<h4>CRC STANDARD MATHEMATICAL
TABLES and FORMULAE</h4>
<h5>Daniel Zwillinger</h5>
<h6>2003 by CRC Press LLC</h6>
<h5>CONTENTS</h5>
Preface 3
Contributors  5
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<b>CHAPTER 1. ANALYSIS </b> 11
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1.1 CONSTANTS 13        
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   1.1.1 TYPES OF NUMBERS 13        
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      1.1.1.1 Natural numbers 13        
      1.1.1.2 Integers 13        
      1.1.1.3 Rational numbers 13        
      1.1.1.4 Real numbers 13        
      1.1.1.5 Complex numbers 13        
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   1.1.2 ROMAN NUMERALS 14        
   1.1.3 ARROW NOTATION 14        
   1.1.4 REPRESENTATION OF NUMBERS 15        
   1.1.5 BINARY PREFIXES 16        
   1.1.6 DECIMAL MULTIPLES AND PREFIXES 16        
   1.1.7 DECIMAL EQUIVALENTS OF COMMON FRACTIONS 17        
   1.1.8 HEXADECIMAL ADDITION AND SUBTRACTION TABLE 18        
   1.1.9 HEXADECIMAL MULTIPLICATION TABLE 18        
   1.1.10 HEXADECIMAL-DECIMAL FRACTION CONVERSION 19        
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1.2 SPECIAL NUMBERS 20        
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   1.2.1 POWERS OF 2 20        
   1.2.2 POWERS OF 16 IN DECIMAL SCALE 22        
   1.2.3 POWERS OF 10 IN HEXADECIMAL SCALE 23        
   1.2.4 SPECIAL CONSTANTS 23        
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      1.2.4.1 The constant ir 23        
      1.2.4.2 The constant e 25        
      1.2.4.3 The constant 7 25        
      1.2.4.4 The constant j 26        
      1.2.4.5 Other constants 26        
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   1.2.5 CONSTANTS IN DIFFERENT BASES 26        
   1.2.6 FACTORIALS 27        
   1.2.7 BERNOULLI POLYNOMIALS AND NUMBERS 29        
   1.2.8 EULER POLYNOMIALS AND NUMBERS 30        
   1.2.9 FIBONACCI NUMBERS 31        
   1.2.10 POWERS OF INTEGERS 31        
   1.2.11 SUMS OF POWERS OF INTEGERS 32        
   1.2.12 NEGATIVE INTEGER POWERS 33        
   1.2.13 DEBRUIJN SEQUENCES 34        
   1.2.14 INTEGER SEQUENCES 35        
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1.3 SERIES AND PRODUCTS 41        
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   1.3.1 DEFINITIONS 41        
   1.3.2 GENERAL PROPERTIES 42        
   1.3.3 CONVERGENCE TESTS 43        
   1.3.4 TYPES OF SERIES 44        
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      1.3.4.1 Bessel series 44        
      1.3.4.2 Dirichlet series 45        
      1.3.4.3 Fourier series 45        
      1.3.4.4 Hypergeometric series 46        
      1.3.4.5 Power series 46        
      1.3.4.6 Taylor series 46        
      1.3.4.7 Telescoping series 47        
      1.3.4.8 Other types of series 48        
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   1.3.5 SUMMATION FORMULAE 50        
   1.3.6 IMPROVING CONVERGENCE: SHANKS 50        
   1.3.7 SUMMABILITY METHODS 51        
   1.3.8 OPERATIONS WITH POWER SERIES 51        
   1.3.9 MISCELLANEOUS SUMS AND SERIES 51        
   1.3.10 INFINITE SERIES 52        
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      1.3.10.1 Algebraic functions 52        
      1.3.10.2 Exponential functions 53        
      1.3.10.3 Logarithmic functions 53        
      1.3.10.4 Trigonometric functions 54        
      1.3.10.5 Inverse trigonometric functions 55        
      1.3.10.6 Hyperbolic functions 55        
      1.3.10.7 Inverse hyperbolic functions 56        
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   1.3.11 INFINITE PRODUCTS 57        
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      1.3.11.1 Weierstrass theorem 57        
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   1.3.12 INFINITE PRODUCTS AND INFINITE SERIES 57        
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1.4 FOURIER SERIES 58        
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   1.4.1 SPECIAL CASES 59        
   1.4.2 ALTERNATE FORMS 60        
   1.4.3 USEFUL SERIES 60        
   1.4.4 EXPANSIONS OF BASIC PERIODIC FUNCTIONS 61        
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1.5 COMPLEX ANALYSIS 63        
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   1.5.1 DEFINITIONS 63        
   1.5.2 OPERATIONS ON COMPLEX NUMBERS 64        
   1.5.3 FUNCTIONS OF A COMPLEX VARIABLE 64        
   1.5.4 CAUCHY-RIEMANN EQUATIONS 65        
   1.5.5 CAUCHY INTEGRAL THEOREM 65        
   1.5.6 CAUCHY INTEGRAL FORMULA 65        
   1.5.7 TAYLOR SERIES EXPANSIONS 65        
   1.5.8 LAURENT SERIES EXPANSIONS 66        
   1.5.9 ZEROS AND SINGULARITIES 66        
   1.5.10 RESIDUES 67        
   1.5.11 THE ARGUMENT PRINCIPLE 68        
   1.5.12 TRANSFORMATIONS AND MAPPINGS 68        
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      1.5.12.1 Bilinear transformations 68        
      1.5.12.2 Table of transformations 69        
      1.5.12.3 Table of conformal mappings 70        
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1.6 INTERVAL ANALYSIS 75        
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   1.6.1 INTERVAL ARITHMETIC RULES 75        
   1.6.2 INTERVAL ARITHMETIC 75        
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1.7 REAL ANALYSIS 76        
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   1.7.1 RELATIONS 76        
   1.7.2 FUNCTIONS (MAPPINGS) 76        
   1.7.3 SETS OF REAL NUMBERS 77        
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      1.7.3.1 Axioms of order 77        
      1.7.3.2 Definitions 78        
      1.7.3.3 Completeness (or least upper bound) axiom 78        
      1.7.3.4 Characterization of the real numbers 78        
      1.7.3.5 Definition of infinity 78        
      1.7.3.6 Inequalities among real numbers 78        
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   1.7.4 TOPOLOGICAL SPACE 79        
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      1.7.4.1 Notes 79        
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   1.7.5 METRIC SPACE 79        
   1.7.6 CONVERGENCE IN E WITH METRIC \x - y\ 80        
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      1.7.6.1 Limit of a sequence 80        
      1.7.6.2 Limit of a function 81        
      1.7.6.3 Limit of a sequence of functions 81        
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   1.7.7 CONTINUITY IN E WITH METRIC \x - y\ 81        
   1.7.8 BANACH SPACE 82        
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      1.7.8.1 Inequalities 83        
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   1.7.9 HILBERT SPACE 84        
   1.7.10 ASYMPTOTIC RELATIONSHIPS 85        
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1.8 GENERALIZED FUNCTIONS 86        
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   1.8.1 DELTA FUNCTION 86        
   1.8.2 OTHER GENERALIZED FUNCTIONS 87        
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HAPTER 2. ALGEBRA 88        
2.1 PROOFS WITHOUT WORDS 90        
2.2 ELEMENTARY ALGEBRA 92        
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   2.2.1 BASIC ALGEBRA 92        
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      2.2.1.1 Algebraic equations 92        
      2.2.1.2 Roots of polynomials 93        
      2.2.1.3 Resultants 94        
      2.2.1.4 Algebraic identities 94        
      2.2.1.5 Laws of exponents 95        
      2.2.1.6 Proportion 95        
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   2.2.2 PROGRESSIONS 95        
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      2.2.2.1 Arithmetic progression 95        
      2.2.2.2 Geometric progression 95        
      2.2.2.3 Means 96        
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   2.2.3 DEMOIVRE'S THEOREM 96        
   2.2.4 PARTIAL FRACTIONS 96        
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      2.2.4.1 Single 97        
      2.2.4.2 Repeated 97        
      2.2.4.3 Single quadratic factor 97        
      2.2.4.4 Repeated quadratic factor 97        
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2.3 POLYNOMIALS 98        
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   2.3.1 QUADRATIC POLYNOMIALS 98        
   2.3.2 CUBIC POLYNOMIALS 98        
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      2.3.2.1 Trigonometric solution of cubic polynomials 98        
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   2.3.3 QUARTIC POLYNOMIALS 99        
   2.3.4 QUARTIC CURVES 99        
   2.3.5 QUINTIC POLYNOMIALS 99        
   2.3.6 TSCHIRNHAUS' TRANSFORMATION 100        
   2.3.7 POLYNOMIAL NORMS 100        
   2.3.8 CYCLOTOMIC POLYNOMIALS 100        
   2.3.9 OTHER POLYNOMIAL PROPERTIES 102        
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2.4 NUMBER THEORY 102        
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   2.4.1 DIVISIBILITY 102        
   2.4.2 CONGRUENCES 103        
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      2.4.2.1 Properties of congruences 104        
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   2.4.3 CHINESE REMAINDER THEOREM 105        
   2.4.4 CONTINUED FRACTIONS 105        
   2.4.5 DIOPHANTINE EQUATIONS 107        
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      2.4.5.1 Pythagorean triples 107        
      2.4.5.2 Pell's equation 108        
      2.4.5.3 Waring's problem 109        
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   2.4.6 GREATEST COMMON DIVISOR 110        
   2.4.7 LEAST COMMON MULTIPLE 110        
   2.4.8 MAGIC SQUARES 110        
   2.4.9 MOBIUS FUNCTION 111        
   2.4.10 PRIME NUMBERS 112        
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      2.4.10.1 Prime formulae 112        
      2.4.10.2 Lucas-Lehmer primality test 113        
      2.4.10.3 Primality test certi cates 113        
      2.4.10.4 Probabilistic primality test 113        
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   2.4.11 PRIME NUMBERS OF SPECIAL FORMS 114        
   2.4.12 PRIME NUMBERS LESS THAN 100,000 117        
   2.4.13 FACTORIZATION TABLE 134        
   2.4.14 FACTORIZATION OF2m - 1 137        
   2.4.15 EULER TOTIENT FUNCTION 137        
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      2.4.15.1 De nitions 137        
      2.4.15.2 Properties of the totient function 137        
      2.4.15.3 Table of totient function values 138        
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2.5 VECTOR ALGEBRA 140        
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   2.5.1 NOTATION FOR VECTORS AND SCALARS 140        
   2.5.2 PHYSICAL VECTORS 140        
   2.5.3 FUNDAMENTAL DEFINITIONS 140        
   2.5.4 LAWS OF VECTOR ALGEBRA 141        
   2.5.5 VECTOR NORMS 142        
   2.5.6 DOT, SCALAR, OR INNER PRODUCT 142        
   2.5.7 VECTOR OR CROSS PRODUCT 144        
   2.5.8 SCALAR AND VECTOR TRIPLE PRODUCTS 145        
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2.6 LINEAR AND MATRIX ALGEBRA 146        
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   2.6.1 DEFINITIONS 146        
   2.6.2 TYPES OF MATRICES 147        
   2.6.3 CONFORMABILITY FOR ADDITION AND 151        
   2.6.4 DETERMINANTS AND PERMANENTS 153        
   2.6.5 MATRIX NORMS 155        
   2.6.6 SINGULARITY, RANK, AND INVERSES 156        
   2.6.7 SYSTEMS OF 2.6.8 2.6.9 TRACES 159        
   2.6.10 GENERALIZED INVERSES 160        
   2.6.11 EIGENSTRUCTURE 161        
   2.6.12 MATRIX DIAGONALIZATION 163        
   2.6.13 MATRIX EXPONENTIALS 164        
   2.6.14 QUADRATIC FORMS 164        
   2.6.15 MATRIX FACTORIZATIONS 165        
   2.6.16 THEOREMS 166        
   2.6.17 THE VECTOR OPERATION 167        
   2.6.18 KRONECKER PRODUCTS 168        
   2.6.19 KRONECKER SUMS 169        
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2.7 ABSTRACT ALGEBRA 169        
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   2.7.1 DEFINITIONS 169        
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      2.7.1.1 Examples of semigroups and monoids 170        
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   2.7.2 GROUPS 170        
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      2.7.2.1 Facts about groups 171        
      2.7.2.2 Examples of groups 172        
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   2.7.3 RINGS 173        
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      2.7.3.1 Denitions 173        
      2.7.3.2 Facts about rings 174        
      2.7.3.3 Examples of rings 174        
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   2.7.4 FIELDS 176        
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      2.7.4.1 Denitions 176        
      2.7.4.2 Examples of elds 176        
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   2.7.5 QUADRATIC FIELDS 176        
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      2.7.5.1 Denitions 176        
      2.7.5.2 Facts about quadratic elds 177        
      2.7.5.3 Examples of quadratic elds 178        
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   2.7.6 FINITE FIELDS 178        
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      2.7.6.1 Facts about nite elds 178        
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   2.7.7 HOMOMORPHISMS AND ISOMORPHISMS 179        
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      2.7.7.1 Denitions 179        
      2.7.7.2 Facts about homomorphisms and isomorphisms 180        
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   2.7.8 MATRIX CLASSES THAT ARE GROUPS 180        
   2.7.9 PERMUTATION GROUPS 181        
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      2.7.9.1 Creating new permutation groups 181        
      2.7.9.2 Polya theory 181        
      2.7.9.3 Polya theory tables 183        
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   2.7.10 TABLES 185        
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      2.7.10.1 Number of non-isomorphic groups of different orders 185        
      2.7.10.2 Number of non-isomorphic Abelian groups of different 186        
      2.7.10.3 Names of groups of small order 187        
      2.7.10.4 Representations of groups of small order 187        
      2.7.10.5 Small nite elds 198        
      2.7.10.6 Addition and multiplication tables for F2, F3, F4, and Fs 199        
      2.7.10.7 Linear characters 200        
      2.7.10.8 List of all sporadic nite simple groups 200        
      2.7.10.9 Indices and power residues 201        
      2.7.10.10 Power residues in Zp 202        
      2.7.10.11 Table of primitive monic polynomials 203        
      2.7.10.12 Table of irreducible polynomials in l®[x] 203        
      2.7.10.13 Table of primitive roots 204        
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HAPTER 3. DISCRETE MATHEMATICS 206        
3.1 SYMBOLIC LOGIC 208        
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   3.1.1 PROPOSITIONAL CALCULUS 208        
   3.1.2 TAUTOLOGIES 208        
   3.1.3 TRUTH TABLES AS FUNCTIONS 209        
   3.1.4 RULES OF INFERENCE 209        
   3.1.5 DEDUCTIONS 210        
   3.1.6 PREDICATE CALCULUS 210        
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3.2 SET THEORY 211        
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   3.2.1 SETS 211        
   3.2.2 SET OPERATIONS AND RELATIONS 211        
   3.2.3 CONNECTION BETWEEN SETS AND PROBABILITY 212        
   3.2.4 VENN DIAGRAMS 212        
   3.2.5 PARADOXES AND THEOREMS OF SET THEORY 212        
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      3.2.5.1 Russell's paradox 212        
      3.2.5.2 In nite sets and the continuum hypothesis 213        
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   3.2.6 INCLUSION/EXCLUSION 213        
   3.2.7 PARTIALLY ORDERED SETS 213        
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3.3 COMBINATORICS 215        
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   3.3.1 SAMPLE SELECTION 215        
   3.3.2 BALLS INTO CELLS 215        
   3.3.3 BINOMIAL COEFFICIENTS 217        
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      3.3.3.1 Pascal's triangle 217        
      3.3.3.2 Binomial coef cient relationships 218        
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   3.3.4 MULTINOMIAL COEFFICIENTS 218        
   3.3.5 ARRANGEMENTS AND DERANGEMENTS 219        
   3.3.6 PARTITIONS 219        
   3.3.7 BELL NUMBERS 220        
   3.3.8 CATALAN NUMBERS 221        
   3.3.9 STIRLING CYCLE NUMBERS 221        
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      3.3.9.1 Properties of Stirling cycle numbers [? 221        
      3.3.9.2 Table of Stirling cycle numbers 222        
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   3.3.10 STIRLING SUBSET NUMBERS 222        
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      3.3.10.1 Properties of Stirling subset numbers {?} 222        
      3.3.10.2 Table of Stirling subset numbers {? 223        
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   3.3.11 TABLES 224        
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      3.3.11.1 Permutations P(,m) 224        
      3.3.11.2 Combinations C(n,m) = 224        
      3.3.11.3 Fractional binomial coef cients 227        
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3.4 GRAPHS 228        
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   3.4.1 NOTATION 228        
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      3.4.1.1 Notation for graphs 228        
      3.4.1.2 Graph invariants 228        
      3.4.1.3 Examples of graphs 228        
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   3.4.2 BASIC DEFINITIONS 228        
   3.4.3 CONSTRUCTIONS 237        
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      3.4.3.1 Operations on graphs 237        
      3.4.3.2 Graphs described by one parameter 238        
      3.4.3.3 Graphs described by two parameters 239        
      3.4.3.4 Graphs described by three or more parameters 239        
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   3.4.4 FUNDAMENTAL RESULTS 240        
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      3.4.4.1 Walks and connectivity 240        
      3.4.4.2 Trees 241        
      3.4.4.3 Circuits and cycles 242        
      3.4.4.4 Cliques and independent sets 242        
      3.4.4.5 Colorings and partitions 243        
      3.4.4.6 Distance 244        
      3.4.4.7 Drawings, embeddings, planarity, and thickness 245        
      3.4.4.8 Vertex degrees 246        
      3.4.4.9 Algebraic methods 246        
      3.4.4.10 Matchings 247        
      3.4.4.11 Enumeration 248        
      3.4.4.12 Descriptions of graphs with few vertices 250        
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   3.4.5 TREE DIAGRAMS 250        
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3.5 COMBINATORIAL DESIGN THEORY 250        
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   3.5.1 t-DESIGNS 250        
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      3.5.1.1 Related designs 253        
      3.5.1.2 The Mathieu 5-design 253        
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   3.5.2 BALANCED INCOMPLETE BLOCK DESIGNS (BIBDS) 254        
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      3.5.2.1 Symmetric designs 254        
      3.5.2.2 Existence table for BIBDs 255        
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   3.5.3 DIFFERENCE SETS 255        
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      3.5.3.1 Some families of cyclic difference sets 255        
      3.5.3.2 Existence table of cyclic difference sets 256        
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   3.5.4 FINITE GEOMETRY 256        
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      3.5.4.1 Af ne planes 256        
      3.5.4.2 Projective planes 257        
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   3.5.5 STEINER TRIPLE SYSTEMS 258        
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      3.5.5.1 Some families of Steiner triple systems 258        
      3.5.5.2 Resolvable Steiner triple systems 258        
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   3.5.6 HADAMARD MATRICES 258        
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      3.5.6.1 Some Hadamard matrices 259        
      3.5.6.2 Designs and Hadamard matrices 259        
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   3.5.7 LATIN SQUARES 260        
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      3.5.7.1 Examples of mutually orthogonal Latin squares 260        
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   3.5.8 ROOM SQUARES 261        
   3.5.9 COSTAS ARRAYS 261        
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3.6 COMMUNICATION THEORY 262        
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   3.6.1 INFORMATION THEORY 262        
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      3.6.1.1 Denitions 262        
      3.6.1.2 Continuous entropy 263        
      3.6.1.3 Channel capacity 264        
      3.6.1.4 Shannon's theorem 264        
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   3.6.2 BLOCK CODING 265        
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      3.6.2.1 Denitions 265        
      3.6.2.2 Coding diagram for 265        
      3.6.2.3 Cyclic codes 265        
      3.6.2.4 Bounds 267        
      3.6.2.5 Table of binary BCH codes 268        
      3.6.2.6 Table of best binary codes 268        
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   3.6.3 SOURCE CODING FOR ENGLISH TEXT 269        
   3.6.4 MORSE CODE 269        
   3.6.5 GRAY CODE 270        
   3.6.6 FINITE FIELDS 270        
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      3.6.6.1 Irreducible polynomials 270        
      3.6.6.2 Table of binary irreducible polynomials 270        
      3.6.6.3 Table of binary primitive polynomials 271        
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   3.6.7 BINARY SEQUENCES 272        
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      3.6.7.1 Barker sequences 272        
      3.6.7.2 Periodic sequences 272        
      3.6.7.3 Them-sequences 272        
      3.6.7.4 Shift registers 273        
      3.6.7.5 Binary sequences with two-valued autocorrelation 273        
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3.7 DIFFERENCE EQUATIONS 274        
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   3.7.1 THE CALCULUS OF FINITE DIFFERENCES 274        
   3.7.2 EXISTENCE AND UNIQUENESS 274        
   3.7.3 Linear independence: general solution 275        
   3.7.4 HOMOGENEOUS EQUATIONS WITH CONSTANT 276        
   3.7.5 NON-HOMOGENEOUS EQUATIONS 277        
   3.7.6 GENERATING FUNCTIONS AND Z TRANSFORMS 277        
   3.7.7 CLOSED-FORM SOLUTIONS FOR SPECIAL EQUATIONS 278        
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      3.7.7.1 First order equation 278        
      3.7.7.2 Riccati equation 279        
      3.7.7.3 Logistic equation 280        
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3.8 DISCRETE DYNAMICAL SYSTEMS AND 281        
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   3.8.1 CHAOTIC ONE-DIMENSIONAL MAPS 281        
   3.8.2 LOGISTIC MAP 281        
   3.8.3 JULIA SETS AND THE MANDELBROT SET 282        
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3.9 GAME THEORY 283        
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   3.9.1 TWO PERSON NON-COOPERATIVE MATRIX GAMES 283        
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      3.9.1.1 The pure zero sum game 284        
      3.9.1.2 The mixed zero sum game 285        
      3.9.1.3 The non-zero sum game 286        
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   3.9.2 VOTING POWER 287        
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      3.9.2.1 Voting power de nitions 287        
      3.9.2.2 Shapley-Shubik power index 287        
      3.9.2.3 Banzhaf power index 288        
      3.9.2.4 Voting power examples 288        
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3.10 OPERATIONS RESEARCH 289        
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   3.10.1 LINEAR PROGRAMMING 289        
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      3.10.1.1 Modeling in LP 290        
      3.10.1.2 Transformation to standard form 291        
      3.10.1.3 Solving LP models: simplex method 292        
      3.10.1.4 Solving LP models: interior point method 293        
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   3.10.2 DUALITY AND COMPLEMENTARY SLACKNESS 294        
   3.10.3 LINEAR INTEGER PROGRAMMING 296        
   3.10.4 BRANCH AND BOUND 296        
   3.10.5 NETWORK FLOW METHODS 297        
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      3.10.5.1 Maximum o w 297        
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   3.10.6 ASSIGNMENT PROBLEM 297        
   3.10.7 DYNAMIC PROGRAMMING 298        
   3.10.8 SHORTEST PATH PROBLEM 299        
   3.10.9 HEURISTIC SEARCH TECHNIQUES 299        
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      3.10.9.1 Simulated annealing (SA) 300        
      3.10.9.2 Tabu search 301        
      3.10.9.3 Genetic algorithms 302        
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<b>CHAPTER 4. GEOMETRY </b> 305
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4.1 COORDINATE SYSTEMS IN THE PLANE 307        
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   4.1.1 CONVENTION 307        
   4.1.2 SUBSTITUTIONS AND TRANSFORMATIONS 307        
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      4.1.2.1 Substitutions 307        
      4.1.2.2 Transformations 308        
      4.1.2.3 Using transformations to change coordinates 308        
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   4.1.3 CARTESIAN COORDINATES IN THE PLANE 309        
   4.1.4 POLAR COORDINATES IN THE PLANE 310        
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      4.1.4.1 Relations between Cartesian and polar coordinates 311        
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   4.1.5 HOMOGENEOUS COORDINATES IN THE PLANE 311        
   4.1.6 OBLIQUE COORDINATES IN THE PLANE 311        
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      4.1.6.1 Relations between two oblique coordinate systems 312        
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4.2 PLANE SYMMETRIES OR ISOMETRIES 313        
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   4.2.1 FORMULAE FOR SYMMETRIES: CARTESIAN 313        
   4.2.2 FORMULAE FOR SYMMETRIES: HOMOGENEOUS 314        
   4.2.3 FORMULAE FOR SYMMETRIES: POLAR COORDINATES 315        
   4.2.4 CRYSTALLOGRAPHIC GROUPS 315        
   4.2.5 CLASSIFYING THE CRYSTALLOGRAPHIC GROUPS 320        
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4.3 OTHER TRANSFORMATIONS OF THE PLANE 320        
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   4.3.1 SIMILARITIES 320        
   4.3.2 AFFINE TRANSFORMATIONS 321        
   4.3.3 PROJECTIVE TRANSFORMATIONS 322        
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4.4 LINES 322        
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   4.4.1 Lines with prescribed properties 323        
   4.4.2 DISTANCES 324        
   4.4.3 ANGLES 324        
   4.4.4 CONCURRENCE AND COLLINEARITY 325        
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4.5 POLYGONS 325        
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   4.5.1 TRIANGLES 326        
   4.5.2 QUADRILATERALS 330        
   4.5.3 REGULAR POLYGONS 331        
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4.6 CONICS 333        
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   4.6.1 ALTERNATIVE CHARACTERIZATION 333        
   4.6.2 THE GENERAL QUADRATIC EQUATION 336        
   4.6.3 ADDITIONAL PROPERTIES OF ELLIPSES 338        
   4.6.4 ADDITIONAL PROPERTIES OF HYPERBOLAS 339        
   4.6.5 ADDITIONAL PROPERTIES OF PARABOLAS 341        
   4.6.6 CIRCLES 341        
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4.7 SPECIAL PLANE CURVES 344        
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   4.7.1 ALGEBRAIC CURVES 344        
   4.7.3 CURVES IN POLAR COORDINATES 350        
   4.7.4 SPIRALS 350        
   4.7.5 THE PEANO CURVE AND FRACTAL CURVES 351        
   4.7.6 FRACTAL OBJECTS 352        
   4.7.7 CLASSICAL CONSTRUCTIONS 353        
// ]
4.8 COORDINATE SYSTEMS IN SPACE 353        
// [
   4.8.1 Conventions 353        
   4.8.1 CARTESIAN COORDINATES IN SPACE 353        
   4.8.2 CYLINDRICAL COORDINATES IN SPACE 354        
   4.8.3 SPHERICAL COORDINATES IN SPACE 354        
   4.8.4 RELATIONS BETWEEN CARTESIAN, CYLINDRICAL, AND 356        
   4.8.5 HOMOGENEOUS COORDINATES IN SPACE 356        
// ]
4.9 SPACE SYMMETRIES OR ISOMETRIES 356        
// [
   4.9.1 FORMULAE FOR SYMMETRIES: CARTESIAN 357        
   4.9.2 FORMULAE FOR SYMMETRIES: HOMOGENEOUS 359        
// ]
4.10 OTHER TRANSFORMATIONS OF SPACE 360        
// [
   4.10.1 SIMILARITIES 360        
   4.10.2 AFFINE TRANSFORMATIONS 360        
   4.10.3 PROJECTIVE TRANSFORMATIONS 361        
// ]
4.11 DIRECTION ANGLES AND DIRECTION 361        
4.12 PLANES 362        
// [
   4.12.1 PLANES WITH PRESCRIBED PROPERTIES 362        
   4.12.2 CONCURRENCE AND COPLANARITY 363        
// ]
4.13 LINES IN SPACE 363        
// [
   4.13.1 DISTANCES 364        
   4.13.2 ANGLES 364        
   4.13.3 CONCURRENCE, COPLANARITY, PARALLELISM 365        
// ]
4.14 POLYHEDRA 365        
// [
   4.14.1 CONVEX REGULAR POLYHEDRA 366        
   4.14.2 POLYHEDRA NETS 368        
// ]
4.15 CYLINDERS 369        
4.16 CONES 369        
4.17 SURFACES OF REVOLUTION: THE TORUS 371        
4.18 QUADRICS 372        
// [
   4.18.1 SPHERES 373        
// [
      4.18.1.1 Spherical cap 375        
      4.18.1.2 Spherical zone (of two bases) 375        
      4.18.1.3 Spherical segment and lune 376        
      4.18.1.4 Volume and area of spheres 376        
// ]
// ]
4.19 SPHERICAL GEOMETRY & TRIGONOMETRY 376        
// [
   4.19.1 RIGHT SPHERICAL TRIANGLES 376        
// [
      4.19.1.1 Napier's rules of circular parts 376        
      4.19.1.2 Rules for determining quadrant 377        
// ]
   4.19.2 OBLIQUE SPHERICAL TRIANGLES 377        
// [
      4.19.2.1 Spherical law of sines 378        
      4.19.2.2 Spherical law of cosines for sides 378        
      4.19.2.3 Spherical law of cosines for angles 378        
      4.19.2.4 Spherical law of tangents 378        
      4.19.2.5 Spherical half angle formulae 378        
      4.19.2.6 Spherical half side formulae 379        
      4.19.2.7 Gauss's formulae 379        
      4.19.2.8 Napier's analogs 379        
      4.19.2.9 Rules for determining quadrant 379        
      4.19.2.10 Summary of solution of oblique spherical triangles 380        
      4.19.2.11 Haversine formulae 380        
      4.19.2.12 Finding the distance between two points on the earth 380        
// ]
// ]
4.20 DIFFERENTIAL GEOMETRY 381        
// [
   4.20.1 CURVES 381        
// [
      4.20.1.1 De nitions 381        
      4.20.1.2 Results 382        
      4.20.1.3 Example 383        
// ]
   4.20.2 SURFACES 384        
// [
      4.20.2.1 Definitions 384        
      4.20.2.2 Results 386        
      4.20.2.3 Example: paraboloid of revolution 388        
// ]
// ]
4.21 ANGLE CONVERSION 389        
4.22 KNOTS UP TO EIGHT CROSSINGS 390        
// ]
 
<b>CHAPTER 5. CONTINUOUS MATHEMATICS </b> 391
// [
 
5.1 DIFFERENTIAL CALCULUS 393        
// [
   5.1.1 LIMITS 393        
   5.1.2 DERIVATIVES 394        
   5.1.3 DERIVATIVES OF COMMON FUNCTIONS 394        
   5.1.4 DERIVATIVE FORMULAE 395        
   5.1.5 DERIVATIVE THEOREMS 396        
   5.1.6 THE TWO-DIMENSIONAL CHAIN RULE 396        
   5.1.7 LHOSPITAL'S RULE 397        
   5.1.8 MAXIMA AND MINIMA OF FUNCTIONS 397        
// [
      5.1.8.1 Lagrange multipliers 397        
// ]
   5.1.9 VECTOR CALCULUS 398        
   5.1.10 MATRIX AND VECTOR DERIVATIVES 399        
// [
      5.1.10.1 De nitions 399        
      5.1.10.2 Properties 401        
// ]
// ]
5.2 DIFFERENTIAL FORMS 403        
// [
   5.2.1 PRODUCTS OF 1 -FORMS 403        
   5.2.2 DIFFERENTIAL 2-FORMS 404        
   5.2.3 THE 2-FORMS IN Rn 404        
   5.2.4 HIGHER DIMENSIONAL FORMS 405        
   5.2.5 THE EXTERIOR DERIVATIVE 405        
   5.2.6 PROPERTIES OF THE EXTERIOR DERIVATIVE 406        
// ]
5.3 INTEGRATION 406        
// [
   5.3.1 DEFINITIONS 406        
   5.3.2 PROPERTIES OF INTEGRALS 407        
   5.3.3 METHODS OF EVALUATING INTEGRALS 408        
// [
      5.3.3.1 Substitution 408        
      5.3.3.2 Partial fraction decomposition 409        
      5.3.3.3 Useful transformations 410        
      5.3.3.4 Integration by parts 410        
      5.3.3.5 Extended integration by parts rule 412        
// ]
   5.3.4 TYPES OF INTEGRALS 412        
// [
      5.3.4.1 Line and surface integrals 412        
      5.3.4.2 Contour integrals 414        
// ]
   5.3.5 INTEGRAL INEQUALITIES 414        
   5.3.6 CONVERGENCE TESTS 415        
   5.3.7 VARIATIONAL PRINCIPLES 415        
   5.3.8 CONTINUITY OF INTEGRAL ANTIDERIVATIVES 415        
   5.3.9 ASYMPTOTIC INTEGRAL EVALUATION 416        
   5.3.10 SPECIAL FUNCTIONS DEFINED BY INTEGRALS 416        
   5.3.11 APPLICATIONS OF INTEGRATION 417        
   5.3.12 MOMENTS OF INERTIA FOR VARIOUS BODIES 418        
   5.3.13 TABLES OF INTEGRALS 419        
// ]
5.4 TABLE OF INDEFINITE INTEGRALS 420        
// [
   5.4.1 ELEMENTARY FORMS 420        
   5.4.2 FORMS CONTAINING a + bx 421        
   5.4.3 FORMS CONTAINING c2±x2 AND x2 - c2 423        
   5.4.4 FORMS CONTAINING a + bx AND c + dx 423        
   5.4.5 FORMS CONTAINING a + bxn 424        
   5.4.6 FORMS CONTAINING c3±x3 426        
   5.4.7 FORMS CONTAINING c4±x4 426        
   5.4.8 FORMS CONTAINING a + bx + cx2 427        
   5.4.9 FORMS CONTAINING SQRT(a + bx) 428        
   5.4.10 FORMS CONTAINING SQRT(a + bx) AND SQRT(c + dx) 429        
   5.4.11 FORMS CONTAINING SQRT(x2±a2) 430        
   5.4.12 FORMS CONTAINING SQRT(a2-x2) 432        
   5.4.13 FORMS CONTAINING SQRT(ax+bx+cx2) 434        
   5.4.14 FORMS CONTAINING SQRT(2ax - x2) 436        
   5.4.15 MISCELLANEOUS ALGEBRAIC FORMS 436        
   5.4.16 FORMS INVOLVING TRIGONOMETRIC FUNCTIONS 438        
   5.4.17 FORMS INVOLVING INVERSE TRIGONOMETRIC 447        
   5.4.18 LOGARITHMIC FORMS 449        
   5.4.19 EXPONENTIAL FORMS 451        
   5.4.20 HYPERBOLIC FORMS 453        
   5.4.21 BESSEL FUNCTIONS 456        
// ]
5.5 TABLE OF DEFINITE INTEGRALS 456        
// [
   5.5.1 TABLE OF SEMI-INTEGRALS 463        
// ]
5.6 ORDINARY DIFFERENTIAL EQUATIONS 464        
// [
   5.6.1 Linear differential equations 464        
// [
      5.6.1.1 Vector representation 464        
      5.6.1.2 Homogeneous solution 464        
      5.6.1.3 Particular solutions 465        
      5.6.1.4 Second-order        
      5.6.1.5 Damping: none, under, over, and critical 468        
// ]
   5.6.2 SOLUTION TECHNIQUES OVERVIEW 469        
   5.6.3 INTEGRATING FACTORS 470        
   5.6.4 VARIATION OF PARAMETERS 470        
   5.6.5 GREEN'S FUNCTIONS 471        
   5.6.6 TABLE OF GREEN'S FUNCTIONS 471        
   5.6.7 TRANSFORM TECHNIQUES 472        
   5.6.8 NAMED ORDINARY DIFFERENTIAL EQUATIONS 473        
   5.6.9 LIAPUNOV'S DIRECT METHOD 474        
   5.6.10 LIE GROUPS 474        
// [
      5.6.10.1 Integrating second-order ordinary differential equations 475        
// ]
   5.6.11 STOCHASTIC DIFFERENTIAL EQUATIONS 475        
   5.6.12 TYPES OF CRITICAL POINTS 476        
// ]
5.7 PARTIAL DIFFERENTIAL EQUATIONS 476        
// [
   5.7.1 CLASSIFICATIONS OF PDES 476        
   5.7.2 NAMED PARTIAL DIFFERENTIAL EQUATIONS 477        
   5.7.3 TRANSFORMING PARTIAL DIFFERENTIAL EQUATIONS 477        
   5.7.4 WELL-POSEDNESSOFPDES 478        
   5.7.5 GREEN'S FUNCTIONS 479        
   5.7.6 QUASI-LINEAR EQUATIONS 480        
   5.7.7 SEPARATION OF VARIABLES 480        
   5.7.8 SOLUTIONS OF LAPLACE'S EQUATION 482        
   5.7.9 SOLUTIONS TO THE WAVE EQUATION 483        
   5.7.10 PARTICULAR SOLUTIONS TO SOME PDES 485        
// ]
5.8 EIGENVALUES 485        
5.9 INTEGRAL EQUATIONS 486        
// [
   5.9.1 DEFINITIONS 486        
// [
      5.9.1.1 Classi cation of integral equations 486        
      5.9.1.2 Classi cation of kernels 486        
// ]
   5.9.2 CONNECTION TO DIFFERENTIAL EQUATIONS 487        
   5.9.3 FREDHOLM ALTERNATIVE 487        
   5.9.4 SPECIAL EQUATIONS WITH SOLUTIONS 488        
// ]
5.10 TENSOR ANALYSIS 490        
// [
   5.10.1 DEFINITIONS 490        
   5.10.2 ALGEBRAIC TENSOR OPERATIONS 491        
   5.10.3 DIFFERENTIATION OF TENSORS 492        
   5.10.4 METRIC TENSOR 494        
   5.10.5 RESULTS 495        
   5.10.6 EXAMPLES OF TENSORS 497        
// ]
5.11 ORTHOGONAL COORDINATE SYSTEMS 500        
// [
   5.11.1 TABLE OF ORTHOGONAL COORDINATE SYSTEMS 502        
// ]
5.12 CONTROL THEORY 505        
// ]
 
<b>CHAPTER 6. SPECIAL FUNCTIONS </b> 507
// [
 
6.1 TRIGONOMETRIC OR CIRCULAR FUNCTIONS 511        
// [
   6.1.1 DEFINITION OF ANGLES 511        
   6.1.2 CHARACTERIZATION OF ANGLES 511        
// [
      6.1.2.1 Relation between radians and degrees 512        
// ]
   6.1.3 CIRCULAR FUNCTIONS 512        
// [
      6.1.3.1 Signs in the four quadrants 513        
// ]
   6.1.4 Circular functions of special angles 514        
   6.1.5 EVALUATING SINES AND COSINES AT MULTIPLES OFir 515        
   6.1.6 SYMMETRY AND PERIODICITY RELATIONSHIPS 515        
   6.1.7 FUNCTIONS IN TERMS OF ANGLES IN THE FIRST 515        
   6.1.8 ONE CIRCULAR FUNCTION IN TERMS OF ANOTHER 516        
   6.1.9 CIRCULAR FUNCTIONS IN TERMS OF EXPONENTIALS 516        
   6.1.10 FUNDAMENTAL IDENTITIES 517        
   6.1.11 ANGLE SUM AND DIFFERENCE RELATIONSHIPS 517        
   6.1.12 DOUBLE-ANGLE FORMULAE 517        
   6.1.13 MULTIPLE-ANGLE FORMULAE 518        
   6.1.14 HALF-ANGLE FORMULAE 518        
   6.1.15 POWERS OF CIRCULAR FUNCTIONS 519        
   6.1.16 PRODUCTS OF SINE AND COSINE 519        
   6.1.17 SUMS OF CIRCULAR FUNCTIONS 519        
// ]
6.2 CIRCULAR FUNCTIONS AND PLANAR 520        
// [
   6.2.1 RIGHT TRIANGLES 520        
   6.2.2 GENERAL PLANE TRIANGLES 520        
   6.2.3 HALF-ANGLE 522        
   6.2.4 SOLUTION OF TRIANGLES 522        
// [
      6.2.4.1 Three sides given 522        
      6.2.4.2 Given two sides F, c) and the included angle {A) 523        
      6.2.4.3 Given two sides F, c) and an angle (C), not the included 523        
      6.2.4.4 Given one side F) and two angles {B, C) 523        
// ]
   6.2.5 TABLES OF TRIGONOMETRIC FUNCTIONS 524        
// ]
6.3 INVERSE CIRCULAR FUNCTIONS 526        
// [
   6.3.1 DEFINITION IN TERMS OF AN INTEGRAL 526        
   6.3.2 PRINCIPAL VALUES OF THE INVERSE CIRCULAR 526        
   6.3.3 FUNDAMENTAL IDENTITIES 527        
   6.3.4 FUNCTIONS OF NEGATIVE ARGUMENTS 527        
   6.3.5 RELATIONSHIP TO INVERSE HYPERBOLIC FUNCTIONS 527        
   6.3.6 SUM AND DIFFERENCE OF TWO INVERSE CIRCULAR 528        
// ]
6.4 CEILING AND FLOOR FUNCTIONS 528        
6.5 EXPONENTIAL FUNCTION 528        
// [
   6.5.1 EXPONENTIATION 528        
   6.5.2 DEFINITION OF e2 529        
   6.5.3 DERIVATIVE AND INTEGRAL OF e* 529        
// ]
6.6 LOGARITHMIC FUNCTIONS 530        
// [
   6.6.1 DEFINITION OF THE NATURAL LOGARITHM 530        
// [
      6.6.1.1 Logarithms to a base other than e 530        
// ]
   6.6.2 LOGARITHM OF SPECIAL VALUES 530        
   6.6.3 RELATING THE LOGARITHM TO THE EXPONENTIAL 531        
   6.6.4 IDENTITIES 531        
   6.6.5 SERIES EXPANSIONS FOR THE NATURAL LOGARITHM 531        
   6.6.6 DERIVATIVE AND INTEGRATION FORMULAE 531        
// ]
6.7 HYPERBOLIC FUNCTIONS 531        
// [
   6.7.1 DEFINITIONS OF THE HYPERBOLIC FUNCTIONS 532        
   6.7.3 HYPERBOLIC FUNCTIONS IN TERMS OF ONE ANOTHER 532        
   6.7.4 RELATIONS AMONG HYPERBOLIC FUNCTIONS 533        
   6.7.5 RELATIONSHIP TO CIRCULAR FUNCTIONS 533        
   6.7.6 SERIES EXPANSIONS 533        
   6.7.7 SYMMETRY RELATIONSHIPS 533        
   6.7.8 SUM AND DIFFERENCE FORMULAE 534        
   6.7.9 MULTIPLE ARGUMENT RELATIONS 534        
   6.7.10 SUMS OF FUNCTIONS 534        
   6.7.11 PRODUCTS OF FUNCTIONS 535        
   6.7.12 HALF-ARGUMENT FORMULAE 535        
   6.7.13 DIFFERENTIATION FORMULAE 535        
// ]
6.8 INVERSE HYPERBOLIC FUNCTIONS 535        
// [
   6.8.1 RANGE OF VALUES 535        
   6.8.2 RELATIONSHIPS AMONG INVERSE HYPERBOLIC 536        
   6.8.3 RELATIONSHIPS WITH LOGARITHMIC FUNCTIONS 537        
   6.8.4 RELATIONSHIPS WITH CIRCULAR FUNCTIONS 537        
   6.8.5 SUM AND DIFFERENCE OF FUNCTIONS 537        
// ]
6.9 GUDERMANNIAN FUNCTION 538        
// [
   6.9.1 FUNDAMENTAL IDENTITIES 538        
   6.9.2 DERIVATIVES OF GUDERMANNIAN 538        
   6.9.3 RELATIONSHIP TO HYPERBOLIC AND CIRCULAR 539        
   6.9.4 NUMERICAL VALUES OF HYPERBOLIC FUNCTIONS 539        
// ]
6.10 ORTHOGONAL POLYNOMIALS 540        
// [
   6.10.1 HERMITE POLYNOMIALS 540        
   6.10.2 JACOB I POLYNOMIALS 541        
   6.10.3 LAGUERRE POLYNOMIALS 541        
   6.10.4 GENERALIZED LAGUERRE POLYNOMIALS 541        
   6.10.5 LEGENDRE POLYNOMIALS 542        
   6.10.6 CHEBYSHEV POLYNOMIALS, FIRST KIND 542        
   6.10.7 CHEBYSHEV POLYNOMIALS, SECOND KIND 543        
   6.10.8 TABLES OF ORTHOGONAL POLYNOMIALS 543        
// [
      6.10.8.1 Table of Jacobi polynomials 545        
// ]
   6.10.9 ZERNIKE POLYNOMIALS 545        
// [
      6.10.9.1 Properties 545        
      6.10.9.2 Tables of Zernike polynomials 546        
// ]
   6.10.10 SPHERICAL HARMONICS 546        
// [
      6.10.10.1 Table of spherical harmonics 548        
// ]
// ]
6.11 GAMMA FUNCTION 548        
// [
   6.11.1 RECURSION FORMULA 548        
   6.11.2 GAMMA FUNCTION OF SPECIAL VALUES 549        
   6.11.3 PROPERTIES 549        
   6.11.4 ASYMPTOTIC EXPANSION 550        
   6.11.5 LOGARITHMIC DERIVATIVE OF THE GAMMA FUNCTION 551        
   6.11.6 NUMERICAL VALUES 551        
// ]
6.12 BETA FUNCTION 552        
// [
   6.12.1 NUMERICAL VALUES OF THE BETA FUNCTION 553        
   6.13.2 ERROR FUNCTION OF SPECIAL VALUES 554        
   6.13.3 EXPANSIONS 554        
   6.13.4 SPECIAL CASES 554        
// ]
6.14 FRESNEL INTEGRALS 555        
// [
   6.14.1 PROPERTIES 555        
   6.14.2 ASYMPTOTIC EXPANSION 556        
   6.14.3 NUMERICAL VALUES OF ERROR FUNCTIONS AND 556        
// ]
6.15 SINE, COSINE, AND EXPONENTIAL 557        
// [
   6.15.1 SINE AND COSINE INTEGRALS 557        
   6.15.2 EXPONENTIAL INTEGRALS 558        
   6.15.3 LOGARITHMIC INTEGRAL 558        
   6.15.4 NUMERICAL VALUES 559        
// ]
6.16 POLYLOGARITHMS 559        
// [
   6.16.1 POLYLOGARITHMS OF SPECIAL VALUES 560        
   6.16.2 POLYLOGARITHM PROPERTIES 560        
// ]
6.17 HYPERGEOMETRIC FUNCTIONS 560        
// [
   6.17.1 SPECIAL CASES 561        
   6.17.2 PROPERTIES 561        
   6.17.3 RECURSION FORMULAE 562        
// ]
6.18 LEGENDRE FUNCTIONS 562        
// [
   6.18.1 DIFFERENTIAL EQUATION: LEGENDRE FUNCTION 562        
   6.18.2 DEFINITION 563        
   6.18.3 SINGULAR POINTS 563        
   6.18.4 RELATIONSHIPS 563        
   6.18.5 RECURSION RELATIONSHIPS 564        
   6.18.6 INTEGRALS 564        
   6.18.7 POLYNOMIAL CASE 564        
   6.18.8 DIFFERENTIAL EQUATION: ASSOCIATED LEGENDRE 565        
   6.18.9 RELATIONSHIPS BETWEEN THE ASSOCIATED AND 566        
   6.18.10 ORTHOGONALITY RELATIONSHIP 566        
   6.18.11 RECURSION RELATIONSHIPS 566        
// ]
6.19 BESSEL FUNCTIONS 567        
// [
   6.19.1 DIFFERENTIAL EQUATION 567        
   6.19.2 SINGULAR POINTS 567        
   6.19.3 RELATIONSHIPS 568        
   6.19.4 SERIES EXPANSIONS 568        
   6.19.5 RECURRENCE RELATIONSHIPS 568        
   6.19.6 BEHAVIOR AS 569        
   6.19.7 INTEGRALS 569        
   6.19.8 FOURIER EXPANSION 569        
   6.19.9 AUXILIARY FUNCTIONS 569        
   6.19.10 INVERSE RELATIONSHIPS 570        
   6.19.11 ASYMPTOTIC EXPANSIONS 570        
   6.19.12 ZEROS OF BESSEL FUNCTIONS 571        
// [
      6.19.12.1 Asymptotic expansions of the zeros 571        
// ]
   6.19.13 HALF ORDER BESSEL FUNCTIONS 571        
// [
      6.19.13.1 Recursion relationships 572        
      6.19.13.2 Differential equation 572        
// ]
   6.19.14 MODIFIED BESSEL FUNCTIONS 572        
   6.19.15 AIRY FUNCTIONS 573        
   6.19.16 NUMERICAL VALUES FOR THE BESSEL FUNCTIONS 575        
// ]
6.20 ELLIPTIC INTEGRALS 576        
// [
   6.20.1 DEFINITIONS 576        
   6.20.2 PROPERTIES 577        
   6.20.3 NUMERICAL VALUES OF THE ELLIPTIC INTEGRALS 578        
// ]
6.21 JACOBIAN ELLIPTIC FUNCTIONS 580        
// [
   6.21.1 PROPERTIES 580        
   6.21.2 DERIVATIVES AND INTEGRALS 581        
   6.21.3 SERIES EXPANSIONS 581        
// ]
6.22 CLEBSCH-GORDAN COEFFICIENTS 582        
6.23 INTEGRAL TRANSFORMS: PRELIMINARIES 584        
6.24 FOURIER TRANSFORM 584        
// [
   6.24.1 EXISTENCE 585        
   6.24.2 PROPERTIES 586        
   6.24.3 INVERSION FORMULA 588        
   6.24.4 POISSON SUMMATION FORMULA 588        
   6.24.5 SHANNON'S SAMPLING THEOREM 589        
   6.24.6 UNCERTAINTY PRINCIPLE 589        
   6.24.7 FOURIER SINE AND COSINE TRANSFORMS 590        
// ]
6.25 DISCRETE FOURIER TRANSFORM (DFT) 590        
// [
   6.25.1 PROPERTIES 591        
// ]
6.26 FAST FOURIER TRANSFORM (FFT) 592        
6.27 MULTIDIMENSIONAL FOURIER TRANSFORM 593        
6.28 LAPLACE TRANSFORM 593        
// [
   6.28.1 EXISTENCE AND DOMAIN OF CONVERGENCE 593        
   6.28.2 PROPERTIES 594        
   6.28.3 INVERSION FORMULAE 596        
// [
      6.28.3.1 Inversion by integration 596        
      6.28.3.2 Inversion by partial fractions 597        
// ]
   6.28.4 CONVOLUTION 597        
// ]
6.29 HANKEL TRANSFORM 597        
// [
   6.29.1 PROPERTIES 598        
// ]
6.30 HARTLEY TRANSFORM 599        
6.31 HUBERT TRANSFORM 599        
// [
   6.31.1 EXISTENCE 600        
   6.31.2 PROPERTIES 600        
   6.31.3 RELATIONSHIP WITH THE FOURIER TRANSFORM 601        
// ]
6.32 Z-TRANSFORM 602        
// [
   6.32.1 EXAMPLES 603        
   6.32.2 PROPERTIES 603        
   6.32.3 INVERSION FORMULA 604        
   6.32.4 CONVOLUTION AND PRODUCT 606        
// ]
6.33 TABLES OF TRANSFORMS 607        
// ]
 
<b>CHAPTER 7. PROBABILITY AND STATISTICS </b> 622
// [
 
7.1 PROBABILITY THEORY 624        
// [
   7.1.1 INTRODUCTION 624        
// [
      7.1.1.1 Denition of probability 624        
      7.1.1.2 Marginal and conditional probability 624        
      7.1.1.3 Probability theorems 625        
      7.1.1.4 Terminology 625        
      7.1.1.5 Characterizing random variables 626        
      7.1.1.6 Generating and characteristic functions 627        
// ]
   7.1.2 MULTIVARIATE DISTRIBUTIONS 628        
// [
      7.1.2.1 Discrete case 628        
      7.1.2.2 Continuous case 628        
      7.1.2.3 Moments 629        
      7.1.2.4 Marginal and conditional distributions 629        
// ]
   7.1.3 RANDOM SUMS OF RANDOM VARIABLES 629        
   7.1.4 TRANSFORMING VARIABLES 630        
   7.1.5 CENTRAL LIMIT THEOREM 630        
   7.1.6 INEQUALITIES 630        
   7.1.7 AVERAGES OVER VECTORS 632        
   7.1.8 GEOMETRIC PROBABILITY 632        
// ]
7.2 CLASSICAL PROBABILITY PROBLEMS 634        
// [
   7.2.1 RAISIN COOKIE PROBLEM 634        
   7.2.2 GAMBLER'S RUIN PROBLEM 634        
   7.2.3 CARD GAMES 635        
   7.2.4 DISTRIBUTION OF DICE SUMS 636        
   7.2.5 BIRTHDAY PROBLEM 636        
// ]
7.3 PROBABILITY DISTRIBUTIONS 637        
// [
   7.3.1 DISCRETE DISTRIBUTIONS 637        
   7.3.2 CONTINUOUS DISTRIBUTIONS 640        
// ]
7.4 QUEUING THEORY 644        
// [
   7.4.1 VARIABLES 644        
   7.4.2 THEOREMS 645        
// ]
7.5 MARKOV CHAINS 647        
// [
   7.5.1 TRANSITION FUNCTION AND MATRIX 647        
// [
      7.5.1.1 Transition function 647        
      7.5.1.2 Transition matrix 647        
// ]
   7.5.2 RECURRENCE 648        
   7.5.3 STATIONARY DISTRIBUTIONS 648        
// [
      7.5.3.1 Example: A simple three-state Markov chain 649        
// ]
   7.5.4 RANDOM WALKS 650        
   7.5.5 EHRENFEST CHAIN 650        
// ]
7.6 RANDOM NUMBER GENERATION 651        
// [
   7.6.1 METHODS OF PSEUDORANDOM NUMBER GENERATION 651        
// [
      7.6.1.1 Methods Of Pseudorandom Number Generation 651        
      7.6.1.2 Shift-register generators 652        
      7.6.1.3 Lagged-Fibonacci generators 653        
      7.6.1.4 Non-linear generators 653        
// ]
   7.6.2 GENERATING NON-UNIFORM RANDOM VARIABLES 654        
// [
      7.6.2.1 Discrete random variables 655        
      7.6.2.2 Testing pseudorandom numbers 656        
// ]
// ]
7.7 CONTROL CHARTS AND RELIABILITY 657        
// [
   7.7.1 CONTROL CHARTS 657        
   7.7.2 ACCEPTANCE SAMPLING 659        
   7.7.3 RELIABILITY 660        
   7.7.4 FAILURE TIME DISTRIBUTIONS 662        
// [
      7.7.4.1 Use of the exponential distribution 662        
// ]
// ]
7.8 RISK ANALYSIS AND DECISION RULES 663        
7.9 STATISTICS 665        
// [
   7.9.1 DESCRIPTIVE STATISTICS 665        
   7.9.2 STATISTICAL ESTIMATORS 668        
// [
      7.9.2.1 Denitions 668        
      7.9.2.2 Consistent estimators 668        
      7.9.2.3 Efcient estimators 669        
      7.9.2.4 Maximum likelihood estimators (MLE) 669        
      7.9.2.5 Method of moments (MOM) 669        
      7.9.2.6 Suf cient statistics 670        
      7.9.2.7 UMVU estimators 670        
      7.9.2.8 Unbiased estimators 670        
// ]
   7.9.3 CRAMER-RAO BOUND 671        
   7.9.4 ORDER STATISTICS 671        
// [
      7.9.4.1 Uniform distribution: 672        
      7.9.4.2 Normal distribution: 672        
// ]
   7.9.5 CLASSIC STATISTICS PROBLEMS 672        
// [
      7.9.5.1 Sample size problem 672        
      7.9.5.2 Large scale testing with infrequent success 672        
// ]
// ]
7.10 CONFIDENCE INTERVALS 673        
// [
   7.10.1 CONFIDENCE INTERVAL: SAMPLE FROM ONE 673        
   7.10.2 CONFIDENCE INTERVAL: SAMPLES FROM TWO 674        
// ]
7.11 TESTS OF HYPOTHESES 676        
// [
   7.11.1 HYPOTHESIS TESTS: PARAMETER FROM ONE 677        
   7.11.2 HYPOTHESIS TESTS: PARAMETERS FROM TWO 680        
   7.11.3 HYPOTHESIS TESTS: DISTRIBUTION OF A 684        
   7.11.4 HYPOTHESIS TESTS: DISTRIBUTIONS OF TWO 687        
   7.11.5 SEQUENTIAL PROBABILITY RATIO TESTS 688        
// ]
7.12 LINEAR REGRESSION       
// [
   7.12.1 linear model yi = bo + b1Xi + e       
   7.12.2 GENERAL MODEL y = bo + b1X1 +... + bnXn + e 692        
// ]
7.13 ANALYSIS OF VARIANCE (ANOVA) 693        
// [
   7.13.1 ONE-FACTOR ANOVA 694        
   7.13.2 UNREPLICATED TWO-FACTOR ANOVA 696        
   7.13.3 REPLICATED TWO-FACTOR ANOVA 699        
// ]
7.14 PROBABILITY TABLES 702        
// [
   7.14.1 CRITICAL VALUES 702        
   7.14.2 TABLE OF THE NORMAL DISTRIBUTION 703        
   7.14.3 PERCENTAGE POINTS, STUDENT'S t-DISTRIBUTION 709        
   7.14.5 PERCENTAGE POINTS, F-DISTRIBUTION 711        
   7.14.6 CUMULATIVE TERMS, BINOMIAL DISTRIBUTION 717        
   7.14.7 CUMULATIVE TERMS, POISSON DISTRIBUTION 719        
   7.14.8 CRITICAL VALUES, KOLMOGOROV-SMIRNOV TEST 724        
   7.14.9 CRITICAL VALUES, TWO SAMPLE 724        
   7.14.10 CRITICAL VALUES, SPEARMAN'S RANK 725        
// ]
7.15 SIGNAL PROCESSING 725        
// [
   7.15.1 ESTIMATION 725        
   7.15.2 KALMAN FILTERS 726        
// [
      7.15.2.1 Discrete Kalman lter 726        
      7.15.2.2 Continuous Kalman Iter 727        
      7.15.2.3 Continuous extended Kalman Iter 727        
      7.15.2.4 Continuous-discrete extended Kalman Iter 728        
// ]
   7.15.3 MATCHED FILTERING (WIENER FILTER) 728        
   7.15.4 WALSH FUNCTIONS 729        
   7.15.5 WAVELETS 730        
// [
      7.15.5.1 De nitions 730        
      7.15.5.2 Generalizations 731        
      7.15.5.3 Wavelet coef cients and gures 732        
// ]
// ]
// ]
 
<b>CHAPTER 8. SCIENTIFIC COMPUTING </b> 734
// [
 
8.1 BASIC NUMERICAL ANALYSIS 735        
// [
   8.1.1 APPROXIMATIONS AND ERRORS 735        
// [
      8.1.1.1 Aitken's A2 method 735        
      8.1.1.2 Richardson's extrapolation 736        
// ]
   8.1.2 SOLUTION TO ALGEBRAIC EQUATIONS 736        
// [
      8.1.2.1 Fixed point iteration 737        
      8.1.2.2 Steffensen's method 737        
      8.1.2.3 Newton-Raphson method (Newton's method) 738        
      8.1.2.4 Modi ed Newton's method 739        
      8.1.2.5 Secant method 739        
      8.1.2.6 Root-bracketing methods 739        
      8.1.2.7 Bisection method 739        
      8.1.2.8 False position (regula falsi) 739        
      8.1.2.9 Horner's method with de ation 740        
      8.1.2.10 Horner's algorithm 740        
// ]
   8.1.3 INTERPOLATION 740        
// [
      8.1.3.1 Lagrange interpolation 740        
      8.1.3.2 Neville's method 741        
      8.1.3.3 Neville's algorithm 741        
      8.1.3.4 Divided differences 742        
      8.1.3.5 Newton's interpolatory divided-difference formula 742        
      8.1.3.6 Inverse interpolation 744        
      8.1.3.7 Hermite interpolation 744        
      8.1.3.8 Hermite interpolating polynomial 744        
// ]
   8.1.4 FITTING EQUATIONS TO DATA 744        
// [
      8.1.4.1 Piecewise polynomial approximation 744        
      8.1.4.2 Algorithm for natural cubic splines 745        
      8.1.4.3 Algorithm for clamped cubic splines 746        
      8.1.4.4 Discrete approximation 746        
      8.1.4.5 Normal equations 746        
      8.1.4.6 Best-t Line 747        
// ]
// ]
8.2 NUMERICAL LINEAR ALGEBRA 747        
// [
   8.2.1 SOLVING LINEAR SYSTEMS       
   8.2.2 GAUSSIAN ELIMINATION 747        
   8.2.3 GAUSSIAN ELIMINATION ALGORITHM 748        
   8.2.4 PIVOTING 748        
// [
      8.2.4.1 Maximal column pivoting 748        
      8.2.4.2 Scaled-column pivoting 749        
      8.2.4.3 Maximal (or complete) pivoting 749        
// ]
   8.2.5 EIGENVALUE COMPUTATION 749        
// [
      8.2.5.1 Power method 749        
      8.2.5.2 Power method algorithm 749        
      8.2.5.3 Inverse power method 750        
      8.2.5.4 Wielandt de ation 751        
// ]
   8.2.6 HOUSEHOLDER'S METHOD 751        
// [
      8.2.6.1 Algorithm for Householder's method 751        
// ]
   8.2.7 QR ALGORITHM 752        
// [
      8.2.7.1 Algorithm for QR 753        
// ]
   8.2.8 NON-LINEAR SYSTEMS AND NUMERICAL OPTIMIZATION 755        
// [
      8.2.8.1 Newton's method 755        
      8.2.8.2 Method of steepest-descent 755        
      8.2.8.3 Algorithm for steepest-descent 756        
// ]
// ]
8.3 NUMERICAL INTEGRATION AND 757        
// [
   8.3.1 NUMERICAL INTEGRATION 757        
// [
      8.3.1.1 Newton-Cotes formulae 757        
      8.3.1.2 Closed Newton-Cotes formulae 757        
      8.3.1.3 Open Newton-Cotes formulae 758        
      8.3.1.4 Composite rules 759        
      8.3.1.5 Romberg integration 760        
      8.3.1.6 Gregory's formula 760        
      8.3.1.7 Gaussian quadrature 762        
      8.3.1.8 Gauss-Legendre quadrature 762        
      8.3.1.9 Gauss-Lag uerre quadrature 763        
      8.3.1.11 Radau quadrature 764        
      8.3.1.12 Lobatto quadrature 764        
      8.3.1.13 Chebyshev quadrature 765        
      8.3.1.14 Multiple integrals 766        
      8.3.1.15 Simpson's double integral over a rectangle 766        
      8.3.1.16 Simpson's double integral algorithm 766        
      8.3.1.17 Gaussian double integral 767        
      8.3.1.18 Gauss-Legendre double integral 767        
      8.3.1.19 Double integrals of polynomials over polygons 768        
      8.3.1.20 Monte-Carlo methods 768        
      8.3.1.21 Hit or miss method 768        
      8.3.1.22 Hit or miss algorithm 769        
      8.3.1.23 Sample-mean Monte-Carlo method 769        
      8.3.1.24 Sample-mean algorithm 770        
      8.3.1.25 Integration in the presence of noise 770        
      8.3.1.26 Weighted Monte-Carlo integration 770        
// ]
   8.3.2 NUMERICAL DIFFERENTIATION 771        
// [
      8.3.2.1 Derivative estimates 771        
      8.3.2.2 Computational molecules 772        
      8.3.2.3 Numerical solution of differential equations 773        
      8.3.2.4 Multistep methods and predictor-corrector methods 774        
      8.3.2.5 Adams-Bashforth n-step (explicit) methods 775        
      8.3.2.6 Adams-Moulton n-step (implicit) methods 775        
      8.3.2.7 Higher-order differential equations and systems 776        
      8.3.2.8 Partial differential equations 777        
      8.3.2.9 Poisson equation 777        
      8.3.2.10 Poisson equation nite-diff erence algorithm 778        
      8.3.2.11 Heat or diffusion equation 780        
      8.3.2.12 Crank-Nicolson algorithm 781        
      8.3.2.13 Wave equation 782        
      8.3.2.14 Wave equation algorithm 782        
// ]
   8.3.3 NUMERICAL SUMMATION 783        
// ]
8.4 PROGRAMMING TECHNIQUES 784        
// ]
 
<b>CHAPTER 9. FINANCIAL ANALYSIS </b> 786
// [
 
9.1 FINANCIAL FORMULAE 786        
// [
   9.1.1 DEFINITION OF FINANCIAL TERMS 786        
   9.1.2 FORMULAE CONNECTING FINANCIAL TERMS 787        
   9.1.3 EXAMPLES 788        
// ]
9.2 FINANCIAL TABLES 790        
// [
   9.2.1 COMPOUND INTEREST: FIND FINAL VALUE 790        
   9.2.2 COMPOUND INTEREST: FIND INTEREST RATE 792        
   9.2.3 COMPOUND INTEREST: FIND ANNUITY 794        
// ]
// ]
 
<b>CHAPTER 10. MISCELLANEOUS </b> 798
// [
 
10.1 UNITS 799        
// [
   10.1.1 SI SYSTEM OF MEASUREMENT 799        
   10.1.2 UNITED STATES CUSTOMARY SYSTEM OF WEIGHTS 800        
   10.1.3 PHYSICAL CONSTANTS 801        
   10.1.4 DIMENSIONAL ANALYSIS/BUCKINGHAM PI 802        
   10.1.5 UNITS OF PHYSICAL QUANTITIES 803        
   10.1.6 CONVERSION: METRIC TO ENGLISH 803        
   10.1.7 CONVERSION: ENGLISH TO METRIC 804        
   10.1.8 MISCELLANEOUS CONVERSIONS 804        
   10.1.9 TEMPERATURE CONVERSION 805        
// ]
10.10 BIOGRAPHIES OF MATHEMATICIANS 817        
10.2 INTERPRETATIONS OF POWERS OF 10 805        
10.3 CALENDAR COMPUTATIONS 806        
// [
   10.3.1 Leap years 798        
   10.3.1 LEAP YEARS 806        
   10.3.2 DAY OF WEEK FOR ANY GIVEN DAY 806        
   10.3.3 NUMBER OF EACH DAY OF THE YEAR 807        
// ]
10.4 AMS CLASSIFICATION SCHEME 808        
10.5 FIELDS MEDALS 809        
10.6 GREEK ALPHABET 810        
10.7 COMPUTER LANGUAGES 810        
// [
   10.7.1 SOFTWARE CONTACT INFORMATION 810        
// ]
10.8 PROFESSIONAL MATHEMATICAL 811        
10.9 ELECTRONIC MATHEMATICAL RESOURCES 814        
 // ]
LIST OF FIGURES 828
LIST OF NOTATION 830