HANDBOOK OF MATHEMATICS
I.N. Bronshtein · K.A. Semendyayev · G. Musiol · H. Muehlig
5th Ed.,With 745s Figures and 142s Tables
TABLE OF CONTENTS
// absolute_page_number=1
Preface to the Fifth English Edition 4
From the Preface to the Fourth English Edition 4
Co-authors 5
// absolute_page_number=0
1 Arithmetic 1
1.1 Elementary Rules for Calculations 1
1.1.1 Numbers 1
1.1.1.1 Natural. Integer, and Rational Numbers 1
1.1.1.2 Irrational and Transcendental Numbers 2
1.1.1.3 Real Numbers 2
1.1.1.4 Continued Fractions 3
1.1.1.5 Commensurability 4
1.1.2 Methods for Proof 4
1.1.2.1 Direct Proof 5
1.1.2.2 Indirect Proof or Proof by Contradiction 5
1.1.2.3 Mathematical Induction 5
1.1.2.4 Constructive Proof 6
1.1.3 Sums and Products 6
1.1.3.1 Sums 6
1.1.3.2 Products 7
1.1.4 Powers, Roots, and Logarithms 7
1.1.4.1 Powers 7
1.1.4.2 Roots 8
1.1.4.3 Logarithms 9
1.1.4.4 Special Logarithms 9
1.1.5 Algebraic Expressions 10
1.1.5.1 Definitions 10
1.1.5.2 Algebraic Expressions in Detail 11
1.1.6 Integral Rational Expressions 11
1.1.6.1 Representation in Polynomial Form 11
1.1.6.2 Factorizing a Polynomial 11
1.1.6.3 Special Formulas 12
1.1.6.4 Binomial Theorem 12
1.1.6.5 Determination of the Greatest Common Divisor of Two Polynomials 14
1.1.7 Rational Expressions 14
1.1.7.1 Reducing to the Simplest Form 14
1.1.7.2 Determination of the Integral Rational Part 15
1.1.7.3 Decomposition into Partial Fractions 15
1.1.7.4 Transformations of Proportions 17
1.1.8 Irrational Expressions 17
1.2 Finite Series 18
1.2.1 Definition of a Finite Series 18
1.2.2 Arithmetic Series 18
1.2.3 Geometric Series 19
1.2.4 Special Finite Series 19
1.2.5 Mean Values 19
1.2.5.1 Arithmetic Mean or Arithmetic Average 19
1.2.5.2 Geometric Mean or Geometric Average 20
1.2.5.3 Harmonic Mean 20
1.2.5.4 Quadratic Mean 20
1.2.5.5 Relations Between the Means of Two Positive Values 20
1.3 Business Mathematics 21
1.3.1 Calculation of Interest or Percentage 21
1.3.2 Calculation of Compound Interest 22
1.3.2.1 Interest 22
1.3.2.2 Compound Interest 22
1.3.3 Amortization Calculus 23
1.3.3.1 Amortization 23
1.3.3.2 Equal Principal Repayments 23
1.3.3.3 Equal Annuities 24
1.3.4 Annuity Calculations 25
1.3.4.1 Annuities 25
1.3.4.2 Future Amount of an Ordinary Annuity 25
1.3.4.3 Balance after n Annuity Payments 25
1.3.5 Depreciation 26
1.4 Inequalities 28
1.4.1 Pure Inequalities 28
1.4.1.1 Definitions 28
1.4.1.2 Properties of Inequalities of Type I and II 29
1.4.2 Special Inequalities 30
1.4.2.1 Triangle Inequality for Real Numbers 30
1.4.2.2 Triangle Inequality for Complex Numbers 30
1.4.2.3 Inequalities for Absolute Values of Differences of Real and Complex Numbers 30
1.4.2.4 Inequality for Arithmetic and Geometric Means 30
1.4.2.5 Inequality for Arithmetic and Quadratic Means 30
1.4.2.6 Inequalities for Different Means of Real Numbers 30
1.4.2.7 Bernoulli's Inequality 30
1.4.2.8 Binomial Inequality 31
1.4.2.9 Cauchy Schwarz Inequality 31
1.4.2.10 Chebyshev Inequality 31
1.4.2.11 Generalized Chebyshev Inequality 32
1.4.2.12 Holder Inequality 32
1.4.2.13 Minkowski Inequality 32
1.4.3 Solution of Linear and Quadratic Inequalities 33
1.4.3.1 General Remarks 33
1.4.3.2 Linear Inequalities 33
1.4.3.3 Quadratic Inequalities 33
1.4.3.4 General Case for Inequalities of Second Degree 33
1.5 Complex Numbers 34
1.5.1 Imaginary and Complex Numbers 34
1.5.1.1 Imaginary Unit 34
1.5.1.2 Complex Numbers 34
1.5.2 Geometric Representation 34
1.5.2.1 Vector Representation 34
1.5.2.2 Equality of Complex Numbers 34
1.5.2.3 Trigonometric Form of Complex Numbers 35
1.5.2.4 Exponential Form of a Complex Number 35
1.5.2.5 Conjugate Complex Numbers 36
1.5.3 Calculation with Complex Numbers 36
1.5.3.1 Addition and Subtraction 36
1.5.3.2 Multiplication 36
1.5.3.3 Division 37
1.5.3.4 General Rules for the Basic Operations 37
1.5.3.5 Taking Powers of Complex Numbers 37
1.5.3.6 Taking of the n-th Root of a Complex Number 38
1.6 Algebraic and Transcendental Equations 38
1.6.1 Transforming Algebraic Equations to Normal Form 38
1.6.1.1 Definition 38
1.6.1.2 Systems of n Algebraic Equations 38
1.6.1.3 Superfluous Roots 39
1.6.2 Equations of Degree at Most Four 39
1.6.2.1 Equations of Degree One (Linear Equations) 39
1.6.2.2 Equations of Degree Two (Quadratic Equations) 39
1.6.2.3 Equations of Degree Three (Cubic Equations) 40
1.6.2.4 Equations of Degree Four 42
1.6.2.5 Equations of Higher Degree 43
1.6.3 Equations of Degree n 43
1.6.3.1 General Properties of Algebraic Equations 43
1.6.3.2 Equations with Real Coefficients 44
1.6.4 Reducing Transcendental Equations to Algebraic Equations 45
1.6.4.1 Definition 45
1.6.4.2 Exponential Equations 45
1.6.4.3 Logarithmic Equations 46
1.6.4.4 Trigonometric Equations 46
1.6.4.5 Equations with Hyperbolic Functions 46
2 Functions 47
2.1 Notion of Functions 47
2.1.1 Definition of a Function 47
2.1.1.1 Function 47
2.1.1.2 Real Functions 47
2.1.1.3 Functions of Several Variables 47
2.1.1.4 Complex Functions 47
2.1.1.5 Further Functions 47
2.1.1.6 Functionals 47
2.1.1.7 Functions and Mappings 48
2.1.2 Methods for Denning a Real Function 48
2.1.2.1 Defining a Function 48
2.1.2.2 Analytic Representation of a Function 48
2.1.3 Certain Types of Functions 49
2.1.3.1 Monotone Functions 49
2.1.3.2 Bounded Functions 50
2.1.3.3 Even Functions 50
2.1.3.4 Odd Functions 50
2.1.3.5 Representation with Even and Odd Functions 50
2.1.3.6 Periodic Functions 50
2.1.3.7 Inverse Functions 51
2.1.4 Limits of Functions 51
2.1.4.1 Definition of the Limit of a Function 51
2.1.4.2 Definition by Limit of Sequences 52
2.1.4.3 Cauchy Condition for Convergence 52
2.1.4.4 Infinity as a Limit of a Function 52
2.1.4.5 Left-Hand and Right-Hand Limit of a Function 52
2.1.4.6 Limit of a Function as x Tends to Infinity 53
2.1.4.7 Theorems About Limits of Functions 53
2.1.4.8 Calculation of Limits 54
2.1.4.9 Order of Magnitude of Functions and Landau Order Symbols 55 2.1.5 Continuityof aFunction 57
2.1.5.1 Notion of Continuity and Discontinuity 57
2.1.5.2 Definition of Continuity 57
2.1.5.3 Most Frequent Types of Discontinuities 57
2.1.5.4 Continuity and Discontinuity of Elementary Functions 58
2.1.5.5 Properties of Continuous Functions 59
2.2 Elementary Functions 60
2.2.1 Algebraic Functions 60
2.2.1.1 Polynomials 60
2.2.1.2 Rational Functions 61
2.2.1.3 Irrational Functions 61
2.2.2 Transcendental Functions 61
2.2.2.1 Exponential Functions 61
2.2.2.2 Logarithmic Functions 61
2.2.2.3 Trigonometric Functions 61
2.2.2.4 Inverse Trigonometric Functions 61
2.2.2.5 Hyperbolic Functions 62
2.2.2.6 Inverse Hyperbolic Functions 62
2.2.3 Composite Functions 62
2.3 Polynomials 62
2.3.1 Linear Function 62
2.3.2 Quadratic Polynomial 62
2.3.3 Cubic Polynomials 63
2.3.4 Polynomials of n-th Degree 63
2.3.5 Parabola of n-th Degree 64
2.4 Rational Functions 64
2.4.1 Special Fractional Linear Function (Inverse Proportionality) 64
2.4.2 Linear Fractional Function 65
2.4.3 Curves of Third Degree, Type I 65
2.4.4 Curves of Third Degree, Type II 66
2.4.5 Curves of Third Degree, Type III 67
2.4.6 Reciprocal Powers 68
2.5 Irrational Functions 69
2.5.1 Square Root of a Linear Binomial 69
2.5.2 Square Root of a Quadratic Polynomial 69
2.5.3 Power Function 70
2.6 Exponential Functions and Logarithmic Functions 71
2.6.1 Exponential Functions 71
2.6.2 Logarithmic Functions 71
2.6.3 Error Curve 72
2.6.4 Exponential Sum 72
2.6.5 Generalized Error Function 73
2.6.6 Product of Power and Exponential Functions 74
2.7 Trigonometric Functions (Functions of Angles) 74
2.7.1 Basic Notion 74
2.7.1.1 Definition and Representation 74
2.7.1.2 Range and Behavior of the Functions "i
2.7.2 Important Formulas for Trigonometric Functions 79
2.7.2.1 Relations Between the Trigonometric Functions of the Same Angle (Addition Theorems) 79
2.7.2.2 Trigonometric Functions of the Sum and Difference of Two Angles 79
2.7.2.3 Trigonometric Functions of an Integer Multiple of an Angle 79
2.7.2.4 Trigonometric Functions of Half-Angles 80
2.7.2.5 Sum and Difference of Two Trigonometric Functions 81
2.7.2.6 Products of Trigonometric Functions 81
2.7.2.7 Powers of Trigonometric Functions 82
2.7.3 Description of Oscillations 82
2.7.3.1 Formulation of the Problem 82
2.7.3.2 Superposition of Oscillations 82
2.7.3.3 Vector Diagram for Oscillations 83
2.7.3.4 Damping of Oscillations 83
2.8 Inverse Trigonometric Functions 84
2.8.1 Definition of the Inverse Trigonometric Functions 84
2.8.2 Reduction to the Principal Value 84
2.8.3 Relations Between the Principal Values 85
2.8.4 Formulas for Negative Arguments 86
2.8.5 Sum and Difference of arcsinx and arcsin y 86
2.8.6 Sum and Difference of arccos x and arccos y 86
2.8.7 Sum and Difference of arctanx and arctany 86
2.8.8 Special Relations for arcsin x, arccos x, arctanx 87
2.9 Hyperbolic Functions 87
2.9.1 Definition of Hyperbolic Functions 87
2.9.2 Graphical Representation of the Hyperbolic Functions 88
2.9.2.1 Hyperbolic Sine 88
2.9.2.2 Hyperbolic Cosine 88
2.9.2.3 Hyperbolic Tangent 88
2.9.2.4 Hyperbolic Cotangent 89
2.9.3 Important Formulas for the Hyperbolic Functions 89
2.9.3.1 Hyperbolic Functions of One Variable 89
2.9.3.2 Expressing a Hyperbolic Function by Another One with the Same Argument 89
2.9.3.3 Formulas for Negative Arguments 89
2.9.3.4 Hyperbolic Functions of the Sum and Difference of Two Arguments (Addition Theorems) 89
2.9.3.5 Hyperbolic Functions of Double Arguments 90
2.9.3.6 De Moivre Formula for Hyperbolic Functions 90
2.9.3.7 Hyperbolic Functions of Half-Argument 90
2.9.3.8 Sum and Difference of Hyperbolic Functions 90
2.9.3.9 Relation Between Hyperbolic and Trigonometric Functions with Complex Arguments z 91
2.10 Area Functions 91
2.10.1 Definitions 91
2.10.1.1 Area Sine 91
2.10.1.2 Area Cosine 91
2.10.1.3 Area Tangent 92
2.10.1.4 Area Cotangent 92
2.10.2 Determination of Area Functions Using Natural Logarithm 92
2.10.3 Relations Between Different Area Functions 93
2.10.4 Sum and Difference of Area Functions 93
2.10.5 Formulas for Negative Arguments 93
2.11 Curves of Order Three (Cubic Curves) 93
2.11.1 Semicubic Parabola 93
2.11.2 Witch of Agnesi 94
2.11.3 Cartesian Folium (Folium of Descartes) 94
2.11.4 Cissoid 95
2.11.5 Strophoide 95
2.12 Curves of Order Four (Quartics) 96
2.12.1 Conchoid of Nicomedes 96
2.12.2 General Conchoid 96
2.12.3 Pascal's Limacon 96
2.12.4 Cardioid ' 98
2.12.5 Cassinian Curve 98
2.12.6 Lemniscate 99
2.13 Cycloids 100
2.13.1 Common (Standard) Cycloid 100
2.13.2 Prolate and Curtate Cycloids or Trochoids 100
2.13.3 Epicycloid 101
2.13.4 Hypocycloid and Astroid 102
2.13.5 Prolate and Curtate Epicycloid and Hypocycloid 102
2.14 Spirals 103
2.14.1 Archimedean Spiral 103
2.14.2 Hyperbolic Spiral 104
2.14.3 Logarithmic Spiral 104
2.14.4 Evolvent of the Circle 104
2.14.5 Clothoid 105
2.15 Various Other Curves 105
2.15.1 Catenary Curve 105
2.15.2 Tractrix 106
2.16 Determination of Empirical Curves 106
2.16.1 Procedure 106
2.16.1.1 Curve-Shape Comparison 106
2.16.1.2 Rectification 107
2.16.1.3 Determination of Parameters 107
2.16.2 Useful Empirical Formulas 107
2.16.2.1 Power Functions 108
2.16.2.2 Exponential Functions 108
2.16.2.3 Quadratic Polynomial 109
2.16.2.4 Rational Linear Function 109
2.16.2.5 Square Root of a Quadratic Polynomial 110
2.16.2.6 General Error Curve 110
2.16.2.7 Curve of Order Three, Type II 110
2.16.2.8 Curve of Order Three, Type III Ill
2.16.2.9 Curve of Order Three, Type I Ill
2.16.2.10 Product of Power and Exponential Functions 112
2.16.2.11 Exponential Sum 112
2.16.2.12 Numerical Example 112
2.17 Scales and Graph Paper 114
2.17.1 Scales 114
2.17.2 Graph Paper 115
2.17.2.1 Semilogarithmic Paper 115
2.17.2.2 Double Logarithmic Paper 115
2.17.2.3 Graph Paper with a Reciprocal Scale 116
2.17.2.4 Remark 116
2.18 Functions of Several Variables 117
2.18.1 Definition and Representation 117
2.18.1.1 Representation of Functions of Several Variables 117
2.18.1.2 Geometric Representation of Functions of Several Variables 117
2.18.2 Different Domains in the Plane 118
2.18.2.1 Domain of a Function 118
2.18.2.2 Two-Dimensional Domains 118
2.18.2.3 Three or Multidimensional Domains 118
2.18.2.4 Methods to Determine a Function 119
2.18.2.5 Various Ways to Define a Function 120
2.18.2.6 Dependence of Functions 121
2.18.3 Limits 122
2.18.3.1 Definition 122
2.18.3.2 Exact Definition 122
2.18.3.3 Generalization for Several Variables 122
2.18.3.4 Iterated Limit 122
2.18.4 Continuity 122
2.18.5 Properties of Continuous Functions 123
2.18.5.1 Theorem on Zeros of Bolzano 123
2.18.5.2 Intermediate Value Theorem 123
2.18.5.3 Theorem About the Boundedness of a Function 123
2.18.5.4 Weierstrass Theorem (About the Existence of Maximum and Minimum) 123
2.19 Nomography 123
2.19.1 Nomograms 123
2.19.2 Net Charts 123
2.19.3 Alignment Charts 124
2.19.3.1 Alignment Charts with Three Straight-Line Scales Through a Point 125
2.19.3.2 Alignment Charts with Two Parallel and One Inclined Straight-Line Scales 125
2.19.3.3 Alignment Charts with Two Parallel Straight Lines and a Curved Scale 126
2.19.4 Net Charts for More Than Three Variables 127
3 Geometry 128
3.1 Plane Geometry 128
3.1.1 Basic Notation 128
3.1.1.1 Point. Line, Ray, Segment 128
3.1.1.2 Angle 128
3.1.1.3 Angle Between Two Intersecting Lines 128
3.1.1.4 Pairs of Angles with Intersecting Parallels 129
3.1.1.5 Angles Measured in Degrees and in Radians 130
3.1.2 Geometrical Definition of Circular and Hyperbolic Functions 130
3.1.2.1 Definition of Circular or Trigonometric Functions 130
3.1.2.2 Definitions of the Hyperbolic Functions 131
3.1.3 Plane Triangles 131
3.1.3.1 Statements about Plane Triangles 131
3.1.3.2 Symmetry 132
3.1.4 Plane Quadrangles 134
3.1.4.1 Parallelogram 134
3.1.4.2 Rectangle and Square 135
3.1.4.3 Rhombus 135
3.1.4.4 Trapezoid 135
3.1.4.5 General Quadrangle 136
3.1.4.6 Inscribed Quadrangle 136
3.1.4.7 Circumscribing Quadrangle 136
3.1.5 Polygons in the Plane 137
3.1.5.1 General Polygon 137
3.1.5.2 Regular Convex Polygons 137
3.1.5.3 Some Regular Convex Polygons 138
3.1.6 The Circle and Related Shapes 138
3.1.6.1 Circle 138
3.1.6.2 Circular Segment and Circular Sector 140
3.1.6.3 Annulus 140
3.2 Plane Trigonometry 141
3.2.1 Triangles 141
3.2.1.1 Calculations in Right-Angled Triangles in the Plane 141
3.2.1.2 Calculations in General Triangles in the Plane 141
3.2.2 Geodesic Applications 143
3.2.2.1 Geodetic Coordinates 143
3.2.2.2 Angles in Geodesy 145
3.2.2.3 Applications in Surveying 14(
3.3 Stereometry 150
3.3.1 Lines and Planes in Space 150
3.3.2 Edge. Corner, Solid Angle 150
3.3.3 Polyeder or Polyhedron 151
3.3.4 Solids Bounded by Curved Surfaces 154
3.4 Spherical Trigonometry 158
3.4.1 Basic Concepts of Geometry on the Sphere 158
3.4.1.1 Curve. Arc, and Angle on the Sphere 158
3.4.1.2 Special Coordinate Systems 160
3.4.1.3 Spherical Lune or Biangle 161
3.4.1.4 Spherical Triangle 161
3.4.1.5 Polar Triangle 162
3.4.1.6 Euler Triangles and Non-Euler Triangles 162
3.4.1.7 Trihedral Angle 163
3.4.2 Basic Properties of Spherical Triangles 163
3.4.2.1 General Statements 163
3.4.2.2 Fundamental Formulas and Applications 164
3.4.2.3 Further Formulas 166
3.4.3 Calculation of Spherical Triangles 167
3.4.3.1 Basic Problems. Accuracy Observations 167
3.4.3.2 Right-Angled Spherical Triangles 168
3.4.3.3 Spherical Triangles with Oblique Angles 169
3.4.3.4 Spherical Curves 172
3.5 Vector Algebra and Analytical Geometry 180
3.5.1 Vector Algebra 180
3.5.1.1 Definition of Vectors 180
3.5.1.2 Calculation Rules for Vectors 181
3.5.1.3 Coordinates of a Vector 182
3.5.1.4 Directional Coefficient 183
3.5.1.5 Scalar Product and Vector Product 183
3.5.1.6 Combination of Vector Products 184
3.5.1.7 Vector Equations 187
3.5.1.8 Covariant and Contravariant Coordinates of a Vector 187
3.5.1.9 Geometric Applications of Vector Algebra 189
3.5.2 Analytical Geometry of the Plane 189
3.5.2.1 Basic Concepts, Coordinate Systems in the Plane 189
3.5.2.2 Coordinate Transformations 190
3.5.2.3 Special Notation in the Plane 191
3.5.2.4 Line 194
3.5.2.5 Circle 197
3.5.2.6 Ellipse 198
3.5.2.7 Hyperbola 200
3.5.2.8 Parabola 203
3.5.2.9 Quadratic Curves (Curves of Second Order or Conic Sections) 205
3.5.3 Analytical Geometry of Space 207
3.5.3.1 Basic Concepts, Spatial Coordinate Systems 207
3.5.3.2 Transformation of Orthogonal Coordinates 210
3.5.3.3 Special Quantities in Space 212
3.5.3.4 Line and Plane in Space 214
3.5.3.5 Surfaces of Second Order. Equations in Normal Form 220
3.5.3.6 Surfaces of Second Order or Quadratic Surfaces. General Theory 223 3.6 Differential Geometry 225
3.6.1 Plane Curves 225
3.6.1.1 Ways to Define a Plane Curve 225
3.6.1.2 Local Elements of a Curve 225
3.6.1.3 Special Points of a Curve 231
3.6.1.4 Asymptotes of Curves 234
3.6.1.5 General Discussion of a Curve Given by an Equation 235
3.6.1.6 Evolutes and Evolvents 236
3.6.1.7 Envelope of a Family of Curves 237
3.6.2 Space Curves 238
3.6.2.1 Ways to Define a Space Curve 238
3.6.2.2 Moving Trihedral 238
3.6.2.3 Curvature and Torsion 240
3.6.3 Surfaces 243
3.6.3.1 Ways to Define a Surface 243
3.6.3.2 Tangent Plane and Surface Normal 244
3.6.3.3 Line Elements of a Surface 245
3.6.3.4 Curvature of a Surface 247
3.6.3.5 Ruled Surfaces and Developable Surfaces 250
3.6.3.6 Geodesic Lines on a Surface 250
4 Linear Algebra 251
4.1 Matrices 251
4.1.1 Notion of Matrix 251
4.1.2 Square Matrices 252
4.1.3 Vectors 253
4.1.4 Arithmetical Operations with Matrices 254
4.1.5 Rules of Calculation for Matrices 257
4.1.6 Vector and Matrix Norms 258
4.1.6.1 Vector Norms 258
4.1.6.2 Matrix Norms 259
4.2 Determinants 259
4.2.1 Definitions 259
4.2.2 Rules of Calculation for Determinants 260
4.2.3 Evaluation of Determinants 261
4.3 Tensors 262
4.3.1 Transformation of Coordinate Systems 262
4.3.2 Tensors in Cartesian Coordinates 262
4.3.3 Tensors with Special Properties 264
4.3.3.1 Tensors of Rank 2 264
4.3.3.2 Invariant Tensors 265
4.3.4 Tensors in Curvilinear Coordinate Systems 266
4.3.4.1 Covariant and Contravariant Basis Vectors 266
4.3.4.2 Covariant and Contravariant Coordinates of Tensors of Rank 1 266
4.3.4.3 Covariant, Contravariant and Mixed Coordinates of Tensors of Rank 2 267
4.3.4.4 Rules of Calculation 268
4.3.5 Pseudotensors 268
4.3.5.1 Symmetry with Respect to the Origin 269
4.3.5.2 Introduction to the Notion of Pseudotensors 270
4.4 Systems of Linear Equations 271
4.4.1 Linear Systems. Pivoting 271
4.4.1.1 Linear Systems 271
4.4.1.2 Pivoting 271
4.4.1.3 Linear Dependence 272
4.4.1.4 Calculation of the Inverse of a Matrix 272
4.4.2 Solution of Systems of Linear Equations 272
4.4.2.1 Definition and Solvability 272
4.4.2.2 Application of Pivoting 274
4.4.2.3 Cramer's Rule 275
4.4.2.4 Gauss's Algorithm 276
4.4.3 Overdetermined Linear Equation Systems 277
4.4.3.1 Overdetermined Linear Systems of Equations and Linear Mean Square Value Problems 277
4.4.3.2 Suggestions for Numerical Solutions of Mean Square Value Problems 278
4.5 Eigenvalue Problems for Matrices 278
4.5.1 General Eigenvalue Problem 278
4.5.2 Special Eigenvalue Problem 278
4.5.2.1 Characteristic Polynomial 278
4.5.2.2 Real Symmetric Matrices. Similarity Transformations 280
4.5.2.3 Transformation of Principal Axes of Quadratic Forms 281
4.5.2.4 Suggestions for the Numerical Calculations of Eigenvalues 283
4.5.3 Singular Value Decomposition 285
5 Algebra and Discrete Mathematics 286
5.1 Logic 286
5.1.1 Propositional Calculus 286
5.1.2 Formulas in Predicate Calculus 289
5.2 Set Theory 290
5.2.1 Concept of Set, Special Sets 290
5.2.2 Operations with Sets 291
5.2.3 Relations and Mappings 294
5.2.4 Equivalence and Order Relations 296
5.2.5 Cardinalityof Sets 298
5.3 Classical Algebraic Structures 298
5.3.1 Operations 298
5.3.2 Semigroups 299
5.3.3 Groups 299
5.3.3.1 Definition and Basic Properties 299
5.3.3.2 Subgroups and Direct Products 300
5.3.3.3 Mappings Between Groups 302
5.3.4 Group Representations 303
5.3.4.1 Definitions 303
5.3.4.2 Particular Representations 303
5.3.4.3 Direct Sum of Representations 305
5.3.4.4 Direct Product of Representations 305
5.3.4.5 Reducible and Irreducible Representations 305
5.3.4.6 Schur's Lemma 1 306
5.3.4.7 Clebsch Gordan Series 306
5.3.4.8 Irreducible Representations of the Symmetric Group S M 306
5.3.5 Applications of Groups 307
5.3.5.1 Symmetry Operations, Symmetry Elements 307
5.3.5.2 Symmetry Groups or Point Groups 308
5.3.5.3 Symmetry Operations with Molecules 308
5.3.5.4 Symmetry Groups in Crystallography 310
5.3.5.5 Symmetry Groups in Quantum Mechanics 312
5.3.5.6 Further Applications of Group Theory in Physics 312
5.3.6 Rings and Fields 313
5.3.6.1 Definitions 313
5.3.6.2 Subrings, Ideals 313
5.3.6.3 Homomorphism, Isomorphism. Homomorphism Theorem 314
5.3.6.4 Finite Fields and Shift Registers 314
5.3.7 Vector Spaces 316
5.3.7.1 Definition 316
5.3.7.2 Linear Dependence 317
5.3.7.3 Linear Mappings 317
5.3.7.4 Subspaces. Dimension Formula 317
5.3.7.5 Euclidean Vector Spaces. Euclidean Norm 318
5.3.7.6 Linear Operators in Vector Spaces 319
5.4 Elementary Number Theory 320
5.4.1 Divisibility 320
5.4.1.1 Divisibility and Elementary Divisibility Rules 320
5.4.1.2 Prime Numbers 320
5.4.1.3 Criteria for Divisibility 322
5.4.1.4 Greatest Common Divisor and Least Common Multiple 323
5.4.1.5 Fibonacci Numbers 325
5.4.2 Linear Diophantine Equations 325
5.4.3 Congruences and Residue Classes 327
5.4.4 Theorems of Fermat, Euler. and Wilson 331
5.4.5 Codes 331
5.5 Cryptology 334
5.5.1 Problem of Cryptology 334
5.5.2 Cryptosystems 334
5.5.3 Mathematical Foundation 334
5.5.4 Security of Cryptosystems 335
5.5.4.1 Methods of Conventional Cryptography 335
5.5.4.2 Linear Substitution Ciphers 336
5.5.4.3 Vigenere Cipher 336
5.5.4.4 Matrix Substitution 336
5.5.5 Methods of Classical Cryptanalysis 337
5.5.5.1 Statistical Analysis 337
5.5.5.2 Kasiski Friedman Test 337
5.5.6 One-Time Pad 338
5.5.7 Public Key Methods 338
5.5.7.1 Diffie Hellman Key Exchange 338
5.5.7.2 One-Way Function 339
5.5.7.3 RSA Method 339
5.5.8 AES Algorithm (Advanced Encryption Standard) 339
5.5.9 IDEA Algorithm (International Data Encryption Algorithm) 340
5.6 Universal Algebra 340
5.6.1 Definition 340
5.6.2 Congruence Relations. Factor Algebras 340
5.6.3 Homomorphism 341
5.6.4 Homomorphism Theorem 341
5.6.5 Varieties 341
5.6.6 Term Algebras. Free Algebras 341
5.7 Boolean Algebras and Switch Algebra 342
5.7.1 Definition 342
5.7.2 Duality Principle 343
5.7.3 Finite Boolean Algebras 343
5.7.4 Boolean Algebras as Orderings 343
5.7.5 Boolean Functions. Boolean Expressions 344
5.7.6 Normal Forms 345
5.7.7 Switch Algebra 346
5.8 Algorithms of Graph Theory 348
5.8.1 Basic Notions and Notation 348
5.8.2 Traverse of Undirected Graphs 351
5.8.2.1 Edge Sequences or Paths 351
5.8.2.2 Euler Trails 352
5.8.2.3 Hamiltonian Cycles 353
5.8.3 Trees and Spanning Trees 354
5.8.3.1 Trees 354
5.8.3.2 Spanning Trees 355
5.8.4 Matchings 356
5.8.5 Planar Graphs 357
5.8.6 Paths in Directed Graphs 357
5.8.7 Transport Networks 358
5.9 Fuzzy Logic 360
5.9.1 Basic Notions of Fuzzy Logic 360
5.9.1.1 Interpretation of Fuzzy Sets 360
5.9.1.2 Membership Functions on the Real Line 361
5.9.1.3 Fuzzy Sets 363
5.9.2 Aggregation of Fuzzy Sets 365
5.9.2.1 Concepts for Aggregation of Fuzzy Sets 365
5.9.2.2 Practical Aggregator Operations of Fuzzy Sets 366
5.9.2.3 Compensatory Operators 368
5.9.2.4 Extension Principle 368
5.9.2.5 Fuzzy Complement 368
5.9.3 Fuzzy-Valued Relations 369
5.9.3.1 Fuzzy Relations 369
5.9.3.2 Fuzzy Product Relation R o S 371
5.9.4 Fuzzy Inference (Approximate Reasoning) 372
5.9.5 Defuzzification Methods 373
5.9.6 Knowledge-Based Fuzzy Systems 374
5.9.6.1 Method of Mamdani 374
5.9.6.2 Method of Sugeno 375
5.9.6.3 Cognitive Systems 375
5.9.6.4 Knowledge-Based Interpolation Systems 377
6 Differentiation 379
6.1 Differentiation of Functions of One Variable 379
6.1.1 Differential Quotient 379
6.1.2 Rules of Differentiation for Functions of One Variable 380
6.1.2.1 Derivatives of the Elementary Functions 380
6.1.2.2 Basic Rules of Differentiation 380
6.1.3 Derivatives of Higher Order 385
6.1.3.1 Definition of Derivatives of Higher Order 385
6.1.3.2 Derivatives of Higher Order of some Elementary Functions 385
6.1.3.3 Leibniz's Formula 385
6.1.3.4 Higher Derivatives of Functions Given in Parametric Form 387
6.1.3.5 Derivatives of Higher Order of the Inverse Function 387
6.1.4 Fundamental Theorems of Differential Calculus 388
6.1.4.1 Monotonicity 388
6.1.4.2 Fermat's Theorem 388
6.1.4.3 Rolle's Theorem 388
6.1.4.4 Mean Value Theorem of Differential Calculus 389
6.1.4.5 Taylor's Theorem of Functions of One Variable 389
6.1.4.6 Generalized Mean Value Theorem of Differential Calculus (Cauchy's Theorem) 390
6.1.5 Determination of the Extreme Values and Inflection Points 390
6.1.5.1 Maxima and Minima 390
6.1.5.2 Necessary Conditions for the Existence of a Relative Extreme Value 390
6.1.5.3 Relative Extreme Values of a Differentiable, Explicit Function 391
6.1.5.4 Determination of Absolute Extrema 392
6.1.5.5 Determination of the Extrema of Implicit Functions 392
6.2 Differentiation of Functions of Several Variables 392
6.2.1 Partial Derivatives 392
6.2.1.1 Partial Derivativeofa Function 392
6.2.1.2 Geometrical Meaning for Functions of Two Variables 393
6.2.1.3 Differentials of x and f(x) 393
6.2.1.4 Basic Properties of the Differential 394
6.2.1.5 Partial Differential 394
6.2.2 Total Differential and Differentials of Higher Order 394
6.2.2.1 Notion of Total Differential of a Function of Several Variables (Complete Differential) 394
6.2.2.2 Derivatives and Differentials of Higher Order 395
6.2.2.3 Taylor's Theorem for Functions of Several Variables 396
6.2.3 Rules of Differentiation for Functions of Several Variables 397
6.2.3.1 Differentiation of Composite Functions 397
6.2.3.2 Differentiation of Implicit Functions 398
6.2.4 Substitution of Variables in Differential Expressions and Coordinate Transformations 399
6.2.4.1 Function of One Variable 399
6.2.4.2 Function of Two Variables 400
6.2.5 Extreme Values of Functions of Several Variables 401
6.2.5.1 Definition 401
6.2.5.2 Geometric Representation 401
6.2.5.3 Determination of Extreme Values of Functions of Two Variables 402
6.2.5.4 Determination of the Extreme Values of a Function of n Variables 402
6.2.5.5 Solution of Approximation Problems 403
6.2.5.6 Extreme Value Problem with Side Conditions 403
7 Infinite Series 404
7.1 Sequences of Numbers 404
7.1.1 Properties of Sequences of Numbers 404
7.1.1.1 Definition of Sequence of Numbers 404
7.1.1.2 Monotone Sequences of Numbers 404
7.1.1.3 Bounded Sequences 404
7.1.2 Limits of Sequences of Numbers 405
7.2 Number Series 406
7.2.1 General Convergence Theorems 406
7.2.1.1 Convergence and Divergence of Infinite Series 406
7.2.1.2 General Theorems about the Convergence of Series 406
7.2.2 Convergence Criteria for Series with Positive Terms 407
7.2.2.1 Comparison Criterion 407
7.2.2.2 D'Alembert's Ratio Test 407
7.2.2.3 Root Test of Cauchy 408
7.2.2.4 Integral Test of Cauchy 408
7.2.3 Absolute and Conditional Convergence 409
7.2.3.1 Definition 409
7.2.3.2 Properties of Absolutely Convergent Series 409
7.2.3.3 Alternating Series 410
7.2.4 Some Special Series 410
7.2.4.1 The Values of Some Important Number Series 410
7.2.4.2 Bernoulli and Euler Numbers 412
7.2.5 Estimation of the Remainder 413
7.2.5.1 Estimation with Majorant 413
7.2.5.2 Alternating Convergent Series 414
7.2.5.3 Special Series 414
7.3 Function Series 414
7.3.1 Definitions 414
7.3.2 Uniform Convergence 414
7.3.2.1 Definition. Weierstrass Theorem 414
7.3.2.2 Properties of Uniformly Convergent Series 415
7.3.3 Power series 416
7.3.3.1 Definition. Convergence 416
7.3.3.2 Calculations with Power Series 416
7.3.3.3 Taylor Series Expansion, Maclaurin Series 417
7.3.4 Approximation Formulas 418
7.3.5 Asymptotic Power Series 419
7.3.5.1 Asymptotic Behavior 419
7.3.5.2 Asymptotic Power Series 420
7.4 Fourier Series 420
7.4.1 Trigonometric Sum and Fourier Series 420
7.4.1.1 Basic Notions 420
7.4.1.2 Most Important Properties of the Fourier Series 421
7.4.2 Determination of Coefficients for Symmetric Functions 422
7.4.2.1 Different Kinds of Symmetries 422
7.4.2.2 Forms of the Expansion into a Fourier Series 423
7.4.3 Determination of the Fourier Coefficients with Numerical Methods 424
7.4.4 Fourier Series and Fourier Integrals 424
7.4.5 Remarks on the Table of Some Fourier Expansions 425
8 Integral Calculus 427
8.1 Indefinite Integrals 427
8.1.1 Primitive Function or Antiderivative 427
8.1.1.1 Indefinite Integrals 428
8.1.1.2 Integrals of Elementary Functions 428
8.1.2 Rules of Integration 429
8.1.3 Integration of Rational Functions 432
8.1.3.1 Integrals of Integer Rational Functions (Polynomials) 432
8.1.3.2 Integrals of Fractional Rational Functions 432
8.1.3.3 Four Cases of Partial Fraction Decomposition 432
8.1.4 Integration of Irrational Functions 435
8.1.4.1 Substitution to Reduce to Integration of Rational Functions 435
8.1.4.2 Integration of Binomial Integrands 436
8.1.4.3 Elliptic Integrals 437
8.1.5 Integration of Trigonometric Functions 438
8.1.5.1 Substitution 438
8.1.5.2 Simplified Methods 438
8.1.6 Integration of Further Transcendental Functions 439
8.1.6.1 Integrals with Exponential Functions 439
8.1.6.2 Integrals with Hyperbolic Functions 440
8.1.6.3 Application of Integration by Parts 440
8.1.6.4 Integrals of Transcendental Functions 440
8.2 Definite Integrals 440
8.2.1 Basic Notions. Rules and Theorems 440
8.2.1.1 Definition and Existence of the Definite Integral 440
8.2.1.2 Properties of Definite Integrals 441
8.2.1.3 Further Theorems about the Limits of Integration 443
8.2.1.4 Evaluation of the Definite Integral 445
8.2.2 Application of Definite Integrals 447
8.2.2.1 General Principles for Application of the Definite Integral 447
8.2.2.2 Applications in Geometry 448
8.2.2.3 Applications in Mechanics and Physics 451
8.2.3 Improper Integrals, Stieltjes and Lebesgue Integrals 453
8.2.3.1 Generalization of the Notion of the Integral 453
8.2.3.2 Integrals with Infinite Integration Limits 454
8.2.3.3 Integrals with Unbounded Integrand 456
8.2.4 Parametric Integrals 459
8.2.4.1 Definition of Parametric Integrals 459
8.2.4.2 Differentiation Under the Symbol of Integration 459
8.2.4.3 Integration Under the Symbol of Integration 459
8.2.5 Integration by Series Expansion. Special Non-Elementary Functions 460
8.3 Line Integrals 462
8.3.1 Line Integrals of the First Type 463
8.3.1.1 Definitions 463
8.3.1.2 Existence Theorem 463
8.3.1.3 Evaluation of the Line Integral of the First Type 463
8.3.1.4 Application of the Line Integral of the First Type 464
8.3.2 Line Integrals of the Second Type 464
8.3.2.1 Definitions 464
8.3.2.2 Existence Theorem 466
8.3.2.3 Calculation of the Line Integral of the Second Type 466
8.3.3 Line Integrals of General Type 467
8.3.3.1 Definition 467
8.3.3.2 Properties of the Line Integral of General Type 467
8.3.3.3 Integral Along a Closed Curve 468
8.3.4 Independence of the Line Integral of the Path of Integration 468
8.3.4.1 Two-Dimensional Case 468
8.3.4.2 Existence of a Primitive Function 469
8.3.4.3 Three-Dimensional Case 469
8.3.4.4 Determination of the Primitive Function 469
8.3.4.5 Zero-Valued Integral Along a Closed Curve 470
8.4 Multiple Integrals 471
8.4.1 Double Integrals 471
8.4.1.1 Notion of the Double Integral 471
8.4.1.2 Evaluation of the Double Integral 472
8.4.1.3 Applications of the Double Integral 474
8.4.2 Triple Integrals 476
8.4.2.1 Notion of the Triple Integral 476
8.4.2.2 Evaluation of the Triple Integral 476
8.4.2.3 Applications of the Triple Integral 479
8.5 Surface Integrals 479
8.5.1 Surface Integral of the First Type 479
8.5.1.1 Notion of the Surface Integral of the First Type 480
8.5.1.2 Evaluation of the Surface Integral of the First Type 481
8.5.1.3 Applications of the Surface Integral of the First Type 482
8.5.2 Surface Integral of the Second Type 483
8.5.2.1 Notion of the Surface Integral of the Second Type 483
8.5.2.2 Evaluation of Surface Integrals of the Second Type 484
8.5.3 Surface Integral in General Form 485
8.5.3.1 Notion of the Surface Integral in General Form 485
8.5.3.2 Properties of the Surface Integrals 485
8.5.3.3 An Application of the Surface Integral 486
9 Differential Equations 487
9.1 Ordinary Differential Equations 487
9.1.1 First-Order Differential Equations 488
9.1.1.1 Existence Theorems. Direction Field 488
9.1.1.2 Important Solution Methods 489
9.1.1.3 Implicit Differential Equations 492
9.1.1.4 Singular Integrals and Singular Points 493
9.1.1.5 Approximation Methods for Solution of First-Order Differential Equations 496
9.1.2 Differential Equations of Higher Order and Systems of Differential Equations 497
9.1.2.1 Basic Results 497
9.1.2.2 Lowering the Order 499 I
9.1.2.3 Linear n-th Order Differential Equations 500
9.1.2.4 Solution of Linear Differential Equations with Constant Coefficients 502
9.1.2.5 Systems of Linear Differential Equations with Constant Coefficients 505
9.1.2.6 Linear Second-Order Differential Equations 507
9.1.3 Boundary Value Problems 514
9.1.3.1 Problem Formulation 514
9.1.3.2 Fundamental Properties of Eigenfunctions and Eigenvalues 515
9.1.3.3 Expansion in Eigenfunctions 516
9.1.3.4 Singular Cases 516
9.2 Partial Differential Equations 517
9.2.1 First-Order Partial Differential Equations 517
9.2.1.1 Linear First-Order Partial Differential Equations 517
9.2.1.2 Non-Linear First-Order Partial Differential Equations 519
9.2.2 Linear Second-Order Partial Differential Equations 522
9.2.2.1 Classification and Properties of Second-Order Differential Equations with Two Independent Variables 522
9.2.2.2 Classification and Properties of Linear Second-Order Differential Equations with more than two Independent Variables 523
9.2.2.3 Integration Methods for Linear Second-Order Partial Differential Equations 524
9.2.3 Some further Partial Differential Equations From Natural Sciences and Engineering 534
9.2.3.1 Formulation of the Problem and the Boundary Conditions 534
9.2.3.2 Wave Equation 536
9.2.3.3 Heat Conduction and Diffusion Equation for Homogeneous Media 537
9.2.3.4 Potential Equation 538
9.2.3.5 Schrodinger's Equation 538
9.2.4 Non-Linear Partial Differential Equations: Solitons, Periodic Patterns and Chaos 546
9.2.4.1 Formulation of the Physical-Mathematical Problem 546
9.2.4.2 Korteweg de Vries Equation (KdV) 548
9.2.4.3 Non-Linear Schrodinger Equation (NLS) 549
9.2.4.4 Sine Gordon Equation (SG) 549
9.2.4.5 Further Non-linear Evolution Equations with Soliton Solutions 551
10 Calculus of Variations 552
10.1 Defining the Problem 552
10.2 Historical Problems 553
10.2.1 Isoperimetric Problem 553
10.2.2 Brachistochrone Problem 553
10.3 Variational Problems of One Variable 553
10.3.1 Simple Variational Problems and Extremal Curves 553
10.3.2 Euler Differential Equation of the Variational Calculus 554
10.3.3 Variational Problems with Side Conditions 555
10.3.4 Variational Problems with Higher-Order Derivatives 556
10.3.5 Variational Problem with Several Unknown Functions 557
10.3.6 Variational Problems using Parametric Representation 557
10.4 Variational Problems with Functions of Several Variables 558
10.4.1 Simple Variational Problem 558
10.4.2 More General Variational Problems 560
10.5 Numerical Solution of Variational Problems 560
10.6 Supplementary Problems 561
10.6.1 First and Second Variation 561
10.6.2 Application in Physics 562
11 Linear Integral Equations 563
11.1 Introduction and Classification 563
11.2 Fredholm Integral Equations of the Second Kind 564
11.2.1 Integral Equations with Degenerate Kernel 564
11.2.2 Successive Approximation Method, Neumann Series 567
11.2.3 Fredholm Solution Method. Fredholm Theorems 569
11.2.3.1 Fredholm Solution Method 569
11.2.3.2 Fredholm Theorems 571
11.2.4 Numerical Methods for Fredholm Integral Equations of the Second Kind 572
11.2.4.1 Approximation of the Integral 572
11.2.4.2 Kernel Approximation 574
11.2.4.3 Collocation Method 576
11.3 Fredholm Integral Equations of the First Kind 577
11.3.1 Integral Equations with Degenerate Kernels 577
11.3.2 Analytic Basis 578
11.3.3 Reduction of an Integral Equation into a Linear System of Equations 580
11.3.4 Solution of the Homogeneous Integral Equation of the First Kind 581
11.3.5 Construction of Two Special Orthonormal Systems for a Given Kernel 582
11.3.6 Iteration Method 584
11.4 Volterra Integral Equations 585
11.4.1 Theoretical Foundations 585
11.4.2 Solution by Differentiation 586
11.4.3 Solution of the Volterra Integral Equation of the Second Kind by Neumann Series 587
11.4.4 Convolution Type Volterra Integral Equations 587
11.4.5 Numerical Methods for Volterra Integral Equations of the Second Kind 588
11.5 Singular Integral Equations 590
11.5.1 Abel Integral Equation 590
11.5.2 Singular Integral Equation with Cauchy Kernel 591
11.5.2.1 Formulation of the Problem 591
11.5.2.2 Existence of a Solution 592
11.5.2.3 Properties of Cauchy Type Integrals 592
11.5.2.4 The Hilbert Boundary Value Problem 593
11.5.2.5 Solution of the Hilbert Boundary Value Problem (in short: Hilbert Problem) 593
11.5.2.6 Solution of the Characteristic Integral Equation 594
12 Functional Analysis 596
12.1 Vector Spaces 596
12.1.1 Notion of a Vector Space 596
12.1.2 Linear and Affine Linear Subsets 597
12.1.3 Linearly Independent Elements 598
12.1.4 Convex Subsets and the Convex Hull 599
12.1.4.1 Convex Sets 599
12.1.4.2 Cones 599
12.1.5 Linear Operators and Functionals 600
12.1.5.1 Mappings 600
12.1.5.2 Homomorphism and Endomorphism 600
12.1.5.3 Isomorphic Vector Spaces 601
12.1.6 Complexification of Real Vector Spaces 601
12.1.7 Ordered Vector Spaces 601
12.1.7.1 Cone and Partial Ordering 601
12.1.7.2 Order Bounded Sets 602
12.1.7.3 Positive Operators 602
12.1.7.4 Vector Lattices 602
12.2 Metric Spaces 604
12.2.1 Notion of a Metric Space 604
12.2.1.1 Balls, Neighborhoods and Open Sets 605
12.2.1.2 Convergence of Sequences in Metric Spaces 606
12.2.1.3 Closed Sets and Closure 606
12.2.1.4 Dense Subsets and Separable Metric Spaces 607
12.2.2 CompleteMetricSpaces 607
12.2.2.1 Cauchy Sequences 607
12.2.2.2 CompleteMetricSpaces 608
12.2.2.3 Some Fundamental Theorems in Complete Metric Spaces 608
12.2.2.4 Some Applications of the Contraction Mapping Principle 608
12.2.2.5 Completion of a Metric Space 610
12.2.3 Continuous Operators 610
12.3 NormedSpaces 611
12.3.1 Notion of a Normed Space 611
12.3.1.1 Axioms of a Normed Space 611
12.3.1.2 Some Properties of Normed Spaces 612
12.3.2 Banach Spaces 612
12.3.2.1 Series in Normed Spaces 612
12.3.2.2 Examples of Banach Spaces 612
12.3.2.3 Sobolev Spaces 613
12.3.3 Ordered Normed Spaces 613
12.3.4 Normed Algebras 614
12.4 HilbertSpaces 615
12.4.1 Notion of a Hilbert Space 615
12.4.1.1 Scalar Product 615
12.4.1.2 Unitary Spaces and Some of their Properties 615
12.4.1.3 Hilbert Space 615
12.4.2 Orthogonality 616
12.4.2.1 Properties of Orthogonality 616
12.4.2.2 Orthogonal Systems 616
12.4.3 Fourier Series in Hilbert Spaces 617
12.4.3.1 Best Approximation 617
12.4.3.2 Parseval Equation, Riesz Fischer Theorem 618
12.4.4 Existence of a Basis, Isomorphic Hilbert Spaces 618
12.5 Continuous Linear Operators and Functionals 619
12.5.1 Boundedness, Norm and Continuity of Linear Operators 619
12.5.1.1 Boundedness and the Norm of Linear Operators 619
12.5.1.2 The Space of Linear Continuous Operators 619
12.5.1.3 Convergence of Operator Sequences 620
12.5.2 Linear Continuous Operators in Banach Spaces 620
12.5.3 Elements of the Spectral Theory of Linear Operators 622
12.5.3.1 Resolvent Set and the Resolvent of an Operator 622
12.5.3.2 Spectrum of an Operator 622
12.5.4 Continuous Linear Functionals 623
12.5.4.1 Definition 623
12.5.4.2 Continuous Linear Functionals in Hilbert Spaces. Riesz Representation Theorem 624
12.5.4.3 Continuous Linear Functionals in L p 624
12.5.5 Extension of a Linear Functional 624
12.5.6 Separation of Convex Sets 625
12.5.7 Second Adjoint Space and Reflexive Spaces 626
12.6 Adjoint Operators in Normed Spaces 626
12.6.1 Adjoint of a Bounded Operator 626
12.6.2 Adjoint Operator of an Unbounded Operator 627
12.6.3 Self-Adjoint Operators 627
12.6.3.1 Positive Definite Operators 628
12.6.3.2 Projectors in a Hilbert Space 628
12.7 Compact Sets and Compact Operators 628
12.7.1 Compact Subsets of a Normed Space 628
12.7.2 Compact Operators 628
12.7.2.1 Definition of Compact Operator 628
12.7.2.2 Properties of Linear Compact Operators 629
12.7.2.3 Weak Convergence of Elements 629
12.7.3 Fredholm Alternative 629
12.7.4 Compact Operators in Hilbert Space 630
12.7.5 Compact Self-Adjoint Operators 630
12.8 Non-Linear Operators 631
12.8.1 Examples of Non-Linear Operators 631
12.8.2 Differentiability of Non-Linear Operators 632
12.8.3 Newton's Method 632
12.8.4 Schauder's Fixed-Point Theorem 633
12.8.5 Leray Schauder Theory 633
12.8.6 Positive Non-Linear Operators 633
12.8.7 Monotone Operators in Banach Spaces 634
12.9 Measure and Lebesgue Integral 635
12.9.1 Sigma Algebra and Measures 635
12.9.2 Measurable Functions 636
12.9.2.1 Measurable Function 636
12.9.2.2 Properties of the Class of Measurable Functions 636
12.9.3 Integration 637
12.9.3.1 Definition of the Integral 637
12.9.3.2 Some Properties of the Integral 637
12.9.3.3 Convergence Theorems 638
12.9.4 p Spaces 639
12.9.5 Distributions 639
12.9.5.1 Formula of Partial Integration 639
12.9.5.2 Generalized Derivative 640
12.9.5.3 Distributions 640
12.9.5.4 Derivative of a Distribution 641
13 Vector Analysis and Vector Fields 642
13.1 Basic Notions of the Theory of Vector Fields 642
13.1.1 Vector Functions of a Scalar Variable 642
13.1.1.1 Definitions 642
13.1.1.2 Derivativeofa Vector Function 642
13.1.1.3 Rules of Differentiation for Vectors 642
13.1.1.4 Taylor Expansion for Vector Functions 643
13.1.2 Scalar Fields 643
13.1.2.1 Scalar Field or Scalar Point Function 643
13.1.2.2 Important Special Cases of Scalar Fields 643
13.1.2.3 Coordinate Definition of a Field 644
13.1.2.4 Level Surfaces and Level Lines of a Field 644
13.1.3 Vector Fields 645
13.1.3.1 Vector Field or Vector Point Function 645
13.1.3.2 Important Cases of Vector Fields 645
13.1.3.3 Coordinate Representation of Vector Fields 646
13.1.3.4 Transformation of Coordinate Systems 647
13.1.3.5 Vector Lines 648
13.2 Differential Operators of Space 649
13.2.1 Directional and Space Derivatives 649
13.2.1.1 Directional Derivative of a Scalar Field 649
13.2.1.2 Directional Derivative of a Vector Field 650
13.2.1.3 Volume Derivative 650
13.2.2 Gradient of a Scalar Field 650
13.2.2.1 Definition of the Gradient 651
13.2.2.2 Gradient and Volume Derivative 651
13.2.2.3 Gradient and Directional Derivative 651
13.2.2.4 Further Properties of the Gradient 651
13.2.2.5 Gradient of the Scalar Field in Different Coordinates 651
13.2.2.6 Rules of Calculations 652
13.2.3 Vector Gradient 652
13.2.4 Divergence of Vector Fields 653
13.2.4.1 Definition of Divergence 653
13.2.4.2 Divergence in Different Coordinates 653
13.2.4.3 Rules for Evaluation of the Divergence 653
13.2.4.4 Divergence of a Central Field 654
13.2.5 Rotation of Vector Fields 654
13.2.5.1 Definitions of the Rotation 654
13.2.5.2 Rotation in Different Coordinates 655
13.2.5.3 Rules for Evaluating the Rotation 655
13.2.5.4 Rotation of a Potential Field 656
13.2.6 Nabla Operator, Laplace Operator 656
13.2.6.1 Nabla Operator 656
13.2.6.2 Rules for Calculations with the Nabla Operator 656
13.2.6.3 Vector Gradient 657
13.2.6.4 Nabla Operator Applied Twice 657
13.2.6.5 Laplace Operator 657
13.2.7 Review of Spatial Differential Operations 658
13.2.7.1 Fundamental Relations and Results (see Table 13.2) 658
13.2.7.2 Rules of Calculation for Spatial Differential Operators 658
13.2.7.3 Expressions of Vector Analysis in Cartesian. Cylindrical, and Spherical Coordinates 659
13.3 Integration in Vector Fields 660
13.3.1 Line Integral and Potential in Vector Fields 660
13.3.1.1 Line Integral in Vector Fields 660
13.3.1.2 Interpretation of the Line Integral in Mechanics 661
13.3.1.3 Properties of the Line Integral 661
13.3.1.4 Line Integral in Cartesian Coordinates 661
13.3.1.5 Integral Along a Closed Curve in a Vector Field 662
13.3.1.6 Conservative or Potential Field 662
13.3.2 Surface Integrals 663
13.3.2.1 Vector of a Plane Sheet 663
13.3.2.2 Evaluation of the Surface Integral 663
13.3.2.3 Surface Integrals and Flow of Fields 664
13.3.2.4 Surface Integrals in Cartesian Coordinates as Surface Integral of Second Type 664
13.3.3 Integral Theorems 665
13.3.3.1 Integral Theorem and Integral Formula of Gauss 665
13.3.3.2 Integral Theorem of Stokes 666
13.3.3.3 Integral Theorems of Green 666
13.4 Evaluation of Fields 667
13.4.1 Pure Source Fields 667
13.4.2 Pure Rotation Field or Zero-Divergence Field 668
13.4.3 Vector Fields with Point-Like Sources 668
13.4.3.1 Coulomb Field of a Point-Like Charge 668
13.4.3.2 Gravitational Field of a Point Mass 669
13.4.4 Superposition of Fields 669
13.4.4.1 Discrete Source Distribution 669
13.4.4.2 Continuous Source Distribution 669
13.4.4.3 Conclusion 669
13.5 Differential Equations of Vector Field Theory 669
13.5.1 Laplace Differential Equation 669
13.5.2 Poisson Differential Equation 670
14 Function Theory 671
14.1 Functions of Complex Variables 671
14.1.1 Continuity, Differentiability 671
14.1.1.1 Definition of a Complex Function 671
14.1.1.2 Limit of a Complex Function 671
14.1.1.3 Continuous Complex Functions 671
14.1.1.4 Differentiability of a Complex Function 671
14.1.2 Analytic Functions 672
14.1.2.1 Definition of Analytic Functions 672
14.1.2.2 Examples of Analytic Functions 672
14.1.2.3 Properties of Analytic Functions 672
14.1.2.4 Singular Points 673
14.1.3 Conformal Mapping 674
14.1.3.1 Notion and Properties of Conformal Mappings 674
14.1.3.2 Simplest Conformal Mappings 675
14.1.3.3 The Schwarz Reflection Principle 681
14.1.3.4 Complex Potential 681
14.1.3.5 Superposition Principle 683
14.1.3.6 Arbitrary Mappings of the Complex Plane 684
14.2 Integration in the Complex Plane 685
14.2.1 Definite and Indefinite Integral 685
14.2.1.1 Definition of the Integral in the Complex Plane 685
14.2.1.2 Properties and Evaluation of Complex Integrals 686
14.2.2 Cauchy Integral Theorem 688
14.2.2.1 Cauchy Integral Theorem for Simply Connected Domains 688
14.2.2.2 Cauchy Integral Theorem for Multiply Connected Domains 688
14.2.3 Cauchy Integral Formulas 689
14.2.3.1 Analytic Function on the Interior of a Domain 689
14.2.3.2 Analytic Function on the Exterior of a Domain 689
14.3 Power Series Expansion of Analytic Functions 689
14.3.1 Convergence of Series with Complex Terms 689
14.3.1.1 Convergence of a Number Sequence with Complex Terms 689
14.3.1.2 Convergence of an Infinite Series with Complex Terms 690
14.3.1.3 Power Series with Complex Terms 690
14.3.2 Taylor Series 691
14.3.3 Principle of Analytic Continuation 691
14.3.4 Laurent Expansion 692
14.3.5 Isolated Singular Points and the Residue Theorem 692
14.3.5.1 Isolated Singular Points 692
14.3.5.2 Meromorphic Functions 693
14.3.5.3 Elliptic Functions 693
14.3.5.4 Residue 693
14.3.5.5 Residue Theorem 694
14.4 Evaluation of Real Integrals by Complex Integrals 694
14.4.1 Application of Cauchy Integral Formulas 694
14.4.2 Application of the Residue Theorem 695
14.4.3 Application of the Jordan Lemma 695
14.4.3.1 Jordan Lemma 695
14.4.3.2 Examples of the Jordan Lemma 696
14.5 Algebraic and Elementary Transcendental Functions 698
14.5.1 Algebraic Functions 698
14.5.2 Elementary Transcendental Functions 698
14.5.3 Description of Curves in Complex Form 701
14.6 Elliptic Functions 702
14.6.1 Relation to Elliptic Integrals 702
14.6.2 Jacobian Functions 703
14.6.3 Theta Function 705
14.6.4 Weierstrass Functions 705
15 Integral Transformations 707
15.1 Notion of Integral Transformation 707
15.1.1 General Definition of Integral Transformations 707
15.1.2 Special IntegralTransformations 707
15.1.3 Inverse Transformations 707
15.1.4 Linearity of Integral Transformations 707
15.1.5 Integral Transformations for Functions of Several Variables 709
15.1.6 Applications of Integral Transformations 709
15.2 Laplace Transformation 710
15.2.1 Properties of the Laplace Transformation 710
15.2.1.1 Laplace Transformation, Original and Image Space 710
15.2.1.2 Rules for the Evaluation of the Laplace Transformation 711
15.2.1.3 Transforms of Special Functions 714
15.2.1.4 Dirac 8 Function and Distributions 717
15.2.2 Inverse Transformation into the Original Space 718
15.2.2.1 Inverse Transformation with the Help of Tables 718
15.2.2.2 Partial Fraction Decomposition 718
15.2.2.3 Series Expansion 719
15.2.2.4 Inverse Integral 720
15.2.3 Solution of Differential Equations using Laplace Transformation 721
15.2.3.1 Ordinary Linear Differential Equations with Constant Coefficients 721
15.2.3.2 Ordinary Linear Differential Equations with Coefficients Depending on the Variable 722
15.2.3.3 Partial Differential Equations 723
15.3 Fourier Transformation 724
15.3.1 Properties of the Fourier Transformation 724
15.3.1.1 Fourier Integral 724
15.3.1.2 Fourier Transformation and Inverse Transformation 725
15.3.1.3 Rules of Calculation with the Fourier Transformation 727
15.3.1.4 Transforms of Special Functions 730
15.3.2 Solution of Differential Equations using the Fourier Transformation 731
15.3.2.1 Ordinary Linear Differential Equations 731
15.3.2.2 Partial Differential Equations 732
15.4 Z-Transformation 733
15.4.1 Properties of the Z-Transformation 734
15.4.1.1 Discrete Functions 734
15.4.1.2 Definition of the Z-Transformation 734
15.4.1.3 Rules of Calculations 735
15.4.1.4 Relation to the Laplace Transformation 736
15.4.1.5 Inverse of the Z-Transformation 737
15.4.2 Applications of the Z-Transformation 738
15.4.2.1 General Solution of Linear Difference Equations 738
15.4.2.2 Second-Order Difference Equations (Initial Value Problem) 739
15.4.2.3 Second-Order Difference Equations (Boundary Value Problem) 740
15.5 Wavelet Transformation 740
15.5.1 Signals 740
15.5.2 Wavelets 741
15.5.3 Wavelet Transformation 741
15.5.4 Discrete Wavelet Transformation 743
15.5.4.1 Fast Wavelet Transformation 743
15.5.4.2 Discrete Haar Wavelet Transformation 743
15.5.5 Gabor Transformation 743
15.6 Walsh Functions 744
15.6.1 Step Functions 744
15.6.2 Walsh Systems 744
16 Probability Theory and Mathematical Statistics 745
16.1 Combinatorics 745
16.1.1 Permutations 745
16.1.2 Combinations 745
16.1.3 Arrangements 746
16.1.4 Collection of the Formulas of Combinatorics (see Table 16.1) 747
16.2 Probability Theory 747
16.2.1 Event, Frequency and Probability 747
16.2.1.1 Events 747
16.2.1.2 Frequencies and Probabilities 748
16.2.1.3 Conditional Probability Bayes Theorem 750
16.2.2 Random Variables. Distribution Functions 751
16.2.2.1 Random Variable 751
16.2.2.2 Distribution Function 751
16.2.2.3 Expected Value and Variance, Chebyshev Inequality 753
16.2.2.4 Multidimensional Random Variable 754
16.2.3 Discrete Distributions 754
16.2.3.1 Binomial Distribution 755
16.2.3.2 Hypergeometric Distribution 756
16.2.3.3 Poisson Distribution 757
16.2.4 Continuous Distributions 758
16.2.4.1 Normal Distribution 758
16.2.4.2 Standard Normal Distribution. Gaussian Error Function 759
16.2.4.3 Logarithmic Normal Distribution 759
16.2.4.4 Exponential Distribution 760
16.2.4.5 Weibull Distribution 761
16.2.4.6 x2 (Chi-Square) Distribution 762
16.2.4.7 Fisher F Distribution 763
16.2.4.8 Student t Distribution 763
16.2.5 Law of Large Numbers. Limit Theorems 764
16.2.6 Stochastic Processes and Stochastic Chains 765
16.2.6.1 Basic Notions. Markov Chains 765
16.2.6.2 Poisson Process 768
16.3 Mathematical Statistics 769
16.3.1 Statistic Function or Sample Function 769
16.3.1.1 Population. Sample. Random Vector 769
16.3.1.2 Statistic Function or Sample Function 770
16.3.2 Descriptive Statistics 772
16.3.2.1 Statistical Summarization and Analysis of Given Data 772
16.3.2.2 Statistical Parameters 773
16.3.3 Important Tests 774
16.3.3.1 Goodness of Fit Test for a Normal Distribution 774
16.3.3.2 Distribution of the Sample Mean 776
16.3.3.3 Confidence Limits for the Mean 777
16.3.3.4 Confidence Interval for the Variance 778
16.3.3.5 Structure of Hypothesis Testing 779
16.3.4 Correlation and Regression 779
16.3.4.1 Linear Correlation of two Measurable Characters 779
16.3.4.2 Linear Regression for two Measurable Characters 780
16.3.4.3 Multidimensional Regression 781
16.3.5 Monte Carlo Methods 783
16.3.5.1 Simulation 783
16.3.5.2 Random Numbers 783
16.3.5.3 Example of a Monte Carlo Simulation 784
16.3.5.4 Application of the Monte Carlo Method in Numerical Mathematics 785
16.3.5.5 Further Applications of the Monte Carlo Method 787
16.4 Calculus of Errors 787
16.4.1 Measurement Error and its Distribution 788
16.4.1.1 Qualitative Characterization of Measurement Errors 788
16.4.1.2 Density Function of the Measurement Error 788
16.4.1.3 Quantitative Characterization of the Measurement Error 790
16.4.1.4 Determining the Result of a Measurement with Bounds on the Error 792
16.4.1.5 Error Estimation for Direct Measurements with the Same Accuracy 793
16.4.1.6 Error Estimation for Direct Measurements with Different Accuracy 793
16.4.2 Error Propagation and Error Analysis 794
16.4.2.1 Gauss Error Propagation Law 794
16.4.2.2 Error Analysis 796
17 Dynamical Systems and Chaos 797
17.1 Ordinary Differential Equations and Mappings 797
17.1.1 Dynamical Systems 797
17.1.1.1 Basic Notions 797
17.1.1.2 Invariant Sets 799
17.1.2 Qualitative Theory of Ordinary Differential Equations 800
17.1.2.1 Existence of Flows, Phase Space Structure 800
17.1.2.2 Linear Differential Equations 801
17.1.2.3 Stability Theory 803
17.1.2.4 Invariant Manifolds 806
17.1.2.5 Poincare Mapping 808
17.1.2.6 Topological Equivalence of Differential Equations 810
17.1.3 Discrete Dynamical Systems 811
17.1.3.1 Steady States, Periodic Orbits and Limit Sets 811
17.1.3.2 Invariant Manifolds 812
17.1.3.3 Topological Conjugacy of Discrete Systems 813
17.1.4 Structural Stability (Robustness) 813
17.1.4.1 Structurally Stable Differential Equations 813
17.1.4.2 Structurally Stable Discrete Systems 814
17.1.4.3 Generic Properties 814
17.2 Quantitative Description of Attractors 816
17.2.1 Probability Measures on Attractors 816
17.2.1.1 Invariant Measure 816
17.2.1.2 Elements of Ergodic Theory 817
17.2.2 Entropies 819
17.2.2.1 Topological Entropy 819
17.2.2.2 Metric Entropy 819
17.2.3 Lyapunov Exponents 820
17.2.4 Dimensions 822
17.2.4.1 Metric Dimensions 822
17.2.4.2 Dimensions Defined by Invariant Measures 824
17.2.4.3 Local Hausdorff Dimension According to Douady and Oesterle 826
17.2.4.4 Examples of Attractors 827
17.2.5 Strange Attractors and Chaos 828
17.2.6 Chaos in One-Dimensional Mappings 829
17.3 Bifurcation Theory and Routes to Chaos 829
17.3.1 Bifurcations in Morse Smale Systems 829
17.3.1.1 Local Bifurcations in Neighborhoods of Steady States 830
17.3.1.2 Local Bifurcations in a Neighborhood of a Periodic Orbit 835
17.3.1.3 Global Bifurcation 838
17.3.2 Transitions to Chaos 839
17.3.2.1 Cascade of Period Doublings 839
17.3.2.2 Intermittency 839
17.3.2.3 Global Homoclinic Bifurcations 840
17.3.2.4 Destruction of a Torus 841
18 Optimization 846
18.1 Linear Programming 846
18.1.1 Formulation of the Problem and Geometrical Representation 846
18.1.1.1 The Form of a Linear Programming Problem 846
18.1.1.2 Examples and Graphical Solutions 847
18.1.2 Basic Notions of Linear Programming. Normal Form 849
18.1.2.1 Extreme Points and Basis 849
18.1.2.2 Normal Form of the Linear Programming Problem 850
18.1.3 Simplex Method 851
18.1.3.1 Simplex Tableau 851
18.1.3.2 Transition to the New Simplex Tableau 852
18.1.3.3 Determination of an Initial Simplex Tableau 854
18.1.3.4 Revised Simplex Method 855
18.1.3.5 Duality in Linear Programming 856
18.1.4 Special Linear Programming Problems 857
18.1.4.1 Transportation Problem 857
18.1.4.2 Assignment Problem 860
18.1.4.3 Distribution Problem 860
18.1.4.4 Travelling Salesman 861
18.1.4.5 Scheduling Problem 861
18.2 Non-linear Optimization 861
18.2.1 Formulation of the Problem, Theoretical Basis 861
18.2.1.1 Formulation of the Problem 861
18.2.1.2 Optimality Conditions 862
18.2.1.3 Duality in Optimization 863
18.2.2 Special Non-linear Optimization Problems 863
18.2.2.1 Convex Optimization 863
18.2.2.2 Quadratic Optimization 864
18.2.3 Solution Methods for Quadratic Optimization Problems 865
18.2.3.1 Wolfe's Method 865
18.2.3.2 Hildreth d'Esopo Method 867
18.2.4 Numerical Search Procedures 867
18.2.4.1 One-Dimensional Search 867
18.2.4.2 Minimum Search in n-Dimensional Euclidean Vector Space 868
18.2.5 Methods for Unconstrained Problems 868
18.2.5.1 Method of Steepest Descent (Gradient Method) 869
18.2.5.2 Application of the Newton Method 869
18.2.5.3 Conjugate Gradient Methods 869
18.2.5.4 Method of Davidon, Fletcher and Powell (DFP) 870
18.2.6 Evolution Strategies 870
18.2.6.1 Mutation Selection Strategy 871
18.2.6.2 Recombination 871
18.2.7 Gradient Methods for Problems with Inequality Type Constraints) 871
18.2.7.1 Method of Feasible Directions 872
18.2.7.2 Gradient Projection Method 873
18.2.8 Penalty Function and Barrier Methods 875
18.2.8.1 Penalty Function Method 875
18.2.8.2 Barrier Method 876
18.2.9 Cutting Plane Methods 877
18.3 Discrete Dynamic Programming 878
18.3.1 Discrete Dynamic Decision Models 878
18.3.1.1 n-Stage Decision Processes 878
18.3.1.2 Dynamic Programming Problem 878
18.3.2 Examples of Discrete Decision Models 879
18.3.2.1 Purchasing Problem 879
18.3.2.2 Knapsack Problem 879
18.3.3 Bellman Functional Equations 879
18.3.3.1 Properties of the Cost Function 879
18.3.3.2 Formulation of the Functional Equations 880
18.3.4 Bellman Optimality Principle 880
18.3.5 Bellman Functional Equation Method 881
18.3.5.1 Determination of Minimal Costs 881
18.3.5.2 Determination of the Optimal Policy 881
18.3.6 Examples of Applications of the Functional Equation Method 881
18.3.6.1 Optimal Purchasing Policy 881
18.3.6.2 Knapsack Problem 882
19 Numerical Analysis 884
19.1 Numerical Solution of Non-Linear Equations in a Single Unknown 884
19.1.1 Iteration Method 884
19.1.1.1 Ordinary Iteration Method 884
19.1.1.2 Newton's Method 885
19.1.1.3 Regula Falsi 886
19.1.2 Solution of Polynomial Equations 887
19.1.2.1 Horner's Scheme 887
19.1.2.2 Positions of the Roots 888
19.1.2.3 Numerical Methods 889
19.2 Numerical Solution of Equation Systems 890
19.2.1 Systems of Linear Equations 890
19.2.1.1 Triangular Decomposition of a Matrix 890
19.2.1.2 Cholesky's Method for a Symmetric Coefficient Matrix 893
19.2.1.3 Orthogonalization Method 893
19.2.1.4 Iteration Methods 895
19.2.2 Non-Linear Equation Systems 896
19.2.2.1 Ordinary Iteration Method 896
19.2.2.2 Newton's Method 897
19.2.2.3 Derivative-Free Gauss Newton Method 897
19.3 Numerical Integration 898
19.3.1 General Quadrature Formulas 898
19.3.2 Interpolation Quadratures 899
19.3.2.1 Rectangular Formula 899
19.3.2.2 Trapezoidal Formula 899
19.3.2.3 Simpson's Formula 900
19.3.2.4 Hermite's Trapezoidal Formula 900
19.3.3 Quadrature Formulas of Gauss 900
19.3.3.1 Gauss Quadrature Formulas 900
19.3.3.2 Lobatto's Quadrature Formulas 901
19.3.4 Method of Romberg 901
19.3.4.1 Algorithm of the Romberg Method 901
19.3.4.2 Extrapolation Principle 902
19.4 Approximate Integration of Ordinary Differential Equations 904
19.4.1 Initial Value Problems 904
19.4.1.1 Euler Polygonal Method 904
19.4.1.2 Runge Kutta Methods 904
19.4.1.3 Multi-Step Methods 905
19.4.1.4 Predictor Corrector Method 906
19.4.1.5 Convergence. Consistency. Stability 907
19.4.2 Boundary Value Problems 908
19.4.2.1 Difference Method 908
19.4.2.2 Approximation by Using Given Functions 909
19.4.2.3 Shooting Method 910
19.5 Approximate Integration of Partial Differential Equations 911
19.5.1 Difference Method 911
19.5.2 Approximation by Given Functions 912
19.5.3 Finite Element Method (FEM) 913
19.6 Approximation. Computation of Adjustment, Harmonic Analysis 917
19.6.1 Polynomial Interpolation 917
19.6.1.1 Newton's Interpolation Formula 917
19.6.1.2 Lagrange's Interpolation Formula 918
19.6.1.3 Aitken Neville Interpolation 918
19.6.2 Approximation in Mean 919
19.6.2.1 Continuous Problems. Normal Equations 919
19.6.2.2 Discrete Problems, Normal Equations, Householder's Method 921
19.6.2.3 Multidimensional Problems 922
19.6.2.4 Non-Linear Least Squares Problems 922
19.6.3 Chebyshev Approximation 923
19.6.3.1 Problem Definition and the Alternating Point Theorem 923
19.6.3.2 Properties of the Chebyshev Polynomials 924
19.6.3.3 Remes Algorithm 925
19.6.3.4 Discrete Chebyshev Approximation and Optimization 926
19.6.4 Harmonic Analysis 927
19.6.4.1 Formulas for Trigonometric Interpolation 927
19.6.4.2 Fast Fourier Transformation (FFT) 928
19.7 Representation of Curves and Surfaces with Splines 931
19.7.1 Cubic Splines 931
19.7.1.1 Interpolation Splines 931
19.7.1.2 Smoothing Splines 932
19.7.2 Bicubic Splines 933
19.7.2.1 Use of Bicubic Splines 933
19.7.2.2 Bicubic Interpolation Splines 933
19.7.2.3 Bicubic Smoothing Splines 935
19.7.3 Bernstein Bezier Representation of Curves and Surfaces 935
19.7.3.1 Principle of the B B Curve Representation 935
19.7.3.2 B B Surface Representation 936
19.8 Using the Computer 936
19.8.1 Internal Symbol Representation 936
19.8.1.1 Number Systems 936
19.8.1.2 Internal Number Representation 938
19.8.2 Numerical Problems in Calculations with Computers 939
19.8.2.1 Introduction, Error Types 939
19.8.2.2 Normalized Decimal Numbers and Round-Off 939
19.8.2.3 Accuracy in Numerical Calculations 941
19.8.3 Libraries of Numerical Methods 944
19.8.3.1 NAG Library 944
19.8.3.2 IMSL Library 945
19.8.3.3 Aachen Library 946
19.8.4 Application of Computer Algebra Systems 946
19.8.4.1 Mathematica 946
19.8.4.2 Maple 949
20 Computer Algebra Systems 953
20.1 Introduction 953
20.1.1 Brief Characterization of Computer Algebra Systems 953
20.1.2 Examples of Basic Application Fields 953
20.1.2.1 Manipulation of Formulas 953
20.1.2.2 Numerical Calculations 954
20.1.2.3 Graphical Representations 955
20.1.2.4 Programming in Computer Algebra Systems 955
20.1.3 Structure of Computer Algebra Systems 955
20.1.3.1 Basic Structure Elements 955
20.2 Mathematica 956
20.2.1 Basic Structure Elements 956
20.2.2 Types of Numbers in Mathematica 957
20.2.2.1 Basic Types of Numbers in Mathematica 957
20.2.2.2 Special Numbers 958
20.2.2.3 Representation and Conversion of Numbers 958
20.2.3 Important Operators 959
20.2.4 Lists 959
20.2.4.1 Notions 959
20.2.4.2 Nested Lists, Arrays or Tables 960
20.2.4.3 Operations with Lists 960
20.2.4.4 Special Lists 961
20.2.5 Vectors and Matrices as Lists 961
20.2.5.1 Creating Appropriate Lists 961
20.2.5.2 Operations with Matrices and Vectors 962
20.2.6 Functions 963
20.2.6.1 Standard Functions 963
20.2.6.2 Special Functions 963
20.2.6.3 Pure Functions 963
20.2.7 Patterns 964
20.2.8 Functional Operations 964
20.2.9 Programming 966
20.2.10 Supplement about Syntax, Information. Messages 966
20.2.10.1 Contexts. Attributes 966
20.2.10.2 Information 967
20.2.10.3 Messages 967
20.3 Maple 968
20.3.1 Basic Structure Elements 968
20.3.1.1 Types and Objects 968
20.3.1.2 Input and Output 969
20.3.2 Types of Numbers in Maple 970
20.3.2.1 Basic Types of Numbers in Maple 970
20.3.2.2 Special Numbers 970
20.3.2.3 Representation and Conversion of Numbers 970
20.3.3 Important Operators in Maple 971
20.3.4 Algebraic Expressions 971
20.3.5 Sequences and Lists 972
20.3.6 Tables, Arrays, Vectors and Matrices 973
20.3.6.1 Tables and Arrays 973
20.3.6.2 One-Dimensional Arrays 974
20.3.6.3 Two-Dimensional Arrays 974
20.3.6.4 Special Commands for Vectors and Matrices 975
20.3.7 Procedures. Functions and Operators 975
20.3.7.1 Procedures 975
20.3.7.2 Functions 975
20.3.7.3 Functional Operators 976
20.3.7.4 Differential Operators 977
20.3.7.5 The Functional Operator map 977
20.3.8 Programming in Maple 977
20.3.9 Supplement about Syntax, Information and Help 978
20.3.9.1 Using the Maple Library 978
20.3.9.2 Environment Variable 978
20.3.9.3 Information and Help 978
20.4 Applications of Computer Algebra Systems 979
20.4.1 Manipulation of Algebraic Expressions 979
20.4.1.1 Mathematica 979
20.4.1.2 Maple 981
20.4.2 Solution of Equations and Systems of Equations 984
20.4.2.1 Mathematica 984
20.4.2.2 Maple 986
20.4.3 Elements of Linear Algebra 988
20.4.3.1 Mathematica 988
20.4.3.2 Maple 989
20.4.4 Differential and Integral Calculus 992
20.4.4.1 Mathematica 992
20.4.4.2 Maple 995
20.5 Graphics in Computer Algebra Systems 998
20.5.1 Graphics with Mathematica 998
20.5.1.1 Basic Elements of Graphics 998
20.5.1.2 Graphics Primitives 999
20.5.1.3 Syntax of Graphical Representation 999
20.5.1.4 Graphical Options 1000
20.5.1.5 Two-Dimensional Curves 1002
20.5.1.6 Parametric Representation of Curves 1003
20.5.1.7 Representation of Surfaces and Space Curves 1003
20.5.2 Graphics with Maple 1005
20.5.2.1 Two-Dimensional Graphics 1005
20.5.2.2 Three-Dimensional Graphics 1008
21 Tables 1010
21.1 Frequently Used Mathematical Constants 1010
21.2 Natural Constants 1010
21.3 Metric Prefixes 1012
21.4 International System of Physical Units (SI-Units) 1012
21.5 Important Series Expansions 1015
21.6 Fourier Series 1020
21.7 Indefinite Integrals 1023
21.7.1 Integral Rational Functions 1023
21.7.1.1 Integrals with X = ax + b 1023
21.7.1.2 Integrals with X = ax2 + bx + c 1025
21.7.1.3 Integrals with X = a2 ± x2 1026
21.7.1.4 Integrals with X = a' ± x3 1028
21.7.1.5 Integrals with X = a4 + x4 1029
21.7.1.6 Integrals with X = a4 - x4 1029
21.7.1.7 Some Cases of Partial Fraction Decomposition 1029
21.7.2 Integrals of Irrational Functions 1030
21.7.2.1 Integrals with pr and a2 ± b2x 1030
21.7.2.2 Other Integrals with pF 1030
21.7.2.3 Integrals with p ax + b 1031
21.7.2.4 Integrals with p ax + b and p fx + g 1032
21.7.2.5 Integrals with po2 - x2 1033
21.7.2.6 Integrals with p x2 + a2 1035
21.7.2.7 Integrals with p x2 - a2 1036
21.7.2.8 Integrals with ax2 + bx + c 1038
21.7.2.9 Integrals with other Irrational Expressions 1040
21.7.2.10 Recursion Formulas for an Integral with Binomial Differential 1040
21.7.3 Integrals of Trigonometric Functions 1041
21.7.3.1 Integrals with Sine Function 1041
21.7.3.2 Integrals with Cosine Function 1043
21.7.3.3 Integrals with Sine and Cosine Function 1045
21.7.3.4 Integrals with Tangent Function 1049
21.7.3.5 Integrals with Cotangent Function 1049
21.7.4 Integrals of other Transcendental Functions 1050
21.7.4.1 Integrals with Hyperbolic Functions 1050
21.7.4.2 Integrals with Exponential Functions 1051
21.7.4.3 Integrals with Logarithmic Functions 1053
21.7.4.4 Integrals with Inverse Trigonometric Functions 1054
21.7.4.5 Integrals with Inverse Hyperbolic Functions 1055
21.8 Definite Integrals 1056
21.8.1 Definite Integrals of Trigonometric Functions 1056
21.8.2 Definite Integrals of Exponential Functions 1057
21.8.3 Definite Integrals of Logarithmic Functions 1058
21.8.4 Definite Integrals of Algebraic Functions 1059
21.9 Elliptic Integrals 1061
21.9.1 Elliptic Integral of the First Kind F{ip,k), k = sina 1061
21.9.2 Elliptic Integral of the Second Kind E ( 0 90
2.8 Domains and ranges of the area functions 92
2.9 For the approximate determination of an empirically given function relation 112
3.1 Names of angles in degree and radian measure 129
3.2 Properties of some regular polygons 139
3.3 Defining quantities of a right angled-triangle in the plane 141
3.4 Defining quantities of a general triangle, basic problems 144
3.5 Conversion between Degrees and Gons 145
3.6 Directional angle in a segment with correct sign for arctan 145
3.7 Regular polyeders with edge length a 154
3.8 Defining quantities of a spherical right-angled triangle 168
3.9 First and second basic problems for spherical oblique triangles 170
3.10 Third basic problem for spherical oblique triangles 171
3.11 Fourth basic problem for spherical oblique triangles 172
3.12 Fifth and sixth basic problemes for a spherical oblique triangle 173
3.13 Scalar product of basis vectors 186
3.14 Vector product of basis vectors 186
3.15 Scalar product of reciprocal basis vectors 186
3.16 Vector product of reciprocal basis vectors 186
3.17 Vector equations 188
3.18 Geometric application of vector algebra 189
3.19 Equation of curves of second order. Central curves ((5 ^ 0) 205
3.20 Equations of curves of second order. Parabolic curves (5 = 0) 206
3.21 Coordinate signs in the octants 208
3.22 Connections between Cartesian, cylindrical, and spherical polar coordinates 210
3.23 Notation for the direction cosines under coordinate transformation 211
3.24 Type of surfaces of second order with 5 ^ 0 (central surfaces) 224
3.25 Type of surfaces of second order with 5 = 0 (paraboloid, cylinder and two planes) 224
3.26 Tangent and normal equations 226
3.27 Vector and coordinate equations of accompanying configurations of a space curve 241
3.28 Vector and coordinate equations of accompanying configurations as functions of the arclength 241
3.29 Equations of the tangent plane and the surface normal 246
5.1 Truth table of propositional calculus 286
5.2 NAND function 288
5.3 NOR function 288
5.4 Primitive Bravais lattice 310
5.5 Bravais lattice, crystal systems, and crystallographic classes 311
5.6 Some Boolean functions with two variables 344
5.7 Tabular representation of a fuzzy set 360
5.8 t- and s-norms, p G IR 367
5.9 Comparison of operations in Boolean logic and in fuzzy logic 369
6.1 Derivatives of elementary functions 381
6.2 Differentiation rules 386
6.3 Derivatives of higher order of some elementary functions 387
7.1 The first Bernoulli numbers 412
7.2 First Euler numbers 413
7.3 Approximation formulas for some frequently used functions 419
8.1 Basic integrals 428
8.2 Important rules of calculation of indefinite integrals 430
8.3 Substitutions for integration of irrational functions I 435
8.4 Substitutions for integration of irrational functions II 436
8.5 Important properties of definite integrals 443
8.6 Line integrals of the first type 465
8.7 Curve elements 465
8.8 Plane elements of area 474
8.9 Applications of the double integral 475
8.10 Elementary volumes 479
8.11 Applications of the triple integral 480
8.12 Elementary regions of curved surfaces 482
11.1 Roots of the Legendre polynomial of the first kind 574
13.1 Relations between the components of a vector in Cartesian, cylindrical, and spherical coordinates 648
13.2 Fundamental relations for spatial differential operators 658
13.3 Expressions of vector analysis in Cartesian, cylindrical, and spherical coordinates 659
13.4 Line, surface, and volume elements in Cartesian, cylindrical, and spherical coordinates 660
14.1 Real and imaginary parts of the trigonometric and hyperbolic functions 700
14.2 Absolute values and arguments of the trigonometric and hyperbolic functions 700
14.3 Periods, roots and poles of Jacobian functions 704
15.1 Overview of integral transformations of functions of one variable 708
15.2 Comparison of the properties of the Fourier and the Laplace transformation 730
16.1 Collection of the formulas of combinatorics 747
16.2 Relations between events 748
16.3 Frequency table 773
16.4 x2 test 776
16.5 Confidence level for the sample mean 777
16.6 Error description of a measurement sequence 794
17.1 Steady state types in three-dimensional phase space 809
19.1 Helping table for FEM 916
19.2 Orthogonal polynomials 920
List of Tables
19.3 Number systems 937
19.4 Parameters for the basic forms 939
19.5 Mathematica, numerical operations 946
19.6 Mathematica, commands for interpolation 947
19.7 Mathematica, numerical solution of differential equations 948
19.8 Maple, options for the command fsolve 950
20.1 Mathematica, Types of numbers 957
20.2 Mathematica, Important operators 959
20.3 Mathematica, Commands for the choice of list elements 960
20.4 Mathematica. Operations with lists 960
20.5 Mathematica, Operation Table 961
20.6 Mathematica, Operations with matrices 962
20.7 Mathematica, Standard functions 963
20.8 Mathematica, Special functions 963
20.9 Maple, Basic types 968
20.10 Maple, Types 968
20.11 Maple, Types of numbers 970
20.12 Maple, Arguments of function convert 971
20.13 Maple, Standard functions 975
20.14 Maple, Special functions 976
20.15 Mathematica, Commands for manipulation of algebraic expressions 979
20.16 Mathematica, Algebraic polynomial operations 980
20.17 Maple, Operations to manipulate algebraic expressions 981
20.18 Mathematica, Operations to solve systems of equations 986
20.19 Maple, Matrix operations 990
20.20 Maple, Operations of the Gaussian algorithm 991
20.21 Mathematica, Operations of differentiation 992
20.22 Mathematica, Commands to solve differential equations 994
20.23 Maple, Options of operation dsolve 997
20.24 Mathematica, Two-dimensional graphic objects 999
20.25 Mathematica, Graphics commands 999
20.26 Mathematica, Some graphical options 1000
20.27 Mathematica, Options for 3D graphics 1005
20.28 Maple, Options for Plot command 1006
20.29 Maple, Options of command plot3d 1008
21.1 Frequently Used Constants 1010
21.2 Natural Constants 1010
21.3 Metric Prefixes 1012
21.4 International System of Physical Units (SI-Units) 1012
21.5 Important Series Expansions 1015
21.6 Fourier Series 1020
21.7 Indefinite Integrals 1023
21.8 Definite Integrals 1056
21.9 Elliptic Integrals 1061
21.10 Gamma Function 1063
21.11 Bessel Functions (Cylindrical Functions) 1064
21.12 Legendre Polynomials of the First Kind 1066
21.13 Laplace Transformation 1067
21.14 Fourier Transformation 1072
21.15 Z- Transformation 1086
21.16 Poisson Distribution 1089
List of Tables XLIII
21.17 Standard Normal Distribution 1091
21.18 x2 Distribution 1093
21.19 Fisher F Distribution 1094
21.20 Student t Distribution 1096
21.21 Random Numbers 1097
// @
Index 1109
Mathematic Symbols 1160