CRC Standard Mathematical Tables and Formulas, 33rd Edition

873 Pages·2017·10.132 MB·English

by  Daniel Zwillinger

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CRC
STANDARD
MATHEMATICAL
TABLES
AND
FORMULAS
33RD EDITION
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Advances in Applied Mathematics
Series Editor: Daniel Zwillinger
Published Titles
Advanced Engineering Mathematics with MATLAB, Fourth Edition
Dean G. Duffy
CRC Standard Curves and Surfaces with Mathematica®, Third Edition
David H. von Seggern
CRC Standard Mathematical Tables and Formulas, 33rd Edition
Dan Zwillinger
Dynamical Systems for Biological Modeling: An Introduction
Fred Brauer and Christopher Kribs
Fast Solvers for Mesh-Based Computations Maciej Paszy´nski
Green’s Functions with Applications, Second Edition Dean G. Duffy
Handbook of Peridynamic Modeling Floriin Bobaru, John T. Foster,
Philippe H. Geubelle, and Stewart A. Silling
Introduction to Financial Mathematics Kevin J. Hastings
Linear and Complex Analysis for Applications John P. D’Angelo
Linear and Integer Optimization: Theory and Practice, Third Edition
Gerard Sierksma and Yori Zwols
Markov Processes James R. Kirkwood
Pocket Book of Integrals and Mathematical Formulas, 5th Edition
Ronald J. Tallarida
Stochastic Partial Differential Equations, Second Edition Pao-Liu Chow
Quadratic Programming with Computer Programs Michael J. Best
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Advances in Applied Mathematics
CRC
STANDARD
MATHEMATICAL
TABLES
AND
FORMULAS
33RD EDITION
Edited by
Dan Zwillinger
“K29784_FM” — 2017/12/6 — 17:41 — page 8 — #8
MATLAB• is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks
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of MATLAB• software or related products does not constitute endorsement or sponsorship by The
MathWorks of a particular pedagogical approach or particular use of the MATLAB• software.
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Table of Contents
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Chapter1
NumbersandElementaryMathematics . . . . . . . . . . . . . . . . . . . 1
1.1 Proofswithoutwords . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Specialnumbers . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Intervalanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5 FractalArithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.6 Max-PlusAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.7 Coupled-analoguesofFunctions . . . . . . . . . . . . . . . . . . 27
1.8 Numbertheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.9 Seriesandproducts . . . . . . . . . . . . . . . . . . . . . . . . . 47
Chapter2
Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.1 Elementaryalgebra . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.2 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.3 Vectoralgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.4 Linearandmatrixalgebra . . . . . . . . . . . . . . . . . . . . . 83
2.5 Abstractalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Chapter3
DiscreteMathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.2 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3.3 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
3.4 Combinatorialdesigntheory . . . . . . . . . . . . . . . . . . . . 172
3.5 Differenceequations . . . . . . . . . . . . . . . . . . . . . . . . 184
Chapter4
Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
4.1 Euclideangeometry . . . . . . . . . . . . . . . . . . . . . . . . 193
4.2 GradesandDegrees . . . . . . . . . . . . . . . . . . . . . . . . 193
4.3 Coordinatesystemsintheplane . . . . . . . . . . . . . . . . . . 194
4.4 Planesymmetriesorisometries . . . . . . . . . . . . . . . . . . 200
4.5 Othertransformationsoftheplane . . . . . . . . . . . . . . . . . 207
4.6 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
4.7 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
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4.8 Surfacesofrevolution:thetorus . . . . . . . . . . . . . . . . . . 219
4.9 Quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
4.10 Sphericalgeometryandtrigonometry . . . . . . . . . . . . . . . 224
4.11 Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
4.12 Specialplanecurves . . . . . . . . . . . . . . . . . . . . . . . . 240
4.13 Coordinatesystemsinspace . . . . . . . . . . . . . . . . . . . . 249
4.14 Spacesymmetriesorisometries . . . . . . . . . . . . . . . . . . 252
4.15 Othertransformationsofspace . . . . . . . . . . . . . . . . . . . 255
4.16 Directionanglesanddirectioncosines . . . . . . . . . . . . . . . 257
4.17 Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
4.18 Linesinspace . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
4.19 Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
4.20 Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
4.21 Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
4.22 Differentialgeometry. . . . . . . . . . . . . . . . . . . . . . . . 267
Chapter5
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
5.1 Differentialcalculus . . . . . . . . . . . . . . . . . . . . . . . . 277
5.2 Differentialforms . . . . . . . . . . . . . . . . . . . . . . . . . 288
5.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
5.4 Tableofindefiniteintegrals . . . . . . . . . . . . . . . . . . . . 305
5.5 Tableofdefiniteintegrals . . . . . . . . . . . . . . . . . . . . . 343
5.6 Ordinarydifferentialequations . . . . . . . . . . . . . . . . . . . 350
5.7 Partialdifferentialequations . . . . . . . . . . . . . . . . . . . . 362
5.8 Integralequations. . . . . . . . . . . . . . . . . . . . . . . . . . 375
5.9 Tensoranalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
5.10 Orthogonalcoordinatesystems. . . . . . . . . . . . . . . . . . . 388
5.11 Realanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
5.12 Generalizedfunctions . . . . . . . . . . . . . . . . . . . . . . . 403
5.13 Complexanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . 405
5.14 SignificantMathematicalEquations . . . . . . . . . . . . . . . . 417
Chapter6
SpecialFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
6.1 Ceilingandfloorfunctions . . . . . . . . . . . . . . . . . . . . . 421
6.2 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
6.3 Exponentialfunction . . . . . . . . . . . . . . . . . . . . . . . . 422
6.4 Logarithmicfunctions . . . . . . . . . . . . . . . . . . . . . . . 422
6.5 Trigonometricfunctions . . . . . . . . . . . . . . . . . . . . . . 424
6.6 Circularfunctionsandplanartriangles. . . . . . . . . . . . . . . 433
6.7 Tablesoftrigonometricfunctions . . . . . . . . . . . . . . . . . 437
6.8 Angleconversion . . . . . . . . . . . . . . . . . . . . . . . . . . 440
6.9 Inversecircularfunctions. . . . . . . . . . . . . . . . . . . . . . 441
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6.10 Hyperbolicfunctions . . . . . . . . . . . . . . . . . . . . . . . . 443
6.11 Inversehyperbolicfunctions . . . . . . . . . . . . . . . . . . . . 447
6.12 Gudermannianfunction . . . . . . . . . . . . . . . . . . . . . . 449
6.13 Orthogonalpolynomials . . . . . . . . . . . . . . . . . . . . . . 451
6.14 Clebsch–Gordancoefficients . . . . . . . . . . . . . . . . . . . . 458
6.15 Besselfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 460
6.16 Betafunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
6.17 Ellipticintegrals . . . . . . . . . . . . . . . . . . . . . . . . . . 470
6.18 Jacobianellipticfunctions . . . . . . . . . . . . . . . . . . . . . 473
6.19 Errorfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
6.20 Fresnelintegrals . . . . . . . . . . . . . . . . . . . . . . . . . . 476
6.21 Gammafunction . . . . . . . . . . . . . . . . . . . . . . . . . . 478
6.22 Hypergeometricfunctions . . . . . . . . . . . . . . . . . . . . . 481
6.23 LambertFunction. . . . . . . . . . . . . . . . . . . . . . . . . . 483
6.24 Legendrefunctions . . . . . . . . . . . . . . . . . . . . . . . . . 484
6.25 Polylogarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
6.26 ProlateSpheroidalWaveFunctions . . . . . . . . . . . . . . . . 489
6.27 Sine,cosine,andexponentialintegrals . . . . . . . . . . . . . . . 490
6.28 WeierstrassEllipticFunction . . . . . . . . . . . . . . . . . . . . 492
6.29 Integraltransforms:List . . . . . . . . . . . . . . . . . . . . . . 493
6.30 Integraltransforms:Preliminaries . . . . . . . . . . . . . . . . . 494
6.31 Fourierintegraltransform . . . . . . . . . . . . . . . . . . . . . 494
6.32 DiscreteFouriertransform(DFT) . . . . . . . . . . . . . . . . . 500
6.33 FastFouriertransform(FFT) . . . . . . . . . . . . . . . . . . . . 502
6.34 MultidimensionalFouriertransforms . . . . . . . . . . . . . . . 502
6.35 Hankeltransform . . . . . . . . . . . . . . . . . . . . . . . . . . 503
6.36 Hartleytransform . . . . . . . . . . . . . . . . . . . . . . . . . . 504
6.37 Hilberttransform . . . . . . . . . . . . . . . . . . . . . . . . . . 505
6.38 Laplacetransform . . . . . . . . . . . . . . . . . . . . . . . . . 508
6.39 Mellintransform . . . . . . . . . . . . . . . . . . . . . . . . . . 512
6.40 Z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
6.41 Tablesoftransforms . . . . . . . . . . . . . . . . . . . . . . . . 517
Chapter7
ProbabilityandStatistics . . . . . . . . . . . . . . . . . . . . . . . . . . 533
7.1 Probabilitytheory . . . . . . . . . . . . . . . . . . . . . . . . . 535
7.2 Classicalprobabilityproblems . . . . . . . . . . . . . . . . . . . 545
7.3 Probabilitydistributions . . . . . . . . . . . . . . . . . . . . . . 553
7.4 Queuingtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
7.5 Markovchains . . . . . . . . . . . . . . . . . . . . . . . . . . . 565
7.6 Randomnumbergeneration . . . . . . . . . . . . . . . . . . . . 568
7.7 Randommatrices . . . . . . . . . . . . . . . . . . . . . . . . . . 574
7.8 Controlchartsandreliability . . . . . . . . . . . . . . . . . . . . 575
7.9 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
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7.10 Confidenceintervals . . . . . . . . . . . . . . . . . . . . . . . . 588
7.11 Testsofhypotheses . . . . . . . . . . . . . . . . . . . . . . . . . 595
7.12 Linearregression . . . . . . . . . . . . . . . . . . . . . . . . . . 609
7.13 Analysisofvariance(ANOVA) . . . . . . . . . . . . . . . . . . 613
7.14 Samplesize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620
7.15 Contingencytables . . . . . . . . . . . . . . . . . . . . . . . . . 623
7.16 Acceptancesampling . . . . . . . . . . . . . . . . . . . . . . . . 626
7.17 Probabilitytables . . . . . . . . . . . . . . . . . . . . . . . . . . 628
Chapter8
ScientificComputing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
8.1 Basicnumericalanalysis . . . . . . . . . . . . . . . . . . . . . . 646
8.2 Numericallinearalgebra . . . . . . . . . . . . . . . . . . . . . . 659
8.3 Numericalintegrationanddifferentiation . . . . . . . . . . . . . 668
8.4 Programmingtechniques . . . . . . . . . . . . . . . . . . . . . . 688
Chapter9
MathematicalFormulasfromtheSciences . . . . . . . . . . . . . . . . . 689
9.1 Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691
9.2 Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692
9.3 Atmosphericphysics . . . . . . . . . . . . . . . . . . . . . . . . 694
9.4 AtomicPhysics . . . . . . . . . . . . . . . . . . . . . . . . . . . 695
9.5 Basicmechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 696
9.6 Beamdynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . 698
9.7 BiologicalModels . . . . . . . . . . . . . . . . . . . . . . . . . 699
9.8 Chemistry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700
9.9 Classicalmechanics . . . . . . . . . . . . . . . . . . . . . . . . 701
9.10 Coordinatesystems–Astronomical . . . . . . . . . . . . . . . . 702
9.11 Coordinatesystems–Terrestrial . . . . . . . . . . . . . . . . . . 703
9.12 Earthquakeengineering . . . . . . . . . . . . . . . . . . . . . . 704
9.13 Economics(Macro) . . . . . . . . . . . . . . . . . . . . . . . . 705
9.14 ElectromagneticTransmission . . . . . . . . . . . . . . . . . . . 707
9.15 Electrostaticsandmagnetism . . . . . . . . . . . . . . . . . . . 708
9.16 ElectromagneticFieldEquations . . . . . . . . . . . . . . . . . . 709
9.17 Electroniccircuits . . . . . . . . . . . . . . . . . . . . . . . . . 710
9.18 Epidemiology. . . . . . . . . . . . . . . . . . . . . . . . . . . . 711
9.19 Fluidmechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 712
9.20 Humanbody . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713
9.21 Modelingphysicalsystems . . . . . . . . . . . . . . . . . . . . . 714
9.22 Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715
9.23 Populationgenetics . . . . . . . . . . . . . . . . . . . . . . . . . 716
9.24 Quantummechanics . . . . . . . . . . . . . . . . . . . . . . . . 717
9.25 Quaternions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719
9.26 Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720
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9.27 Relativisticmechanics . . . . . . . . . . . . . . . . . . . . . . . 721
9.28 Solidmechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 722
9.29 Statisticalmechanics . . . . . . . . . . . . . . . . . . . . . . . . 723
9.30 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 724
Chapter10
Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725
10.1 Calendarcomputations . . . . . . . . . . . . . . . . . . . . . . . 727
10.2 Cellularautomata. . . . . . . . . . . . . . . . . . . . . . . . . . 728
10.3 Communicationtheory . . . . . . . . . . . . . . . . . . . . . . . 729
10.4 Controltheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 734
10.5 Computerlanguages . . . . . . . . . . . . . . . . . . . . . . . . 736
10.6 CompressiveSensing . . . . . . . . . . . . . . . . . . . . . . . . 737
10.7 ConstrainedLeastSquares . . . . . . . . . . . . . . . . . . . . . 738
10.8 Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739
10.9 Discretedynamicalsystemsandchaos. . . . . . . . . . . . . . . 740
10.10 Ellipticcurves . . . . . . . . . . . . . . . . . . . . . . . . . . . 743
10.11 Financialformulas . . . . . . . . . . . . . . . . . . . . . . . . . 746
10.12 Gametheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
10.13 Knottheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757
10.14 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759
10.15 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761
10.16 Momentsofinertia . . . . . . . . . . . . . . . . . . . . . . . . . 766
10.17 Music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767
10.18 Operationsresearch . . . . . . . . . . . . . . . . . . . . . . . . 769
10.19 ProofMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . 781
10.20 Recreationalmathematics . . . . . . . . . . . . . . . . . . . . . 782
10.21 Riskanalysisanddecisionrules . . . . . . . . . . . . . . . . . . 783
10.22 Signalprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . 785
10.23 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794
10.24 Votingpower . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801
10.25 Greekalphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . 803
10.26 Braillecode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803
10.27 Morsecode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803
10.28 BarCodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804
ListofReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805
ListofFigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809
ListofNotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819