И.Н. Бронштейн и др. | Справочник по математике (2015) [PDF] [En]

List of Tables XLII

1 Arithmetics 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 Factoring Polynomials 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 Partial Fraction Decomposition 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 201.2.5.5 Relations Between the Means of Two Positive Values 20
1.3 Business Mathematics 21
1.3.1 Calculation of Interest or Percentage 21
1.3.1.1 Percentage or Interest 21
1.3.1.2 Increment 21
1.3.1.3 Discount or Reduction 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
1.3.5.1 Methods of Depreciation 26
1.3.5.2 Straight-Line Method 26
1.3.5.3 Arithmetically Declining Balance Depreciation 26
1.3.5.4 Digital Declining Balance Depreciation 27
1.3.5.5 Geometrically Declining Balance Depreciation 27
1.3.5.6 Depreciation with Different Types of Depreciation Account 28
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 H¨older 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 35
1.5.2.3 Trigonometric Form of Complex Numbers 35
1.5.2.4 Exponential Form of a Complex Number 36
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 37
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 38
1.5.3.6 Taking 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 39
1.6.1.3 Extraneous 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) 40
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 Coeficients 44
1.6.4 Reducing Transcendental Equations to Algebraic Equations 45
1.6.4.1 Definition 45
1.6.4.2 Exponential Equations 46
1.6.4.3 Logarithmic Equations 46
1.6.4.4 Trigonometric Equations 46
1.6.4.5 Equations with Hyperbolic Functions 47

2 Functions 48
2.1 Notion of Functions 48
2.1.1 Definition of a Function 48
2.1.1.1 Function 48
2.1.1.2 Real Functions 48
2.1.1.3 Functions of Several Variables 48
2.1.1.4 Complex Functions 48
2.1.1.5 Further Functions 48
2.1.1.6 Functionals 48
2.1.1.7 Functions and Mappings 49
2.1.2 Methods for Defining a Real Function 49
2.1.2.1 Defining a Function 49
2.1.2.2 Analytic Representation of a Function 49
2.1.3 Certain Types of Functions 50
2.1.3.1 Monotone Functions 50
2.1.3.2 Bounded Functions 51
2.1.3.3 Extreme Values of Functions 51
2.1.3.4 Even Functions 51
2.1.3.5 Odd Functions 51
2.1.3.6 Representation with Even and Odd Functions 52
2.1.3.7 Periodic Functions 52
2.1.3.8 Inverse Functions 52
2.1.4 Limits of Functions 53
2.1.4.1 Definition of the Limit of a Function 53
2.1.4.2 Definition by Limit of Sequences 53
2.1.4.3 Cauchy Condition for Convergence 53
2.1.4.4 Infinity as a Limit of a Function 53
2.1.4.5 Left-Hand and Right-Hand Limit of a Function 54
2.1.4.6 Limit of a Function as x Tends to Infinity 54
2.1.4.7 Theorems About Limits of Functions 55
2.1.4.8 Calculation of Limits 55
2.1.4.9 Order of Magnitude of Functions and Landau Order Symbols 57
2.1.5 Continuity of a Function 58
2.1.5.1 Notion of Continuity and Discontinuity 58
2.1.5.2 Definition of Continuity 58
2.1.5.3 Most Frequent Types of Discontinuities 59
2.1.5.4 Continuity and Discontinuity of Elementary Functions 60
2.1.5.5 Properties of Continuous Functions 60
2.2 Elementary Functions 62
2.2.1 Algebraic Functions 62
2.2.1.1 Polynomials 62
2.2.1.2 Rational Functions 62
2.2.1.3 Irrational Functions 62
2.2.2 Transcendental Functions 62
2.2.2.1 Exponential Functions 62
2.2.2.2 Logarithmic Functions 63
2.2.2.3 Trigonometric Functions 63
2.2.2.4 Inverse Trigonometric Functions 63
2.2.2.5 Hyperbolic Functions 63
2.2.2.6 Inverse Hyperbolic Functions 63
2.2.3 Composite Functions 63
2.3 Polynomials 63
2.3.1 Linear Function 63
2.3.2 Quadratic Polynomial 64
2.3.3 Cubic Polynomials 64
2.3.4 Polynomials of n-th Degree 65
2.3.5 Parabola of n-th Degree 66
2.4 Rational Functions 66
2.4.1 Special Fractional Linear Function (Inverse Proportionality) 66
2.4.2 Linear Fractional Function 66
2.4.3 Curves of Third Degree, Type I 67
2.4.4 Curves of Third Degree, Type II 67
2.4.5 Curves of Third Degree, Type III 68
2.4.6 Reciprocal Powers 70
2.5 Irrational Functions 71
2.5.1 Square Root of a Linear Binomial 71
2.5.2 Square Root of a Quadratic Polynomial 71
2.5.3 Power Function 71
2.6 Exponential Functions and Logarithmic Functions 72
2.6.1 Exponential Functions 72
2.6.2 Logarithmic Functions 73
2.6.3 Error Curve 73
2.6.4 Exponential Sum 74
2.6.5 Generalized Error Function 74
2.6.6 Product of Power and Exponential Functions 75
2.7 Trigonometric Functions (Functions of Angles) 76
2.7.1 Basic Notions 76
2.7.1.1 Definition and Representation 76
2.7.1.2 Range and Behavior of the Functions 79
2.7.2 Important Formulas for Trigonometric Functions 81
2.7.2.1 Relations Between the Trigonometric Functions 81
2.7.2.2 Trigonometric Functions of the Sum and Difference of Two Angles (Addition Theorems) 81
2.7.2.3 Trigonometric Functions of an Integer Multiple of an Angle 81
2.7.2.4 Trigonometric Functions of Half-Angles 82
2.7.2.5 Sum and Difference of Two Trigonometric Functions 83
2.7.2.6 Products of Trigonometric Functions 83
2.7.2.7 Powers of Trigonometric Functions 83
2.7.3 Description of Oscillations 84
2.7.3.1 Formulation of the Problem 84
2.7.3.2 Superposition of Oscillations 84
2.7.3.3 Vector Diagram for Oscillations 85
2.7.3.4 Damping of Oscillations 85
2.8 Cyclometric or Inverse Trigonometric Functions 85
2.8.1 Definition of the Inverse Trigonometric Functions 85
2.8.2 Reduction to the Principal Value 86
2.8.3 Relations Between the Principal Values 87
2.8.4 Formulas for Negative Arguments 87
2.8.5 Sum and Difference of arcsin x and arcsin y 88
2.8.6 Sum and Difference of arccos x and arccos y 88
2.8.7 Sum and Difference of arctan x and arctan y 88
2.8.8 Special Relations for arcsin x, arccos x, arctan x 88
2.9 Hyperbolic Functions 89
2.9.1 Definition of Hyperbolic Functions 89
2.9.2 Graphical Representation of the Hyperbolic Functions 89
2.9.2.1 Hyperbolic Sine 89
2.9.2.2 Hyperbolic Cosine 89
2.9.2.3 Hyperbolic Tangent 90
2.9.2.4 Hyperbolic Cotangent 90
2.9.3 Important Formulas for the Hyperbolic Functions 91
2.9.3.1 Hyperbolic Functions of One Variable 91
2.9.3.2 Expressing a Hyperbolic Function by Another One with the Same Argument 91
2.9.3.3 Formulas for Negative Arguments 91
2.9.3.4 Hyperbolic Functions of the Sum and Difference of Two Arguments (Addition Theorems) 91
2.9.3.5 Hyperbolic Functions of Double Arguments 92
2.9.3.6 De Moivre Formula for Hyperbolic Functions 92
2.9.3.7 Hyperbolic Functions of Half-Argument 92
2.9.3.8 Sum and Difference of Hyperbolic Functions 92
2.9.3.9 Relation Between Hyperbolic and Trigonometric Functions with Complex Arguments z 92
2.10 Area Functions 93
2.10.1 Definitions 93
2.10.1.1 Area Sine 93
2.10.1.2 Area Cosine 93
2.10.1.3 Area Tangent 94
2.10.1.4 Area Cotangent 94
2.10.2 Determination of Area Functions Using Natural Logarithm 94
2.10.3 Relations Between Different Area Functions 94
2.10.4 Sum and Difference of Area Functions 95
2.10.5 Formulas for Negative Arguments 95
2.11 Curves of Order Three (Cubic Curves) 95
2.11.1 Semicubic Parabola 95
2.11.2 Witch of Agnesi 95
2.11.3 Cartesian Folium (Folium of Descartes) 96
2.11.4 Cissoid 96
2.11.5 Strophoide 97
2.12 Curves of Order Four (Quartics) 97
2.12.1 Conchoid of Nicomedes 97
2.12.2 General Conchoid 98
2.12.3 Pascal’s Lima¸con 98
2.12.4 Cardioid 99
2.12.5 Cassinian Curve 100
2.12.6 Lemniscate 101
2.13 Cycloids 101
2.13.1 Common (Standard) Cycloid 101
2.13.2 Prolate and Curtate Cycloids or Trochoids 102
2.13.3 Epicycloid 102
2.13.4 Hypocycloid and Astroid 103
2.13.5 Prolate and Curtate Epicycloid and Hypocycloid 104
2.14 Spirals 105
2.14.1 Archimedean Spiral 105
2.14.2 Hyperbolic Spiral 105
2.14.3 Logarithmic Spiral 106
2.14.4 Evolvent of the Circle 106
2.14.5 Clothoid 107
2.15 Various Other Curves 107
2.15.1 Catenary Curve 107
2.15.2 Tractrix 108
2.16 Determination of Empirical Curves 108
2.16.1 Procedure 108
2.16.1.1 Curve-Shape Comparison 108
2.16.1.2 Rectification 108
2.16.1.3 Determination of Parameters 109
2.16.2 Useful Empirical Formulas 109
2.16.2.1 Power Functions 109
2.16.2.2 Exponential Functions 110
2.16.2.3 Quadratic Polynomial 111
2.16.2.4 Rational Linear Functions 111
2.16.2.5 Square Root of a Quadratic Polynomial 111
2.16.2.6 General Error Curve 112
2.16.2.7 Curve of Order Three, Type II 112
2.16.2.8 Curve of Order Three, Type III 112
2.16.2.9 Curve of Order Three, Type I 113
2.16.2.10 Product of Power and Exponential Functions 113
2.16.2.11 Exponential Sum 113
2.16.2.12 Numerical Example 114
2.17 Scales and Graph Paper 115
2.17.1 Scales 115
2.17.2 Graph Paper 116
2.17.2.1 Semilogarithmic Paper 116
2.17.2.2 Double Logarithmic Paper 117
2.17.2.3 Graph Paper with a Reciprocal Scale 117
2.17.2.4 Remark 118
2.18 Functions of Several Variables 118
2.18.1 Definition and Representation 118
2.18.1.1 Representation of Functions of Several Variables 118
2.18.1.2 Geometric Representation of Functions of Several Variables 118
2.18.2 Different Domains in the Plane 119
2.18.2.1 Domain of a Function 119
2.18.2.2 Two-Dimensional Domains 119
2.18.2.3 Three or Multidimensional Domains 120
2.18.2.4 Methods to Determine a Function 120
2.18.2.5 Various Forms for the Analytical Representation of a Function 121
2.18.2.6 Dependence of Functions 122
2.18.3 Limits 123
2.18.3.1 Definition 123
2.18.3.2 Exact Definition 123
2.18.3.3 Generalization for Several Variables 123
2.18.3.4 Iterated Limit 124
2.18.4 Continuity 124
2.18.5 Properties of Continuous Functions 124
2.18.5.1 Theorem on Zeros of Bolzano 124
2.18.5.2 Intermediate Value Theorem 124
2.18.5.3 Theorem About the Boundedness of a Function 124
2.18.5.4 Weierstrass Theorem (About the Existence of Maximum and Minimum) 125
2.19 Nomography 125
2.19.1 Nomograms 125
2.19.2 Net Charts 125
2.19.3 Alignment Charts 126
2.19.3.1 Alignment Charts with Three Straight-Line Scales Through a Point 126
2.19.3.2 Alignment Charts with Two Parallel Inclined Straight-Line Scales and One Inclined Straight-Line Scale 127
2.19.3.3 Alignment Charts with Two Parallel Straight Lines and a Curved Scale 127
2.19.4 Net Charts for More Than Three Variables 128

3 Geometry 129
3.1 Plane Geometry 129
3.1.1 Basic Notations 129
3.1.1.1 Point, Line, Ray, Segment 129
3.1.1.2 Angle 129
3.1.1.3 Angle Between Two Intersecting Lines 130
3.1.1.4 Pairs of Angles with Intersecting Parallels 130
3.1.1.5 Angles Measured in Degrees and in Radians 131
3.1.2 Geometrical Definition of Circular and Hyperbolic Functions 131
3.1.2.1 Definition of Circular or Trigonometric Functions 131
3.1.2.2 Definitions of the Hyperbolic Functions 132
3.1.3 Plane Triangles 132
3.1.3.1 Statements about Plane Triangles 132
3.1.3.2 Symmetry 133
3.1.4 Plane Quadrangles 135
3.1.4.1 Parallelogram 135
3.1.4.2 Rectangle and Square 136
3.1.4.3 Rhombus 136
3.1.4.4 Trapezoid 136
3.1.4.5 General Quadrangle 136
3.1.4.6 Inscribed Quadrangle 137
3.1.4.7 Circumscribing Quadrangle 137
3.1.5 Polygons in the Plane 138
3.1.5.1 General Polygon 138
3.1.5.2 Regular Convex Polygons 138
3.1.5.3 Some Regular Convex Polygons 139
3.1.6 The Circle and Related Shapes 139
3.1.6.1 Circle 139
3.1.6.2 Circular Segment and Circular Sector 141
3.1.6.3 Annulus 141
3.2 Plane Trigonometry 142
3.2.1 Triangles 142
3.2.1.1 Calculations in Right-Angled Triangles in the Plane 142
3.2.1.2 Calculations in General (Oblique) Triangles in the Plane 142
3.2.2 Geodesic Applications 144
3.2.2.1 Geodesic Coordinates 144
3.2.2.2 Angles in Geodesy 146
3.2.2.3 Applications in Surveying 148
3.3 Stereometry 151
3.3.1 Lines and Planes in Space 151
3.3.2 Edge, Corner, Solid Angle 152
3.3.3 Polyeder or Polyhedron 153
3.3.4 Solids Bounded by Curved Surfaces 156
3.4 Spherical Trigonometry 160
3.4.1 Basic Concepts of Geometry on the Sphere 160
3.4.1.1 Curve, Arc, and Angle on the Sphere 160
3.4.1.2 Special Coordinate Systems 162
3.4.1.3 Spherical Lune or Biangle 163
3.4.1.4 Spherical Triangle 163
3.4.1.5 Polar Triangle 164
3.4.1.6 Euler Triangles and Non-Euler Triangles 164
3.4.1.7 Trihedral Angle 164
3.4.2 Basic Properties of Spherical Triangles 165
3.4.2.1 General Statements 165
3.4.2.2 Fundamental Formulas and Applications 165
3.4.2.3 Further Formulas 168
3.4.3 Calculation of Spherical Triangles 169
3.4.3.1 Basic Problems, Accuracy Observations 169
3.4.3.2 Right-Angled Spherical Triangles 169
3.4.3.3 Spherical Triangles with Oblique Angles 171
3.4.3.4 Spherical Curves 174
3.5 Vector Algebra and Analytical Geometry 181
3.5.1 Vector Algebra 181
3.5.1.1 Definition of Vectors 181
3.5.1.2 Calculation Rules for Vectors 182
3.5.1.3 Coordinates of a Vector 183
3.5.1.4 Directional Coeficient 184
3.5.1.5 Scalar Product and Vector Product 184
3.5.1.6 Combination of Vector Products 185
3.5.1.7 Vector Equations 188
3.5.1.8 Covariant and Contravariant Coordinates of a Vector 188
3.5.1.9 Geometric Applications of Vector Algebra 190
3.5.2 Analytical Geometry of the Plane 190
3.5.2.1 Basic Concepts, Coordinate Systems in the Plane 190
3.5.2.2 Coordinate Transformations 191
3.5.2.3 Special Notations and Points in the Plane 192
3.5.2.4 Areas 194
3.5.2.5 Equation of a Curve 195
3.5.2.6 Line 195
3.5.2.7 Circle 198
3.5.2.8 Ellipse 199
3.5.2.9 Hyperbola 201
3.5.2.10 Parabola 204
3.5.2.11 Quadratic Curves (Curves of Second Order or Conic Sections) 206
3.5.3 Analytical Geometry of Space 209
3.5.3.1 Basic Concepts 209
3.5.3.2 Spatial Coordinate Systems 210
3.5.3.3 Transformation of Orthogonal Coordinates 212
3.5.3.4 Rotations with Direction Cosines 213
3.5.3.5 Cardan Angles 214
3.5.3.6 Euler’s angles 215
3.5.3.7 Special Quantities in Space 216
3.5.3.8 Equation of a Surface 217
3.5.3.9 Equation of a Space Curve 218
3.5.3.10 Line and Plane in Space 218
3.5.3.11 Lines in Space 221
3.5.3.12 Intersection Points and Angles of Lines and Planes in Space 223
3.5.3.13 Surfaces of Second Order, Equations in Normal Form 224
3.5.3.14 Surfaces of Second Order or Quadratic Surfaces, General Theory 228
3.5.4 Geometric Transformations and Coordinate Transformations 229
3.5.4.1 Geometric 2D Transformations 229
3.5.4.2 Homogeneous Coordinates, Matrix Representation 231
3.5.4.3 Coordinate Transformation 231
3.5.4.4 Composition of Transformations 232
3.5.4.5 3D–Transformations 233
3.5.4.6 Deformation Transformations 236
3.5.5 Planar Projections 237
3.5.5.1 Classification of the projections 237
3.5.5.2 Local or Projection Coordinate System 238
3.5.5.3 Principal Projections 238
3.5.5.4 Axonometric Projection 238
3.5.5.5 Isometric Projection 239
3.5.5.6 Oblique Parallel Projection 240
3.5.5.7 Perspective Projection 241
3.6 Differential Geometry 243
3.6.1 Plane Curves 243
3.6.1.1 Ways to Define a Plane Curve 243
3.6.1.2 Local Elements of a Curve 243
3.6.1.3 Special Points of a Curve 249
3.6.1.4 Asymptotes of Curves 252
3.6.1.5 General Discussion of a Curve Given by an Equation 253
3.6.1.6 Evolutes and Evolvents 254
3.6.1.7 Envelope of a Family of Curves 255
3.6.2 Space Curves 256
3.6.2.1 Ways to Define a Space Curve 256
3.6.2.2 Moving Trihedral 256
3.6.2.3 Curvature and Torsion 258
3.6.3 Surfaces 261
3.6.3.1 Ways to Define a Surface 261
3.6.3.2 Tangent Plane and Surface Normal 262
3.6.3.3 Line Elements of a Surface 263
3.6.3.4 Curvature of a Surface 265
3.6.3.5 Ruled Surfaces and Developable Surfaces 268
3.6.3.6 Geodesic Lines on a Surface 268

4 Linear Algebra 269
4.1 Matrices 269
4.1.1 Notion of Matrix 269
4.1.2 Square Matrices 270
4.1.3 Vectors 271
4.1.4 Arithmetical Operations with Matrices 272
4.1.5 Rules of Calculation for Matrices 275
4.1.6 Vector and Matrix Norms 276
4.1.6.1 Vector Norms 277
4.1.6.2 Matrix Norms 277
4.2 Determinants 278
4.2.1 Definitions 278
4.2.2 Rules of Calculation for Determinants 278
4.2.3 Evaluation of Determinants 279
4.3 Tensors 280
4.3.1 Transformation of Coordinate Systems 280
4.3.2 Tensors in Cartesian Coordinates 281
4.3.3 Tensors with Special Properties 283
4.3.3.1 Tensors of Rank 2 283
4.3.3.2 Invariant Tensors 283
4.3.4 Tensors in Curvilinear Coordinate Systems 284
4.3.4.1 Covariant and Contravariant Basis Vectors 284
4.3.4.2 Covariant and Contravariant Coordinates of Tensors of Rank 1 285
4.3.4.3 Covariant, Contravariant and Mixed Coordinates of Tensors of Rank 2 286
4.3.4.4 Rules of Calculation 287
4.3.5 Pseudotensors 287
4.3.5.1 Symmetry with Respect to the Origin 287
4.3.5.2 Introduction to the Notion of Pseudotensors 288
4.4 Quaternions and Applications 289
4.4.1 Quaternions 290
4.4.1.1 Definition and Representation 290
4.4.1.2 Matrix Representation of Quaternions 291
4.4.1.3 Calculation Rules 292
4.4.2 Representation of Rotations in IR3 294
4.4.2.1 Rotations of an Object About the Coordinate Axes 295
4.4.2.2 Cardan-Angles 295
4.4.2.3 Euler Angles 296
4.4.2.4 Rotation Around an Arbitrary Zero Point Axis 296
4.4.2.5 Rotation and Quaternions 297
4.4.2.6 Quaternions and Cardan Angles 298
4.4.2.7 Eficiency of the Algorithms 301
4.4.3 Applications of Quaternions 302
4.4.3.1 3D Rotations in Computer Graphics 302
4.4.3.2 Interpolation by Rotation matrices 303
4.4.3.3 Stereographic Projection 303
4.4.3.4 Satellite navigation 304
4.4.3.5 Vector Analysis 305
4.4.3.6 Normalized Quaternions and Rigid Body Motion 306
4.5 Systems of Linear Equations 307
4.5.1 Linear Systems, Pivoting 307
4.5.1.1 Linear Systems 307
4.5.1.2 Pivoting 307
4.5.1.3 Linear Dependence 308
4.5.1.4 Calculation of the Inverse of a Matrix 308
4.5.2 Solution of Systems of Linear Equations 308
4.5.2.1 Definition and Solvability 308
4.5.2.2 Application of Pivoting 310
4.5.2.3 Cramer’s Rule 311
4.5.2.4 Gauss’s Algorithm 312
4.5.3 Overdetermined Linear Systems of Equations 313
4.5.3.1 Overdetermined Linear Systems of Equations and Linear Least Squares Problems 313
4.5.3.2 Suggestions for Numerical Solutions of Least Squares Problems 314
4.6 Eigenvalue Problems for Matrices 314
4.6.1 General Eigenvalue Problem 314
4.6.2 Special Eigenvalue Problem 315
4.6.2.1 Characteristic Polynomial 315
4.6.2.2 Real Symmetric Matrices, Similarity Transformations 316
4.6.2.3 Transformation of Principal Axes of Quadratic Forms 317
4.6.2.4 Suggestions for the Numerical Calculations of Eigenvalues 319
4.6.3 Singular Value Decomposition 321

5 Algebra and Discrete Mathematics 323
5.1 Logic 323
5.1.1 Propositional Calculus 323
5.1.2 Formulas in Predicate Calculus 326
5.2 Set Theory 327
5.2.1 Concept of Set, Special Sets 327
5.2.2 Operations with Sets 328
5.2.3 Relations and Mappings 331
5.2.4 Equivalence and Order Relations 333
5.2.5 Cardinality of Sets 335
5.3 Classical Algebraic Structures 335
5.3.1 Operations 335
5.3.2 Semigroups 336
5.3.3 Groups 336
5.3.3.1 Definition and Basic Properties 336
5.3.3.2 Subgroups and Direct Products 337
5.3.3.3 Mappings Between Groups 339
5.3.4 Group Representations 340
5.3.4.1 Definitions 340
5.3.4.2 Particular Representations 342
5.3.4.3 Direct Sum of Representations 343
5.3.4.4 Direct Product of Representations 344
5.3.4.5 Reducible and Irreducible Representations 344
5.3.4.6 Schur’s Lemma 1 345
5.3.4.7 Clebsch-Gordan Series 345
5.3.4.8 Irreducible Representations of the Symmetric Group S M 345
5.3.5 Applications of Groups 345
5.3.5.1 Symmetry Operations, Symmetry Elements 346
5.3.5.2 Symmetry Groups or Point Groups 346
5.3.5.3 Symmetry Operations with Molecules 347
5.3.5.4 Symmetry Groups in Crystallography 348
5.3.5.5 Symmetry Groups in Quantum Mechanics 350
5.3.5.6 Further Applications of Group Theory in Physics 351
5.3.6 Lie Groups and Lie Algebras 351
5.3.6.1 Introduction 351
5.3.6.2 Matrix-Lie Groups 352
5.3.6.3 Important Applications 355
5.3.6.4 Lie Algebra 356
5.3.6.5 Applications in Robotics 358
5.3.7 Rings and Fields 361
5.3.7.1 Definitions 361
5.3.7.2 Subrings, Ideals 362
5.3.7.3 Homomorphism, Isomorphism, Homomorphism Theorem 362
5.3.7.4 Finite Fields and Shift Registers 363
5.3.8 Vector Spaces 365
5.3.8.1 Definition 365
5.3.8.2 Linear Dependence 366
5.3.8.3 Linear Operators 366
5.3.8.4 Subspaces, Dimension Formula 367
5.3.8.5 Euclidean Vector Spaces, Euclidean Norm 367
5.3.8.6 Bilinear Mappings, Bilinear Forms 368
5.4 Elementary Number Theory 370
5.4.1 Divisibility 370
5.4.1.1 Divisibility and Elementary Divisibility Rules 370
5.4.1.2 Prime Numbers 370
5.4.1.3 Criteria for Divisibility 372
5.4.1.4 Greatest Common Divisor and Least Common Multiple 373
5.4.1.5 Fibonacci Numbers 375
5.4.2 Linear Diophantine Equations 375
5.4.3 Congruences and Residue Classes 377
5.4.4 Theorems of Fermat, Euler, and Wilson 381
5.4.5 Prime Number Tests 382
5.4.6 Codes 383
5.4.6.1 Control Digits 383
5.4.6.2 Error correcting codes 385
5.5 Cryptology 386
5.5.1 Problem of Cryptology 386
5.5.2 Cryptosystems 387
5.5.3 Mathematical Foundation 387
5.5.4 Security of Cryptosystems 388
5.5.4.1 Methods of Conventional Cryptography 388
5.5.4.2 Linear Substitution Ciphers 389
5.5.4.3 Vigen`ere Cipher 389
5.5.4.4 Matrix Substitution 389
5.5.5 Methods of Classical Cryptanalysis 389
5.5.5.1 Statistical Analysis 390
5.5.5.2 Kasiski-Friedman Test 390
5.5.6 One-Time Pad 390
5.5.7 Public Key Methods 391
5.5.7.1 Difie-Hellman Key Exchange 391
5.5.7.2 One-Way Function 391
5.5.7.3 RSA Codes and RSA Method 392
5.5.8 DES Algorithm (Data Encryption Standard) 393
5.5.9 IDEA Algorithm (International Data Encryption Algorithm) 393
5.6 Universal Algebra 394
5.6.1 Definition 394
5.6.2 Congruence Relations, Factor Algebras 394
5.6.3 Homomorphism 394
5.6.4 Homomorphism Theorem 395
5.6.5 Varieties 395
5.6.6 Term Algebras, Free Algebras 395
5.7 Boolean Algebras and Switch Algebra 395
5.7.1 Definition 395
5.7.2 Duality Principle 396
5.7.3 Finite Boolean Algebras 397
5.7.4 Boolean Algebras as Orderings 397
5.7.5 Boolean Functions, Boolean Expressions 397
5.7.6 Normal Forms 399
5.7.7 Switch Algebra 399
5.8 Algorithms of Graph Theory 401
5.8.1 Basic Notions and Notation 401
5.8.2 Traverse of Undirected Graphs 404
5.8.2.1 Edge Sequences or Paths 404
5.8.2.2 Euler Trails 405
5.8.2.3 Hamiltonian Cycles 406
5.8.3 Trees and Spanning Trees 407
5.8.3.1 Trees 407
5.8.3.2 Spanning Trees 408
5.8.4 Matchings 409
5.8.5 Planar Graphs 410
5.8.6 Paths in Directed Graphs 410
5.8.7 Transport Networks 411
5.9 Fuzzy Logic 413
5.9.1 Basic Notions of Fuzzy Logic 413
5.9.1.1 Interpretation of Fuzzy Sets 413
5.9.1.2 Membership Functions on the Real Line 414
5.9.1.3 Fuzzy Sets 416
5.9.2 Connections (Aggregations) of Fuzzy Sets 418
5.9.2.1 Concepts for Aggregations of Fuzzy Sets 418
5.9.2.2 Practical Aggregation Operations of Fuzzy Sets 419
5.9.2.3 Compensatory Operators 421
5.9.2.4 Extension Principle 421
5.9.2.5 Fuzzy Complement 421
5.9.3 Fuzzy-Valued Relations 422
5.9.3.1 Fuzzy Relations 422
5.9.3.2 Fuzzy Product Relation R ◦ S 424
5.9.4 Fuzzy Inference (Approximate Reasoning) 425
5.9.5 Defuzzification Methods 426
5.9.6 Knowledge-Based Fuzzy Systems 427
5.9.6.1 Method of Mamdani 427
5.9.6.2 Method of Sugeno 428
5.9.6.3 Cognitive Systems 428
5.9.6.4 Knowledge-Based Interpolation Systems 430

6 Differentiation 432
6.1 Differentiation of Functions of One Variable 432
6.1.1 Differential Quotient 432
6.1.2 Rules of Differentiation for Functions of One Variable 433
6.1.2.1 Derivatives of the Elementary Functions 433
6.1.2.2 Basic Rules of Differentiation 433
6.1.3 Derivatives of Higher Order 438
6.1.3.1 Definition of Derivatives of Higher Order 438
6.1.3.2 Derivatives of Higher Order of some Elementary Functions 438
6.1.3.3 Leibniz’s Formula 438
6.1.3.4 Higher Derivatives of Functions Given in Parametric Form 440
6.1.3.5 Derivatives of Higher Order of the Inverse Function 440
6.1.4 Fundamental Theorems of Differential Calculus 441
6.1.4.1 Monotonicity 441
6.1.4.2 Fermat’s Theorem 441
6.1.4.3 Rolle’s Theorem 441
6.1.4.4 Mean Value Theorem of Differential Calculus 442
6.1.4.5 Taylor’s Theorem of Functions of One Variable 442
6.1.4.6 Generalized Mean Value Theorem of Differential Calculus (Cauchy’s Theorem) 443
6.1.5 Determination of the Extreme Values and Inflection Points 443
6.1.5.1 Maxima and Minima 443
6.1.5.2 Necessary Conditions for the Existence of a Relative Extreme Value 443
6.1.5.3 Determination of the Relative Extreme Values and the Inflection Points of a Differentiable, Explicit Function y = f(x) 444
6.1.5.4 Determination of Absolute Extrema 445
6.1.5.5 Determination of the Extrema of Implicit Functions 445
6.2 Differentiation of Functions of Several Variables 445
6.2.1 Partial Derivatives 445
6.2.1.1 Partial Derivative of a Function 445
6.2.1.2 Geometrical Meaning for Functions of Two Variables 446
6.2.1.3 Differentials of x and f(x) 446
6.2.1.4 Basic Properties of the Differential 447
6.2.1.5 Partial Differential 447
6.2.2 Total Differential and Differentials of Higher Order 447
6.2.2.1 Notion of Total Differential of a Function of Several Variables (Complete Differential) 447
6.2.2.2 Derivatives and Differentials of Higher Order 448
6.2.2.3 Taylor’s Theorem for Functions of Several Variables 449
6.2.3 Rules of Differentiation for Functions of Several Variables 450
6.2.3.1 Differentiation of Composite Functions 450
6.2.3.2 Differentiation of Implicit Functions 451
6.2.4 Substitution of Variables in Differential Expressions and Coordinate Transformations 452
6.2.4.1 Function of One Variable 452
6.2.4.2 Function of Two Variables 453
6.2.5 Extreme Values of Functions of Several Variables 454
6.2.5.1 Definition of a Relative Extreme Value 454
6.2.5.2 Geometric Representation 454
6.2.5.3 Determination of Extreme Values of Differentiable Functions of Two Variables 455
6.2.5.4 Determination of the Extreme Values of a Function of n Variables 455
6.2.5.5 Solution of Approximation Problems 456
6.2.5.6 Extreme Value Problem with Side Conditions 456

7 Infinite Series 457
7.1 Sequences of Numbers 457
7.1.1 Properties of Sequences of Numbers 457
7.1.1.1 Definition of Sequence of Numbers 457
7.1.1.2 Monotone Sequences of Numbers 457
7.1.1.3 Bounded Sequences of Numbers 457
7.1.2 Limits of Sequences of Numbers 458
7.2 Number Series 459
7.2.1 General Convergence Theorems 459
7.2.1.1 Convergence and Divergence of Infinite Series 459
7.2.1.2 General Theorems about the Convergence of Series 460
7.2.2 Convergence Criteria for Series with Positive Terms 460
7.2.2.1 Comparison Criterion 460
7.2.2.2 D’Alembert’s Ratio Test 461
7.2.2.3 Root Test of Cauchy 461
7.2.2.4 Integral Test of Cauchy 462
7.2.3 Absolute and Conditional Convergence 462
7.2.3.1 Definition 462
7.2.3.2 Properties of Absolutely Convergent Series 463
7.2.3.3 Alternating Series 463
7.2.4 Some Special Series 464
7.2.4.1 The Values of Some Important Number Series 464
7.2.4.2 Bernoulli and Euler Numbers 465
7.2.5 Estimation of the Remainder 466
7.2.5.1 Estimation with Majorant 466
7.2.5.2 Alternating Convergent Series 467
7.2.5.3 Special Series 467
7.3 Function Series 467
7.3.1 Definitions 467
7.3.2 Uniform Convergence 468
7.3.2.1 Definition, Weierstrass Theorem 468
7.3.2.2 Properties of Uniformly Convergent Series 468
7.3.3 Power series 469
7.3.3.1 Definition, Convergence 469
7.3.3.2 Calculations with Power Series 470
7.3.3.3 Taylor Series Expansion, Maclaurin Series 471
7.3.4 Approximation Formulas 472
7.3.5 Asymptotic Power Series 472
7.3.5.1 Asymptotic Behavior 472
7.3.5.2 Asymptotic Power Series 472
7.4 Fourier Series 474
7.4.1 Trigonometric Sum and Fourier Series 474
7.4.1.1 Basic Notions 474
7.4.1.2 Most Important Properties of the Fourier Series 475
7.4.2 Determination of Coeficients for Symmetric Functions 476
7.4.2.1 Different Kinds of Symmetries 476
7.4.2.2 Forms of the Expansion into a Fourier Series 477
7.4.3 Determination of the Fourier Coeficients with Numerical Methods 477
7.4.4 Fourier Series and Fourier Integrals 478
7.4.5 Remarks on the Table of Some Fourier Expansions 479

8 Integral Calculus 480
8.1 Indefinite Integrals 480
8.1.1 Primitive Function or Antiderivative 480
8.1.1.1 Indefinite Integrals 481
8.1.1.2 Integrals of Elementary Functions 481
8.1.2 Rules of Integration 482
8.1.3 Integration of Rational Functions 485
8.1.3.1 Integrals of Integer Rational Functions (Polynomials) 485
8.1.3.2 Integrals of Fractional Rational Functions 485
8.1.3.3 Four Cases of Partial Fraction Decomposition 485
8.1.4 Integration of Irrational Functions 488
8.1.4.1 Substitution to Reduce to Integration of Rational Functions 488
8.1.4.2 Integration of Binomial Integrands 489
8.1.4.3 Elliptic Integrals 490
8.1.5 Integration of Trigonometric Functions 491
8.1.5.1 Substitution 491
8.1.5.2 Simplified Methods 491
8.1.6 Integration of Further Transcendental Functions 492
8.1.6.1 Integrals with Exponential Functions 492
8.1.6.2 Integrals with Hyperbolic Functions 493
8.1.6.3 Application of Integration by Parts 493
8.1.6.4 Integrals of Transcendental Functions 493
8.2 Definite Integrals 493
8.2.1 Basic Notions, Rules and Theorems 493
8.2.1.1 Definition and Existence of the Definite Integral 493
8.2.1.2 Properties of Definite Integrals 494
8.2.1.3 Further Theorems about the Limits of Integration 496
8.2.1.4 Evaluation of the Definite Integral 498
8.2.2 Applications of Definite Integrals 500
8.2.2.1 General Principle for Applications of the Definite Integral 500
8.2.2.2 Applications in Geometry 501
8.2.2.3 Applications in Mechanics and Physics 504
8.2.3 Improper Integrals, Stieltjes and Lebesgue Integrals 506
8.2.3.1 Generalization of the Notion of the Integral 506
8.2.3.2 Integrals with Infinite Integration Limits 507
8.2.3.3 Integrals with Unbounded Integrand 509
8.2.4 Parametric Integrals 512
8.2.4.1 Definition of Parametric Integrals 512
8.2.4.2 Differentiation Under the Symbol of Integration 512
8.2.4.3 Integration Under the Symbol of Integration 512
8.2.5 Integration by Series Expansion, Special Non-Elementary Functions 513
8.3 Line Integrals 515
8.3.1 Line Integrals of the First Type 516
8.3.1.1 Definitions 516
8.3.1.2 Existence Theorem 516
8.3.1.3 Evaluation of the Line Integral of the First Type 516
8.3.1.4 Application of the Line Integral of the First Type 517
8.3.2 Line Integrals of the Second Type 517
8.3.2.1 Definitions 517
8.3.2.2 Existence Theorem 519
8.3.2.3 Calculation of the Line Integral of the Second Type 519
8.3.3 Line Integrals of General Type 519
8.3.3.1 Definition 519
8.3.3.2 Properties of the Line Integral of General Type 520
8.3.3.3 Integral Along a Closed Curve 521
8.3.4 Independence of the Line Integral of the Path of Integration 521
8.3.4.1 Two-Dimensional Case 521
8.3.4.2 Existence of a Primitive Function 521
8.3.4.3 Three-Dimensional Case 522
8.3.4.4 Determination of the Primitive Function 522
8.3.4.5 Zero-Valued Integral Along a Closed Curve 523
8.4 Multiple Integrals 523
8.4.1 Double Integrals 524
8.4.1.1 Notion of the Double Integral 524
8.4.1.2 Evaluation of the Double Integral 524
8.4.1.3 Applications of the Double Integral 527
8.4.2 Triple Integrals 527
8.4.2.1 Notion of the Triple Integral 527
8.4.2.2 Evaluation of the Triple Integral 529
8.4.2.3 Applications of the Triple Integral 531
8.5 Surface Integrals 532
8.5.1 Surface Integral of the First Type 532
8.5.1.1 Notion of the Surface Integral of the First Type 532
8.5.1.2 Evaluation of the Surface Integral of the First Type 533
8.5.1.3 Applications of the Surface Integral of the First Type 535
8.5.2 Surface Integral of the Second Type 535
8.5.2.1 Notion of the Surface Integral of the Second Type 535
8.5.2.2 Evaluation of Surface Integrals of the Second Type 537
8.5.3 Surface Integral in General Form 537
8.5.3.1 Notion of the Surface Integral in General Form 537
8.5.3.2 Properties of the Surface Integrals 538

9 Differential Equations 540
9.1 Ordinary Differential Equations 540
9.1.1 First-Order Differential Equations 540
9.1.1.1 Existence Theorems, Direction Field 540
9.1.1.2 Important Solution Methods 542
9.1.1.3 Implicit Differential Equations 545
9.1.1.4 Singular Integrals and Singular Points 546
9.1.1.5 Approximation Methods for Solution of First-Order Differential Equations 549
9.1.2 Differential Equations of Higher Order and Systems of Differential Equations 550
9.1.2.1 Basic Results 550
9.1.2.2 Lowering the Order 552
9.1.2.3 Linear n-th Order Differential Equations 553
9.1.2.4 Solution of Linear Differential Equations with Constant Coeficients 555
9.1.2.5 Systems of Linear Differential Equations with Constant Coeficients 558
9.1.2.6 Linear Second-Order Differential Equations 560
9.1.3 Boundary Value Problems 569
9.1.3.1 Problem Formulation 569
9.1.3.2 Fundamental Properties of Eigenfunctions and Eigenvalues 569
9.1.3.3 Expansion in Eigenfunctions 570
9.1.3.4 Singular Cases 570
9.2 Partial Differential Equations 571
9.2.1 First-Order Partial Differential Equations 571
9.2.1.1 Linear First-Order Partial Differential Equations 571
9.2.1.2 Non-Linear First-Order Partial Differential Equations 573
9.2.2 Linear Second-Order Partial Differential Equations 576
9.2.2.1 Classification and Properties of Second-Order Differential Equations with Two Independent Variables 576
9.2.2.2 Classification and Properties of Linear Second-Order Differential Equations with more than two Independent Variables 578
9.2.2.3 Integration Methods for Linear Second-Order Partial Differential Equations 579
9.2.3 Some further Partial Differential Equations From Natural Sciences and Engineering 589
9.2.3.1 Formulation of the Problem and the Boundary Conditions 589
9.2.3.2 Wave Equation 590
9.2.3.3 Heat Conduction and Diffusion Equation for Homogeneous Media 591
9.2.3.4 Potential Equation 592
9.2.4 Schroedinger’s Equation 592
9.2.4.1 Notion of the Schroedinger Equation 592
9.2.4.2 Time-Dependent Schroedinger Equation 593
9.2.4.3 Time-Independent Schroedinger Equation 594
9.2.4.4 Statistical Interpretation of the Wave Function 594
9.2.4.5 Force-Free Motion of a Particle in a Block 597
9.2.4.6 Particle Movement in a Symmetric Central Field (see 13.1.2.2, p. 702) 598
9.2.4.7 Linear Harmonic Oscillator 601
9.2.5 Non-Linear Partial Differential Equations: Solitons, Periodic Patterns, Chaos 603
9.2.5.1 Formulation of the Physical-Mathematical Problem 603
9.2.5.2 Korteweg de Vries Equation (KdV) 605
9.2.5.3 Non-Linear Schroedinger Equation (NLS) 606
9.2.5.4 Sine-Gordon Equation (SG) 607
9.2.5.5 Further Non-linear Evolution Equations with Soliton Solutions 608

10 Calculus of Variations 610
10.1 Defining the Problem 610
10.2 Historical Problems 611
10.2.1 Isoperimetric Problem 611
10.2.2 Brachistochrone Problem 611
10.3 Variational Problems of One Variable 611
10.3.1 Simple Variational Problems and Extremal Curves 611
10.3.2 Euler Differential Equation of the Variational Calculus 612
10.3.3 Variational Problems with Side Conditions 614
10.3.4 Variational Problems with Higher-Order Derivatives 614
10.3.5 Variational Problem with Several Unknown Functions 615
10.3.6 Variational Problems using Parametric Representation 615
10.4 Variational Problems with Functions of Several Variables 617
10.4.1 Simple Variational Problem 617
10.4.2 More General Variational Problems 618
10.5 Numerical Solution of Variational Problems 618
10.6 Supplementary Problems 619
10.6.1 First and Second Variation 619
10.6.2 Application in Physics 620

11 Linear Integral Equations 621
11.1 Introduction and Classification 621
11.2 Fredholm Integral Equations of the Second Kind 622
11.2.1 Integral Equations with Degenerate Kernel 622
11.2.2 Successive Approximation Method, Neumann Series 625
11.2.3 Fredholm Solution Method, Fredholm Theorems 627
11.2.3.1 Fredholm Solution Method 627
11.2.3.2 Fredholm Theorems 629
11.2.4 Numerical Methods for Fredholm Integral Equations of the Second Kind 630
11.2.4.1 Approximation of the Integral 630
11.2.4.2 Kernel Approximation 632
11.2.4.3 Collocation Method 634
11.3 Fredholm Integral Equations of the First Kind 635
11.3.1 Integral Equations with Degenerate Kernels 635
11.3.2 Analytic Basis 636
11.3.3 Reduction of an Integral Equation into a Linear System of Equations 638
11.3.4 Solution of the Homogeneous Integral Equation of the First Kind 639
11.3.5 Construction of Two Special Orthonormal Systems for a Given Kernel 640
11.3.6 Iteration Method 642
11.4 Volterra Integral Equations 643
11.4.1 Theoretical Foundations 643
11.4.2 Solution by Differentiation 644
11.4.3 Solution of the Volterra Integral Equation of the Second Kind by Neumann Series 645
11.4.4 Convolution Type Volterra Integral Equations 645
11.4.5 Numerical Methods for Volterra Integral Equations of the Second Kind 646
11.5 Singular Integral Equations 648
11.5.1 Abel Integral Equation 648
11.5.2 Singular Integral Equation with Cauchy Kernel 649
11.5.2.1 Formulation of the Problem 649
11.5.2.2 Existence of a Solution 650
11.5.2.3 Properties of Cauchy Type Integrals 650
11.5.2.4 The Hilbert Boundary Value Problem 651
11.5.2.5 Solution of the Hilbert Boundary Value Problem (in short: Hilbert Problem) 651
11.5.2.6 Solution of the Characteristic Integral Equation 652

12 Functional Analysis 654
12.1 Vector Spaces 654
12.1.1 Notion of a Vector Space 654
12.1.2 Linear and Afine Linear Subsets 655
12.1.3 Linearly Independent Elements 656
12.1.4 Convex Subsets and the Convex Hull 657
12.1.4.1 Convex Sets 657
12.1.4.2 Cones 657
12.1.5 Linear Operators and Functionals 658
12.1.5.1 Mappings 658
12.1.5.2 Homomorphism and Endomorphism 658
12.1.5.3 Isomorphic Vector Spaces 659
12.1.6 Complexification of Real Vector Spaces 659
12.1.7 Ordered Vector Spaces 659
12.1.7.1 Cone and Partial Ordering 659
12.1.7.2 Order Bounded Sets 660
12.1.7.3 Positive Operators 660
12.1.7.4 Vector Lattices 660
12.2 Metric Spaces 662
12.2.1 Notion of a Metric Space 662
12.2.1.1 Balls, Neighborhoods and Open Sets 663
12.2.1.2 Convergence of Sequences in Metric Spaces 664
12.2.1.3 Closed Sets and Closure 664
12.2.1.4 Dense Subsets and Separable Metric Spaces 665
12.2.2 Complete Metric Spaces 665
12.2.2.1 Cauchy Sequences 665
12.2.2.2 Complete Metric Spaces 666
12.2.2.3 Some Fundamental Theorems in Complete Metric Spaces 666
12.2.2.4 Some Applications of the Contraction Mapping Principle 666
12.2.2.5 Completion of a Metric Space 668
12.2.3 Continuous Operators 668
12.3 Normed Spaces 669
12.3.1 Notion of a Normed Space 669
12.3.1.1 Axioms of a Normed Space 669
12.3.1.2 Some Properties of Normed Spaces 670
12.3.2 Banach Spaces 670
12.3.2.1 Series in Normed Spaces 670
12.3.2.2 Examples of Banach Spaces 670
12.3.2.3 Sobolev Spaces 671
12.3.3 Ordered Normed Spaces 671
12.3.4 Normed Algebras 672
12.4 Hilbert Spaces 673
12.4.1 Notion of a Hilbert Space 673
12.4.1.1 Scalar Product 673
12.4.1.2 Unitary Spaces and Some of their Properties 673
12.4.1.3 Hilbert Space 673
12.4.2 Orthogonality 674
12.4.2.1 Properties of Orthogonality 674
12.4.2.2 Orthogonal Systems 674
12.4.3 Fourier Series in Hilbert Spaces 675
12.4.3.1 Best Approximation 675
12.4.3.2 Parseval Equation, Riesz-Fischer Theorem 676
12.4.4 Existence of a Basis, Isomorphic Hilbert Spaces 676
12.5 Continuous Linear Operators and Functionals 677
12.5.1 Boundedness, Norm and Continuity of Linear Operators 677
12.5.1.1 Boundedness and the Norm of Linear Operators 677
12.5.1.2 The Space of Linear Continuous Operators 677
12.5.1.3 Convergence of Operator Sequences 678
12.5.2 Linear Continuous Operators in Banach Spaces 678
12.5.3 Elements of the Spectral Theory of Linear Operators 680
12.5.3.1 Resolvent Set and the Resolvent of an Operator 680
12.5.3.2 Spectrum of an Operator 680
12.5.4 Continuous Linear Functionals 681
12.5.4.1 Definition 681
12.5.4.2 Continuous Linear Functionals in Hilbert Spaces, Riesz Representation Theorem 682
12.5.4.3 Continuous Linear Functionals in L p 682
12.5.5 Extension of a Linear Functional 682
12.5.6 Separation of Convex Sets 683
12.5.7 Second Adjoint Space and Reflexive Spaces 684
12.6 Adjoint Operators in Normed Spaces 684
12.6.1 Adjoint of a Bounded Operator 684
12.6.2 Adjoint Operator of an Unbounded Operator 685
12.6.3 Self-Adjoint Operators 685
12.6.3.1 Positive Definite Operators 686
12.6.3.2 Projectors in a Hilbert Space 686
12.7 Compact Sets and Compact Operators 686
12.7.1 Compact Subsets of a Normed Space 686
12.7.2 Compact Operators 686
12.7.2.1 Definition of Compact Operator 686
12.7.2.2 Properties of Linear Compact Operators 687
12.7.2.3 Weak Convergence of Elements 687
12.7.3 Fredholm Alternative 687
12.7.4 Compact Operators in Hilbert Space 688
12.7.5 Compact Self-Adjoint Operators 688
12.8 Non-Linear Operators 689
12.8.1 Examples of Non-Linear Operators 689
12.8.2 Differentiability of Non-Linear Operators 690
12.8.3 Newton’s Method 690
12.8.4 Schauder’s Fixed-Point Theorem 691
12.8.5 Leray-Schauder Theory 692
12.8.6 Positive Non-Linear Operators 692
12.8.7 Monotone Operators in Banach Spaces 693
12.9 Measure and Lebesgue Integral 693
12.9.1 Set Algebras and Measures 693
12.9.2 Measurable Functions 695
12.9.2.1 Measurable Function 695
12.9.2.2 Properties of the Class of Measurable Functions 695
12.9.3 Integration 696
12.9.3.1 Definition of the Integral 696
12.9.3.2 Some Properties of the Integral 696
12.9.3.3 Convergence Theorems 697
12.9.4 Lp Spaces 697
12.9.5 Distributions 698
12.9.5.1 Formula of Partial Integration 698
12.9.5.2 Generalized Derivative 699
12.9.5.3 Distributions 699
12.9.5.4 Derivative of a Distribution 700

13 Vector Analysis and Vector Fields 701
13.1 Basic Notions of the Theory of Vector Fields 701
13.1.1 Vector Functions of a Scalar Variable 701
13.1.1.1 Definitions 701
13.1.1.2 Derivative of a Vector Function 701
13.1.1.3 Rules of Differentiation for Vectors 701
13.1.1.4 Taylor Expansion for Vector Functions 702
13.1.2 Scalar Fields 702
13.1.2.1 Scalar Field or Scalar Point Function 702
13.1.2.2 Important Special Cases of Scalar Fields 702
13.1.2.3 Coordinate Representation of Scalar Fields 703
13.1.2.4 Level Surfaces and Level Lines of a Field 703
13.1.3 Vector Fields 704
13.1.3.1 Vector Field or Vector Point Function 704
13.1.3.2 Important Cases of Vector Fields 705
13.1.3.3 Coordinate Representation of Vector Fields 706
13.1.3.4 Transformation of Coordinate Systems 706
13.1.3.5 Vector Lines 708
13.2 Differential Operators of Space 708
13.2.1 Directional and Space Derivatives 708
13.2.1.1 Directional Derivative of a Scalar Field 708
13.2.1.2 Directional Derivative of a Vector Field 708
13.2.1.3 Volume Derivative 709
13.2.2 Gradient of a Scalar Field 710
13.2.2.1 Definition of the Gradient 710
13.2.2.2 Gradient and Directional Derivative 710
13.2.2.3 Gradient and Volume Derivative 710
13.2.2.4 Further Properties of the Gradient 710
13.2.2.5 Gradient of the Scalar Field in Different Coordinates 710
13.2.2.6 Rules of Calculations 711
13.2.3 Vector Gradient 711
13.2.4 Divergence of Vector Fields 712
13.2.4.1 Definition of Divergence 712
13.2.4.2 Divergence in Different Coordinates 712
13.2.4.3 Rules for Evaluation of the Divergence 713
13.2.4.4 Divergence of a Central Field 713
13.2.5 Rotation of Vector Fields 713
13.2.5.1 Definitions of the Rotation 713
13.2.5.2 Rotation in Different Coordinates 714
13.2.5.3 Rules for Evaluating the Rotation 715
13.2.5.4 Rotation of a Potential Field 715
13.2.6 Nabla Operator, Laplace Operator 715
13.2.6.1 Nabla Operator 715
13.2.6.2 Rules for Calculations with the Nabla Operator 716
13.2.6.3 Vector Gradient 716
13.2.6.4 Nabla Operator Applied Twice 716
13.2.6.5 Laplace Operator 716
13.2.7 Review of Spatial Differential Operations 717
13.2.7.1 Rules of Calculation for Spatial Differential Operators 717
13.2.7.2 Expressions of Vector Analysis in Cartesian, Cylindrical, and Spherical Coordinates 718
13.2.7.3 Fundamental Relations and Results (see Table 13.3) 719
13.3 Integration in Vector Fields 719
13.3.1 Line Integral and Potential in Vector Fields 719
13.3.1.1 Line Integral in Vector Fields 719
13.3.1.2 Interpretation of the Line Integral in Mechanics 720
13.3.1.3 Properties of the Line Integral 720
13.3.1.4 Line Integral in Cartesian Coordinates 721
13.3.1.5 Integral Along a Closed Curve in a Vector Field 721
13.3.1.6 Conservative Field or Potential Field 721
13.3.2 Surface Integrals 722
13.3.2.1 Vector of a Plane Sheet 722
13.3.2.2 Evaluation of the Surface Integral 722
13.3.2.3 Surface Integrals and Flow of Fields 723
13.3.2.4 Surface Integrals in Cartesian Coordinates as Surface Integral of Second Type 723
13.3.3 Integral Theorems 724
13.3.3.1 Integral Theorem and Integral Formula of Gauss 724
13.3.3.2 Integral Theorem of Stokes 725
13.3.3.3 Integral Theorems of Green 725
13.4 Evaluation of Fields 726
13.4.1 Pure Source Fields 726
13.4.2 Pure Rotation Field or Zero-Divergence Field 727
13.4.3 Vector Fields with Point-Like Sources 727
13.4.3.1 Coulomb Field of a Point-Like Charge 727
13.4.3.2 Gravitational Field of a Point Mass 728
13.4.4 Superposition of Fields 728
13.4.4.1 Discrete Source Distribution 728
13.4.4.2 Continuous Source Distribution 728
13.4.4.3 Conclusion 729
13.5 Differential Equations of Vector Field Theory 729
13.5.1 Laplace Differential Equation 729
13.5.2 Poisson Differential Equation 729

14 Function Theory 731
14.1 Functions of Complex Variables 731
14.1.1 Continuity, Differentiability 731
14.1.1.1 Definition of a Complex Function 731
14.1.1.2 Limit of a Complex Function 731
14.1.1.3 Continuous Complex Functions 731
14.1.1.4 Differentiability of a Complex Function 731
14.1.2 Analytic Functions 732
14.1.2.1 Definition of Analytic Functions 732
14.1.2.2 Examples of Analytic Functions 732
14.1.2.3 Properties of Analytic Functions 732
14.1.2.4 Singular Points 733
14.1.3 Conformal Mapping 734
14.1.3.1 Notion and Properties of Conformal Mappings 734
14.1.3.2 Simplest Conformal Mappings 735
14.1.3.3 Schwarz Reflection Principle 741
14.1.3.4 Complex Potential 741
14.1.3.5 Superposition Principle 744
14.1.3.6 Arbitrary Mappings of the Complex Plane 745
14.2 Integration in the Complex Plane 745
14.2.1 Definite and Indefinite Integral 745
14.2.1.1 Definition of the Integral in the Complex Plane 745
14.2.1.2 Properties and Evaluation of Complex Integrals 746
14.2.2 Cauchy Integral Theorem 747
14.2.2.1 Cauchy Integral Theorem for Simply Connected Domains 747
14.2.2.2 Cauchy Integral Theorem for Multiply Connected Domains 748
14.2.3 Cauchy Integral Formulas 748
14.2.3.1 Analytic Function on the Interior of a Domain 748
14.2.3.2 Analytic Function on the Exterior of a Domain 749
14.3 Power Series Expansion of Analytic Functions 749
14.3.1 Convergence of Series with Complex Terms 749
14.3.1.1 Convergence of a Number Sequence with Complex Terms 749
14.3.1.2 Convergence of an Infinite Series with Complex Terms 749
14.3.1.3 Power Series with Complex Terms 750
14.3.2 Taylor Series 751
14.3.3 Principle of Analytic Continuation 751
14.3.4 Laurent Expansion 752
14.3.5 Isolated Singular Points and the Residue Theorem 752
14.3.5.1 Isolated Singular Points 752
14.3.5.2 Meromorphic Functions 753
14.3.5.3 Elliptic Functions 753
14.3.5.4 Residue 753
14.3.5.5 Residue Theorem 754
14.4 Evaluation of Real Integrals by Complex Integrals 754
14.4.1 Application of Cauchy Integral Formulas 754
14.4.2 Application of the Residue Theorem 755
14.4.3 Application of the Jordan Lemma 755
14.4.3.1 Jordan Lemma 755
14.4.3.2 Examples of the Jordan Lemma 756
14.5 Algebraic and Elementary Transcendental Functions 758
14.5.1 Algebraic Functions 758
14.5.2 Elementary Transcendental Functions 758
14.5.3 Description of Curves in Complex Form 760
14.6 Elliptic Functions 762
14.6.1 Relation to Elliptic Integrals 762
14.6.2 Jacobian Functions 763
14.6.3 Theta Functions 764
14.6.4 Weierstrass Functions 765

15 Integral Transformations 767
15.1 Notion of Integral Transformation 767
15.1.1 General Definition of Integral Transformations 767
15.1.2 Special Integral Transformations 767
15.1.3 Inverse Transformations 767
15.1.4 Linearity of Integral Transformations 767
15.1.5 Integral transformations for functions of several variables 769
15.1.6 Applications of Integral Transformations 769
15.2 Laplace Transformation 770
15.2.1 Properties of the Laplace Transformation 770
15.2.1.1 Laplace Transformation, Original and Image Space 770
15.2.1.2 Rules for the Evaluation of the Laplace Transformation 771
15.2.1.3 Transforms of Special Functions 774
15.2.1.4 Dirac δ-Function and Distributions 777
15.2.2 Inverse Transformation into the Original Space 778
15.2.2.1 Inverse Transformation with the Help of Tables 778
15.2.2.2 Partial Fraction Decomposition 778
15.2.2.3 Series Expansion 779
15.2.2.4 Inverse Integral 780
15.2.3 Solution of Differential Equations using Laplace Transformation 781
15.2.3.1 Ordinary Linear Differential Equations with Constant Coeficients 781
15.2.3.2 Ordinary Linear Differential Equations with Coeficients Depending on the Variable 782
15.2.3.3 Partial Differential Equations 783
15.3 Fourier Transformation 784
15.3.1 Properties of the Fourier Transformation 784
15.3.1.1 Fourier Integral 784
15.3.1.2 Fourier Transformation and Inverse Transformation 785
15.3.1.3 Rules of Calculation with the Fourier Transformation 787
15.3.1.4 Transforms of Special Functions 790
15.3.2 Solution of Differential Equations using the Fourier Transformation 791
15.3.2.1 Ordinary Linear Differential Equations 792
15.3.2.2 Partial Differential Equations 792
15.4 Z-Transformation 794
15.4.1 Properties of the Z-Transformation 794
15.4.1.1 Discrete Functions 794
15.4.1.2 Definition of the Z-Transformation 794
15.4.1.3 Rules of Calculations 795
15.4.1.4 Relation to the Laplace Transformation 796
15.4.1.5 Inverse of the Z-Transformation 797
15.4.2 Applications of the Z-Transformation 798
15.4.2.1 General Solution of Linear Difference Equations 798
15.4.2.2 Second-Order Difference Equations (Initial Value Problem) 799
15.4.2.3 Second-Order Difference Equations (Boundary Value Problem) 800
15.5 Wavelet Transformation 800
15.5.1 Signals 800
15.5.2 Wavelets 801
15.5.3 Wavelet Transformation 801
15.5.4 Discrete Wavelet Transformation 803
15.5.4.1 Fast Wavelet Transformation 803
15.5.4.2 Discrete Haar Wavelet Transformation 803
15.5.5 Gabor Transformation 803
15.6 Walsh Functions 804
15.6.1 Step Functions 804
15.6.2 Walsh Systems 804

16 Probability Theory and Mathematical Statistics 805
16.1 Combinatorics 805
16.1.1 Permutations 805
16.1.2 Combinations 805
16.1.3 Arrangements 806
16.1.4 Collection of the Formulas of Combinatorics (see Table 16.1) 807
16.2 Probability Theory 807
16.2.1 Event, Frequency and Probability 807
16.2.1.1 Events 807
16.2.1.2 Frequencies and Probabilities 808
16.2.1.3 Conditional Probability, Bayes Theorem 810
16.2.2 Random Variables, Distribution Functions 811
16.2.2.1 Random Variable 811
16.2.2.2 Distribution Function 811
16.2.2.3 Expected Value and Variance, Chebyshev Inequality 813
16.2.2.4 Multidimensional Random Variable 814
16.2.3 Discrete Distributions 814
16.2.3.1 Binomial Distribution 815
16.2.3.2 Hypergeometric Distribution 816
16.2.3.3 Poisson Distribution 817
16.2.4 Continuous Distributions 818
16.2.4.1 Normal Distribution 818
16.2.4.2 Standard Normal Distribution, Gaussian Error Function 819
16.2.4.3 Logarithmic Normal Distribution 819
16.2.4.4 Exponential Distribution 820
16.2.4.5 Weibull Distribution 821
16.2.4.6 χ 2 (Chi-Square) Distribution 822
16.2.4.7 Fisher F Distribution 823
16.2.4.8 Student t Distribution 824
16.2.5 Law of Large Numbers, Limit Theorems 825
16.2.6 Stochastic Processes and Stochastic Chains 825
16.2.6.1 Basic Notions, Markov Chains 826
16.2.6.2 Poisson Process 828
16.3 Mathematical Statistics 830
16.3.1 Statistic Function or Sample Function 830
16.3.1.1 Population, Sample, Random Vector 830
16.3.1.2 Statistic Function or Sample Function 831
16.3.2 Descriptive Statistics 832
16.3.2.1 Statistical Summarization and Analysis of Given Data 832
16.3.2.2 Statistical Parameters 833
16.3.3 Important Tests 834
16.3.3.1 Goodness of Fit Test for a Normal Distribution 834
16.3.3.2 Distribution of the Sample Mean 836
16.3.3.3 Confidence Limits for the Mean 837
16.3.3.4 Confidence Interval for the Variance 838
16.3.3.5 Structure of Hypothesis Testing 839
16.3.4 Correlation and Regression 839
16.3.4.1 Linear Correlation of two Measurable Characters 839
16.3.4.2 Linear Regression for two Measurable Characters 841
16.3.4.3 Multidimensional Regression 842
16.3.5 Monte Carlo Methods 843
16.3.5.1 Simulation 843
16.3.5.2 Random Numbers 843
16.3.5.3 Example of a Monte Carlo Simulation 845
16.3.5.4 Application of the Monte Carlo Method in Numerical Mathematics 845
16.3.5.5 Further Applications of the Monte Carlo Method 847
16.4 Calculus of Errors 848
16.4.1 Measurement Error and its Distribution 848
16.4.1.1 Qualitative Characterization of Measurement Errors 848
16.4.1.2 Density Function of the Measurement Error 848
16.4.1.3 Quantitative Characterization of the Measurement Error 850
16.4.1.4 Determining the Result of a Measurement with Bounds on the Error 853
16.4.1.5 Error Estimation for Direct Measurements with the Same Accuracy 853
16.4.1.6 Error Estimation for Direct Measurements with Different Accuracy 854
16.4.2 Error Propagation and Error Analysis 854
16.4.2.1 Gauss Error Propagation Law 855
16.4.2.2 Error Analysis 856

17 Dynamical Systems and Chaos 857
17.1 Ordinary Differential Equations and Mappings 857
17.1.1 Dynamical Systems 857
17.1.1.1 Basic Notions 857
17.1.1.2 Invariant Sets 859
17.1.2 Qualitative Theory of Ordinary Differential Equations 860
17.1.2.1 Existence of Flows, Phase Space Structure 860
17.1.2.2 Linear Differential Equations 861
17.1.2.3 Stability Theory 863
17.1.2.4 Invariant Manifolds 866
17.1.2.5 Poincar´e Mapping 868
17.1.2.6 Topological Equivalence of Differential Equations 870
17.1.3 Discrete Dynamical Systems 871
17.1.3.1 Steady States, Periodic Orbits and Limit Sets 871
17.1.3.2 Invariant Manifolds 872
17.1.3.3 Topological Conjugation of Discrete Systems 873
17.1.4 Structural Stability (Robustness) 873
17.1.4.1 Structurally Stable Differential Equations 873
17.1.4.2 Structurally Stable Time Discrete Systems 874
17.1.4.3 Generic Properties 874
17.2 Quantitative Description of Attractors 876
17.2.1 Probability Measures on Attractors 876
17.2.1.1 Invariant Measure 876
17.2.1.2 Elements of Ergodic Theory 877
17.2.2 Entropies 879
17.2.2.1 Topological Entropy 879
17.2.2.2 Metric Entropy 879
17.2.3 Lyapunov Exponents 880
17.2.4 Dimensions 882
17.2.4.1 Metric Dimensions 882
17.2.4.2 Dimensions Defined by Invariant Measures 884
17.2.4.3 Local Hausdorff Dimension According to Douady and Oesterl´e 886
17.2.4.4 Examples of Attractors 887
17.2.5 Strange Attractors and Chaos 888
17.2.6 Chaos in One-Dimensional Mappings 889
17.2.7 Reconstruction of Dynamics from Time Series 889
17.2.7.1 Foundations, Reconstruction with Basic Properties 889
17.2.7.2 Reconstructions with Prevalent Properties 891
17.3 Bifurcation Theory and Routes to Chaos 892
17.3.1 Bifurcations in Morse-Smale Systems 892
17.3.1.1 Local Bifurcations in Neighborhoods of Steady States 892
17.3.1.2 Local Bifurcations in a Neighborhood of a Periodic Orbit 897
17.3.1.3 Global Bifurcation 901
17.3.2 Transitions to Chaos 901
17.3.2.1 Cascade of Period Doublings 901
17.3.2.2 Intermittency 902
17.3.2.3 Global Homoclinic Bifurcations 902
17.3.2.4 Destruction of a Torus 904

18 Optimization 909
18.1 Linear Programming 909
18.1.1 Formulation of the Problem and Geometrical Representation 909
18.1.1.1 The Form of a Linear Programming Problem 909
18.1.1.2 Examples and Graphical Solutions 910
18.1.2 Basic Notions of Linear Programming, Normal Form 911
18.1.2.1 Extreme Points and Basis 911
18.1.2.2 Normal Form of the Linear Programming Problem 913
18.1.3 Simplex Method 914
18.1.3.1 Simplex Tableau 914
18.1.3.2 Transition to the New Simplex Tableau 915
18.1.3.3 Determination of an Initial Simplex Tableau 916
18.1.3.4 Revised Simplex Method 917
18.1.3.5 Duality in Linear Programming 919
18.1.4 Special Linear Programming Problems 920
18.1.4.1 Transportation Problem 920
18.1.4.2 Assignment Problem 923
18.1.4.3 Distribution Problem 923
18.1.4.4 Travelling Salesman 923
18.1.4.5 Scheduling Problem 924
18.2 Non-linear Optimization 924
18.2.1 Formulation of the Problem, Theoretical Basis 924
18.2.1.1 Formulation of the Problem 924
18.2.1.2 Optimality Conditions 924
18.2.1.3 Duality in Optimization 926
18.2.2 Special Non-linear Optimization Problems 926
18.2.2.1 Convex Optimization 926
18.2.2.2 Quadratic Optimization 926
18.2.3 Solution Methods for Quadratic Optimization Problems 928
18.2.3.1 Wolfe’s Method 928
18.2.3.2 Hildreth-d’Esopo Method 929
18.2.4 Numerical Search Procedures 930
18.2.4.1 One-Dimensional Search 930
18.2.4.2 Minimum Search in n-Dimensional Euclidean Vector Space 930
18.2.5 Methods for Unconstrained Problems 931
18.2.5.1 Method of Steepest Descent 931
18.2.5.2 Application of the Newton Method 931
18.2.5.3 Conjugate Gradient Methods 932
18.2.5.4 Method of Davidon, Fletcher and Powell (DFP) 932
18.2.6 Evolution Strategies 933
18.2.6.1 Evolution Principles 933
18.2.6.2 Evolution Algorithms 933
18.2.6.3 Classification of Evolution Strategies 934
18.2.6.4 Generating Random Numbers 934
18.2.6.5 Application of Evolution Strategies 934
18.2.6.6 (1 + 1)-Mutation-Selection Strategy 934
18.2.6.7 Population Strategies 935
18.2.7 Gradient Methods for Problems with Inequality Type Constraints 936
18.2.7.1 Method of Feasible Directions 937
18.2.7.2 Gradient Projection Method 938
18.2.8 Penalty Function and Barrier Methods 940
18.2.8.1 Penalty Function Method 940
18.2.8.2 Barrier Method 941
18.2.9 Cutting Plane Methods 942
18.3 Discrete Dynamic Programming 943
18.3.1 Discrete Dynamic Decision Models 943
18.3.1.1 n-Stage Decision Processes 943
18.3.1.2 Dynamic Programming Problem 943
18.3.2 Examples of Discrete Decision Models 944
18.3.2.1 Purchasing Problem 944
18.3.2.2 Knapsack Problem 944
18.3.3 Bellman Functional Equations 944
18.3.3.1 Properties of the Cost Function 944
18.3.3.2 Formulation of the Functional Equations 945
18.3.4 Bellman Optimality Principle 945
18.3.5 Bellman Functional Equation Method 946
18.3.5.1 Determination of Minimal Costs 946
18.3.5.2 Determination of the Optimal Policy 946
18.3.6 Examples for Applications of the Functional Equation Method 946
18.3.6.1 Optimal Purchasing Policy 946
18.3.6.2 Knapsack Problem 947

19 Numerical Analysis 949
19.1 Numerical Solution of Non-Linear Equations in a Single Unknown 949
19.1.1 Iteration Method 949
19.1.1.1 Ordinary Iteration Method 949
19.1.1.2 Newton’s Method 950
19.1.1.3 Regula Falsi 951
19.1.2 Solution of Polynomial Equations 952
19.1.2.1 Horner’s Scheme 952
19.1.2.2 Positions of the Roots 953
19.1.2.3 Numerical Methods 954
19.2 Numerical Solution of Systems of Equations 955
19.2.1 Systems of Linear Equations 955
19.2.1.1 Triangular Decomposition of a Matrix 955
19.2.1.2 Cholesky’s Method for a Symmetric Coeficient Matrix 958
19.2.1.3 Orthogonalization Method 958
19.2.1.4 Iteration Methods 960
19.2.2 System of Non-Linear Equations 961
19.2.2.1 Ordinary Iteration Method 961
19.2.2.2 Newton’s Method 962
19.2.2.3 Derivative-Free Gauss-Newton Method 962
19.3 Numerical Integration 963
19.3.1 General Quadrature Formulas 963
19.3.2 Interpolation Quadratures 964
19.3.2.1 Rectangular Formula 964
19.3.2.2 Trapezoidal Formula 964
19.3.2.3 Simpson’s Formula 965
19.3.2.4 Hermite’s Trapezoidal Formula 965
19.3.3 Quadrature Formulas of Gauss 965
19.3.3.1 Gauss Quadrature Formulas 965
19.3.3.2 Lobatto’s Quadrature Formulas 966
19.3.4 Method of Romberg 966
19.3.4.1 Algorithm of the Romberg Method 966
19.3.4.2 Extrapolation Principle 967
19.4 Approximate Integration of Ordinary Differential Equations 969
19.4.1 Initial Value Problems 969
19.4.1.1 Euler Polygonal Method 969
19.4.1.2 Runge-Kutta Methods 969
19.4.1.3 Multi-Step Methods 970
19.4.1.4 Predictor-Corrector Method 971
19.4.1.5 Convergence, Consistency, Stability 972
19.4.2 Boundary Value Problems 973
19.4.2.1 Difference Method 973
19.4.2.2 Approximation by Using Given Functions 974
19.4.2.3 Shooting Method 975
19.5 Approximate Integration of Partial Differential Equations 976
19.5.1 Difference Method 976
19.5.2 Approximation by Given Functions 977
19.5.3 Finite Element Method (FEM) 978
19.6 Approximation, Computation of Adjustment, Harmonic Analysis 982
19.6.1 Polynomial Interpolation 982
19.6.1.1 Newton’s Interpolation Formula 982
19.6.1.2 Lagrange’s Interpolation Formula 983
19.6.1.3 Aitken-Neville Interpolation 983
19.6.2 Approximation in Mean 984
19.6.2.1 Continuous Problems, Normal Equations 984
19.6.2.2 Discrete Problems, Normal Equations, Householder’s Method 985
19.6.2.3 Multidimensional Problems 986
19.6.2.4 Non-Linear Least Squares Problems 987
19.6.3 Chebyshev Approximation 988
19.6.3.1 Problem Definition and the Alternating Point Theorem 988
19.6.3.2 Properties of the Chebyshev Polynomials 989
19.6.3.3 Remes Algorithm 990
19.6.3.4 Discrete Chebyshev Approximation and Optimization 991
19.6.4 Harmonic Analysis 992
19.6.4.1 Formulas for Trigonometric Interpolation 992
19.6.4.2 Fast Fourier Transformation (FFT) 993
19.7 Representation of Curves and Surfaces with Splines 996
19.7.1 Cubic Splines 996
19.7.1.1 Interpolation Splines 996
19.7.1.2 Smoothing Splines 997
19.7.2 Bicubic Splines 998
19.7.2.1 Use of Bicubic Splines 998
19.7.2.2 Bicubic Interpolation Splines 998
19.7.2.3 Bicubic Smoothing Splines 1000
19.7.3 Bernstein–B´ezier Representation of Curves and Surfaces 1000
19.7.3.1 Principle of the B–B Curve Representation 1000
19.7.3.2 B–B Surface Representation 1001
19.8 Using the Computer 1001
19.8.1 Internal Symbol Representation 1001
19.8.1.1 Number Systems 1001
19.8.1.2 Internal Number Representation INR 1003
19.8.2 Numerical Problems in Calculations with Computers 1004
19.8.2.1 Introduction, Error Types 1004
19.8.2.2 Normalized Decimal Numbers and Round-Off 1005
19.8.2.3 Accuracy in Numerical Calculations 1006
19.8.3 Libraries of Numerical Methods 1009
19.8.3.1 NAG Library 1009
19.8.3.2 IMSL Library 1010
19.8.3.3 Aachen Library 1011
19.8.4 Application of Interactive Program Systems and Computeralgebra Systems 1011
19.8.4.1 Matlab 1011
19.8.4.2 Mathematica 1016
19.8.4.3 Maple 1019

20 Computer Algebra Systems-Example Mathematica 1023
20.1 Introduction 1023
20.1.1 Brief Characterization of Computer Algebra Systems 1023
20.1.1.1 General Purpose of Computer Algebra Systems 1023
20.1.1.2 Restriction to Mathematica 1023
20.1.1.3 Two Introducing Examples of Basic Application Fields 1023
20.2 Important Structure Elements of Mathematica 1024
20.2.1 Basic Structure Elements of Mathematica 1024
20.2.2 Types of Numbers in Mathematica 1025
20.2.2.1 Basic Types of Numbers 1025
20.2.2.2 Special Numbers 1026
20.2.2.3 Representation and Conversion of Numbers 1026
20.2.3 Important Operators 1026
20.2.4 Lists 1027
20.2.4.1 Notions 1027
20.2.4.2 Nested Lists 1028
20.2.4.3 Operations with Lists 1028
20.2.4.4 Tables 1029
20.2.5 Vectors and Matrices as Lists 1029
20.2.5.1 Creating Appropriate Lists 1029
20.2.5.2 Operations with Matrices and Vectors 1030
20.2.6 Functions 1031
20.2.6.1 Standard Functions 1031
20.2.6.2 Special Functions 1031
20.2.6.3 Pure Functions 1031
20.2.7 Patterns 1032
20.2.8 Functional Operations 1032
20.2.9 Programming 1034
20.2.10 Supplement about Syntax, Information, Messages 1035
20.2.10.1 Contexts, Attributes 1035
20.2.10.2 Information 1035
20.2.10.3 Messages 1035
20.3 Important Applications with Mathematica 1036
20.3.1 Manipulation of Algebraic Expressions 1036
20.3.1.1 Multiplication of Expressions 1036
20.3.1.2 Factorization of Polynomials 1037
20.3.1.3 Operations with Polynomials 1037
20.3.1.4 Partial Fraction Decomposition 1037
20.3.1.5 Manipulation of Non-Polynomial Expressions 1038
20.3.2 Solution of Equations and Systems of Equations 1038
20.3.2.1 Equations as Logical Expressions 1038
20.3.2.2 Solution of Polynomial Equations 1039
20.3.2.3 Solution of Transcendental Equations 1039
20.3.2.4 Solution of Systems of Equations 1040
20.3.3 Linear Systems of Equations and Eigenvalue Problems 1040
20.3.4 Differential and Integral Calculus 1042
20.3.4.1 Calculation of Derivatives 1042
20.3.4.2 Indefinite Integrals 1043
20.3.4.3 Definite Integrals and Multiple Integrals 1044
20.3.4.4 Solution of Differential Equations 1044
20.4 Graphics with Mathematica 1045
20.4.1 Basic Elements of Graphics 1045
20.4.2 Graphics Primitives 1046
20.4.3 Graphical Options 1047
20.4.4 Syntax of Graphical Representation 1047
20.4.4.1 Building Graphic Objects 1047
20.4.4.2 Graphical Representation of Functions 1047
20.4.5 Two-Dimensional Curves 1049
20.4.5.1 Exponential Functions 1049
20.4.5.2 Function y = x + Arcoth x 1049
20.4.5.3 Bessel Functions (see 9.1.2.6, 2., p. 562) 1050
20.4.6 Parametric Representation of Curves 1050
20.4.7 Representation of Surfaces and Space Curves 1051
20.4.7.1 Graphical Representation of Surfaces 1051
20.4.7.2 Options for 3D Graphics 1051
20.4.7.3 Three-Dimensional Objects in Parametric Representation 1051

21 Tables 1053
21.1 Frequently Used Mathematical Constants 1053
21.2 Important Natural Constants 1053
21.3 Metric Prefixes 1054
21.4 International System of Physical Units (SI Units) 1055
21.5 Important Series Expansions 1057
21.6 Fourier Series 1062
21.7 Indefinite Integrals 1065
21.7.1 Integral Rational Functions 1065
21.7.1.1 Integrals with X = ax + b 1065
21.7.1.2 Integrals with X = ax2 + bx + c 1067
21.7.1.3 Integrals with X = a2 ± x2 1068
21.7.1.4 Integrals with X = a3 ± x3 1070
21.7.1.5 Integrals with X = a4 + x4 1071
21.7.1.6 Integrals with X = a4 − x4 1071
21.7.1.7 Some Cases of Partial Fraction Decomposition 1071
21.7.2 Integrals of Irrational Functions 1072
21.7.2.1 Integrals with √x and a2 ± b2x 1072
21.7.2.2 Other Integrals with √x 1072
21.7.2.3 Integrals with √ax + b 1073
21.7.2.4 Integrals with √ax + b and √fx + g 1074
21.7.2.5 Integrals with √a2 − x2 1075
21.7.2.6 Integrals with √x2 + a2 1077
21.7.2.7 Integrals with √x2 − a2 1078
21.7.2.8 Integrals with √ax2 + bx + c 1080
21.7.2.9 Integrals with other Irrational Expressions 1082
21.7.2.10 Recursion Formulas for an Integral with Binomial Differential 1082
21.7.3 Integrals of Trigonometric Functions 1083
21.7.3.1 Integrals with Sine Function 1083
21.7.3.2 Integrals with Cosine Function 1085
21.7.3.3 Integrals with Sine and Cosine Function 1087
21.7.3.4 Integrals with Tangent Function 1091
21.7.3.5 Integrals with Cotangent Function 1091
21.7.4 Integrals of other Transcendental Functions 1092
21.7.4.1 Integrals with Hyperbolic Functions 1092
21.7.4.2 Integrals with Exponential Functions 1093
21.7.4.3 Integrals with Logarithmic Functions 1095
21.7.4.4 Integrals with Inverse Trigonometric Functions 1096
21.7.4.5 Integrals with Inverse Hyperbolic Functions 1097
21.8 Definite Integrals 1098
21.8.1 Definite Integrals of Trigonometric Functions 1098
21.8.2 Definite Integrals of Exponential Functions 1099
21.8.3 Definite Integrals of Logarithmic Functions 1100
21.8.4 Definite Integrals of Algebraic Functions 1101
21.9 Elliptic Integrals 1103
21.9.1 Elliptic Integral of the First Kind F(ϕ, k), k = sin α 1103
21.9.2 Elliptic Integral of the Second Kind E(ϕ, k), k = sin α 1103
21.9.3 Complete Elliptic Integral, k = sin α 1104
21.10 Gamma Function 1105
21.11 Bessel Functions (Cylindrical Functions) 1106
21.12 Legendre Polynomials of the First Kind 1108
21.13 Laplace Transformation 1109
21.14 Fourier Transformation 1114
21.14.1 Fourier Cosine Transformation 1114
21.14.2 Fourier Sine Transformation 1120
21.14.3 Fourier Transformation 1125
21.14.4 Exponential Fourier Transformation 1127
21.15 Z Transformation 1128
21.16 Poisson Distribution 1131
21.17 Standard Normal Distribution 1133
21.17.1 Standard Normal Distribution for 0.00 ≤ x ≤ 1.99 1133
21.17.2 Standard Normal Distribution for 2.00 ≤ x ≤ 3.90 1134
21.18 χ 2 Distribution 1135
21.19 Fisher F Distribution 1136
21.20 Student t Distribution 1138
21.21 Random Numbers 1139

22 Bibliography 1140

Index 1152
Mathematic Symbols 1208