C interest rate . . . . . . . . . . . . . . . . 779 iid independent and identically distributed . . . . . . . . . . . . . . 619 inf greatest lower bound . . . . . . . . . . . . . 68 in mum greatest lower bound . . . . . . . 68 

H

J

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

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

?

? 

D

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

 D 

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

D  D

? ?

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

?

I I rst fundamental form . . . . . . 377 identity matrix . . . . . . . . . . . . . 138 = A B  mutual information . . 254 I Roman numeral (1) . . . . . . . . . . . . 4 ICG inversive congruential generator 646 = = second fundamental form . . . . . . . 377 Im imaginary part of a complex number 53 = identity matrix . . . . . . . . . . . . . . . . . 138 Inv number of invariant elements . . . 172 IVP initial-value problem . . . . . . . . . . 265 = =

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half order Bessel function 563 
zero of Bessel function . . . 563 



 Hankel function . . . . . . . . . 559 
  Hankel function . . . . . . . . . 559 ?
-stage hyperexponential service time . . . . . . . . . . . . 637 ? harmonic numbers . . . . . . . . . 32 ?  Hermite polynomials . . 532 " Hankel transform . . . . . . . . . 589 H.M. harmonic mean . . . . . . . . . . . . . . 660 Hz hertz . . . . . . . . . . . . . . . . . . . . . . . . . . 792 hav trigonometric function . . . . . 372, 505 @ metric coef cients . . . . . . . . . . . . . . 492 ?

 



K K Gaussian curvature . . . . . . . . 377 system capacity . . . . . . . . . . . 637 K Kelvin (degrees) . . . . . . . . . . . 792

3 3

3 

complete graph . . . . . . . . . . .229  complete bipartite graph 230 3 ½  complete multipartite graph . . . . . . . . . . . . . . . . . . 230 3 empty graph . . . . . . . . . . . . . 229 Ki kibi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 k k curvature vector . . . . . . . . . . . . 374 k unit vector . . . . . . . . . . . . . . . . . 494  unit vector . . . . . . . . . . . . . . . . . 135 k Boltzmann constant . . . . . . . . .794 dimension of a code . . . . . . . . 258   kernel . . . . . . . . . . . . . . . . 478 3

3

 

k geodesic curvature . . . . . . . . 377 k normal curvature vector . . . . 377  block size . . . . . . . . . . . . . . . . .241 kg kilogram . . . . . . . . . . . . . . . . . . . . . . 792

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

 

norm . . . . . . . . . . . . . . . . . . . . 133 norm . . . . . . . . . . . . . . . . . . . . 133  average number of customers 638  norm . . . . . . . . . . . . . . . . . . . . . . 73  Lie group . . . . . . . . . . . . . . . . 466  space of measurable functions 73 LCG linear congruential generator . . 644 LCL lower control limit . . . . . . . . . . . . 650 LCM least common multiple . . . . . . . 101   logarithm . . . . . . . . . . . . . . . . . . 551 
 dilogarithm . . . . . . . . . . . . . . . . 551 LIFO last in, rst out . . . . . . . . . . . . . . 637   polylogarithm . . . . . . . . . . . . . . 551  logarithmic integral . . . . . . . . . . . 550 LP linear programming . . . . . . . . . . . . 280 LTPD lot tolerance percent defective 652

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




M M Mandelbrot set . . . . . . . . . . . . 273 exponential service time . . . 637 E number of codewords . . . . . . 258 E F  measure of a polynomial 93 $ Mellin transform . . . . . . . . . . 612 M mass . . . . . . . . . . . . . . . . . . . . . 796 M Roman numeral (1000) . . . . . . . 4 MA5 moving average . . . . . . . . . . . . 719 M.D. mean deviation . . . . . . . . . . . . . . 660

MFLG multiplicative lagged-Fibonacci generator . . . . . . . . . . . . . . . 646 E00! queue . . . . . . . . . . . . . . . . . . . . 639 E00202 queue . . . . . . . . . . . . . . . . . . 639 E00 queue . . . . . . . . . . . . . . . . . . . 639 Mi mebi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 MLE maximum likelihood estimator 662 E0E0! queue . . . . . . . . . . . . . . . . . . . 638 E0E02 queue . . . . . . . . . . . . . . . . . . . . 639 E M¨ obius ladder graph . . . . . . . . . . 229 MOLS mutually orthogonal Latin squares . . . . . . . . . . . . . . . . . 251 MOM method of moments . . . . . . . . . 662 MTBF mean time between failures . . 655 m  mortgage amount . . . . . . . . . . 779  number in the source . . . . . . . 637 m meter . . . . . . . . . . . . . . . . . . . . . 792 mid midrange . . . . . . . . . . . . . . . . . . . . . 660 mod modular arithmetic . . . . . . . . . . . . . 94 mol mole . . . . . . . . . . . . . . . . . . . . . . . . . 792

N N number of zeros . . . . . . . . . . . . 58 null space . . . . . . . . . . . . . 149 G " ( normal random variable 619 N unit normal vector . . . . . . . . . .378 % normal vector . . . . . . . . . . . . . 377  natural numbers . . . . . . . . . . . . . . 3 N newton . . . . . . . . . . . . . . . . . . . . 792 G  number of monic irreducible polynomials . . . . . . . . . . . . 261 n n principal normal unit vector . 374  unit normal vector . . . . . . . . . . 135 n  code length . . . . . . . . . . . . . . . . 258  number of time periods . . . . . 779  order of a plane . . . . . . . . . . . . 248 G

G 

O

E E

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

H

P P number of poles . . . . . . . . . . . . 58 principal . . . . . . . . . . . . . . . . . . 779 F 1   conditional probability 617 F ; probability of event ; . . 617 F #  auxiliary function . . . . . 561 F   -permutation . . . . . . . 215 F   -permutation . . . . . . . . 206 F   Markov transition function 640 F & ' Riemann F function . . . . . 465 F F

F 

  chromatic polynomial . . 221 path (type of graph) . . . . . . . 229 F  Lagrange interpolating polynomial . . . . . . . . . . . . . 733 F  Legendre function . . . . . 465 F  Legendre polynomials . . 534    F  Jacobi polynomials . 533 F  Legendre function . . . . . .554  F  associated Legendre functions . . . . . . . . . . . . . . . 557 Pa pascal . . . . . . . . . . . . . . . . . . . . . . . . . 792 Per  period of a sequence . . . . . . . 644 Pi pebi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 PID principal ideal domain . . . . . . . . . 165 F   -step Markov transition matrix . . . . . . . . . . . . . . . . . 641  F   permutations with replacement 206 PRI priority service . . . . . . . . . . . . . . . . 637 PRNG pseudorandom number generator 644 p , partitions . . . . . . . . . . . . . . . 210 ," product of prime numbers . 106 p1 crystallographic group . . . . . . 309, 311 p2 crystallographic group . . . . . . . . . . 310 p3 crystallographic group . . . . . . . . . . 311 p31m crystallographic group . . . . . . . 311 p3m1 crystallographic group . . . . . . . 311 p4 crystallographic group . . . . . . . . . . 310 p4g crystallographic group . . . . . . . . . 310 p4m crystallographic group . . . . . . . . 310 p6 crystallographic group . . . . . . . . . . 311 p6m crystallographic group . . . . . . . . 311 per permanent . . . . . . . . . . . . . . . . . . . . 145 pg crystallographic group . . . . . . . . . . 309

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



p  joint probability distribution 254 ,
discrete probability . . . . . . . . 619 ,
 partitions . . . . . . . . . . . . . . 207 ,  restricted partitions . . . . 210 , proportion of time . . . . . . . . . 638

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

F

J

F

J# 

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J 

cube (type of graph) . . . . . . 229 Legendre function . . . . . 465 J  Legendre function . . . . . 554  J  associated Legendre functions 557 7 nome . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 J

J



R R Ricci tensor . . . . . . . . . . . 485, 488 Riemann tensor . . . . . . . . . . . . 488 K curvature tensor . . . . . . . . . . . 485 K radius (circumscribed circle) 319, 513 K range . . . . . . . . . . . . . . . . . . . . . 650 K rate of a code . . . . . . . . . . . . . . 255 K range space . . . . . . . . . . . . 149 K* .  risk function . . . . . . . . .657 K reliability function . . . . . . 655
continuity in . . . . . . . . . . . . . . . . 71
convergence in . . . . . . . . . . . . . . 70
real numbers . . . . . . . . . . . . 3, 167 K K

K 

 reliability of a component . . 653  reliability of parallel system 653 K reliability of series system . 653 K radius of the earth . . . . . . . . 372
real  element vectors . . . . . . . . . . 131
 real    matrices . . . . . . . . 137 Re real part of a complex number . . . . 53 R.M.S. root mean square . . . . . . . . . . . 660 K

K





RSS random service . . . . . . . . . . . . . . . 637 r 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 . . . . . . . . . . 226 rad radian . . . . . . . . . . . . . . . . . . . . . . . . 792  replication number . . . . . . . . . . . . . . 241
 Rademacher functions . . . . . . . 722

S S sample space . . . . . . . . . . . . . . 617 torsion tensor . . . . . . . . . . . . . . 485  Fresnel integral . . . . . . . . . 547  symmetric group . . . . . . . . . . 163   Stirling number second kind . . . . . . . . . . . . . . . . . . . 213 ( /  symmetric part of a tensor 484 S siemen . . . . . . . . . . . . . . . . . . . . 792

 


area of circumscribed polygon 324


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

T

 

 

T

T

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

 Chebyshev polynomials 534  isomorphism class of trees 241 Ti tebi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 TN  Toeplitz network . . . . . . . . . 230  trace of matrix  . . . . . . . . . . . . 150 -  design nomenclature . . . . . 241 /

/

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

  Stirling number rst kind 213

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

sample standard deviation . . . 660

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

© 2003 by CRC Press LLC

 

critical value . . . . . . . . . . . . . . 695 ! transition probabilities . . . . 255 tan trigonometric function . . . . . . . . . . 505 tanh hyperbolic function . . . . . . . . . . . 524 t unit tangent vector . . . . . . . . . . . . . . . 374 



U U universe . . . . . . . . . . . . . . . . . . 201 matrix group . . . . . . . . . . . 172 L 
 uniform random variable 619 L  Chebyshev polynomials 535 UCL upper control limit . . . . . . . . . . . 650 UFD unique factorization domain . . . 165 UMVU type of estimator . . . . . . . . . . . 663 URL Uniform Resource Locators . . . 803 8 traf c intensity . . . . . . . . . . . . . . . . . . 638 8 unit step function . . . . . . . . . . . . 595 8 distance . . . . . . . . . . . . . . . . . . . . . . . 492 L

L 

V V

Y 

B 

Klein four group . . . . . . . . . . . 179 vertex set . . . . . . . . . . . . . . . . . 219 V Roman numeral (5) . . . . . . . . . . . 4 V volt . . . . . . . . . . . . . . . . . . . . . . . 792 % vector operation . . . . . . . . . . . . . . .158 :   volume of a sphere . . . . . . . . . . 368 vers trigonometric function . . . . . . . . . 505 :

 

:

W

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

homogeneous solution . . 456 half order Bessel function 563   particular solution . . . . . . 456 
zero of Bessel function . . . 563

"

 





Z Z

W

queue discipline . . . . . . . . . . . 637 center of a graph . . . . . . . 221 4  instantaneous hazard rate .655  integers . . . . . . . . . . . . . . . . . 3, 167 ) 4 -transform . . . . . . . . . . . . . . . 594 4

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

4 

M 

 root of unity . . . . . . . . . . . . . 582  average time . . . . . . . . . . . . .638 M wheel (type of graph) . . . . . 229 M  Walsh functions . . . . . . . 722 Wb weber . . . . . . . . . . . . . . . . . . . . . . . . 792 M M

X X in nitesimal generator . . . . . 466 set of points . . . . . . . . . . . . . . . 241 X Roman numeral (10) . . . . . . . . . . 4  A rst prolongation . . . . . . . . . . . . . 466 
 A second prolongation . . . . . . . . . . 466 th  C order statistic . . . . . . . . . . . . . . 659  rectangular coordinates . . . . . . . . . . 492 A A

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4  4


4 semidirect group product

187  integers modulo  . . . . . . . . 167  a group . . . . . . . . . . . . . . . . . . 163  integers modulo , . . . . . . . . . 167  complex number . . . . . . . . . . . . . . . . . . 53  critical value . . . . . . . . . . . . . . . . . . . 695