Handbook of mathematics:
Gespeichert in:
| Format: | Buch |
|---|---|
| Sprache: | English German Russian |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2004
|
| Ausgabe: | 4. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XLII, 1153 S. graph. Darst. |
| ISBN: | 3540434917 |
Internformat
MARC
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| 130 | 0 | |a Spravočnik po matematike | |
| 245 | 1 | 0 | |a Handbook of mathematics |c I.N. Bronshtein ... |
| 250 | |a 4. ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2004 | |
| 300 | |a XLII, 1153 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Mathematics |v Handbooks, manuals, etc | |
| 650 | 0 | 7 | |a Mathematik |0 (DE-588)4037944-9 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Mathematik |0 (DE-588)4037944-9 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Bronštejn, Ilʹja N. |d 1903-1976 |e Sonstige |0 (DE-588)104989742 |4 oth | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-012823489 | ||
Datensatz im Suchindex
| _version_ | 1804132793248120832 |
|---|---|
| adam_text | Titel: Handbook of mathematics
Autor: Bronštejn, Ilʹja N
Jahr: 2004
Contents VII
Contents
List of Tables XXXIX
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
VIII Contents
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 Inequalitj ......................... 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
Contents IX
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 Functional.............................. 47
2.1.1.7 Functions and Mappings....................... 48
2.1.2 Methods for Defining 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
contents
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 Continuity of a Function ............................ 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 re-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.................... gg
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
2.71.2 Range and Behavior of the Functions ...............
77
2.7.2 Important Formulas for Trigonometric Functions............... 7g
Contents XI
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 arcsin x and arcsin!/.................. 86
2.8.6 Sum and Difference of arccos x and arccos y.................. 86
2.8.7 Sum and Difference of arctan x and arctan y.................. 86
2.8.8 Special Relations for arcsin x, arccos x, arctan x................ 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 Com-
plex 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
XII Contents
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 Pascals Limaçon ................................ 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.................... HO
2.16.2.8 Curve of Order Three. Type III ................... HI
2.16.2.9 Curve of Order Three, Type I .................... HI
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................................... H^
2.17.2.1 Somilogarithmic Paper........................ Hg
2.17.2.2 Double Logarithmic Paper...................... 14g
2.17.2.3 Graph Paper with a Reciprocal Scale ................ !jg
Contents XIII
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 Geometrie 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 Mini-
mum) ................................. 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. Rav, 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
XIV Contents
I
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 j
3.1.5.1 General Polygon ........................... 137 j
3.1.5.2 Regular Convex Polygons....................... 137 s
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...................... 147
3.3 Stereometry....................................... 1501
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......................... 18fj
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
Contents XV
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
XVI Contents
4.2.1 Definitions.................................... 259
4.2.2 Subdeterminants ................................ 259
4.2.3 Rules of Calculation for Determinants..................... 260
4.2.4 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
4.4.24 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 Prepositional 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
Contents XVII
5.2.5 Cardinality of Sets................................ 298
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 Sju...... 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
5.3.7 Vector Spaces.................................. 314
5.3.7.1 Definition............................... 314
5.3.7.2 Linear Dependence.......................... 315
5.3.7.3 Linear Mappings ........................... 315
5.3.7.4 Subspaces, Dimension Formula.................... 315
5.3.7.5 Euclidean Vector Spaces, Euclidean Norm.............. 316
5.3.7.6 Linear Operators in Vector Spaces.................. 316
Elementary Number Theory .............................. 318
5.4.1 Divisibility.................................... 318
5.4.1.1 Divisibility and Elementary Divisibility Rules............ 318
5.4.1.2 Prime Numbers............................ 318
5.4.1.3 Criteria for Divisibility........................ 320
5.4.1.4 Greatest Common Divisor and Least Common Multiple...... 321
5.4.1.5 Fibonacci Numbers.......................... 323
5.4.2 Linear Diophantine Equations......................... 323
5.4.3 Congruences and Residue Classes ....................... 325
5.4.4 Theorems of Fermât, Euler, and Wilson.................... 329
5.4.5 Codes...................................... 329
Cryptology........................................ 332
5.5.1 Problem of Cryptology............................. 332
5.5.2 Cryptosystems.................................. 332
5.5.3 Mathematical Foundation ........................... 332
5.5.4 Security of Cryptosystems ........................... 333
5.5.4.1 Methods of Conventional Cryptography............... 333
XVIII Contents
5.5.4.2 Linear Substitution Ciphers..................... 334
5.5.4.3 Vigenère Cipher............................ 334
5.5.4.4 Matrix Substitution.......................... 334
5.5.5 Methods of Classical Cryptanalysis ...................... 335
5.5.5.1 Statistical Analysis.......................... 335
5.5.5.2 Kasiski-Friedman Test........................ 335
5.5.6 One-Time Pad.................................. 336
5.5.7 Public Key Methods............................... 336
5.5.7.1 Diffie-Hellman Key Exchange.................... 336
5.5.7.2 One-Way Function.......................... 337
5.5.7.3 RSA Method ............................. 337
5.5.8 DES Algorithm (Data Encryption Standard) ................. 337
5.5.9 IDEA Algorithm (International Data Encryption Algorithm) ........ 338
5.6 Universal Algebra.................................... 338
5.6.1 Definition.................................... 338
5.6.2 Congruence Relations, Factor Algebras.................... 338
5.6.3 Homomorphism................................. 339
5.6.4 Homomorphism Theorem............................ 339
5.6.5 Varieties..................................... 339
5.6.6 Term Algebras, Free Algebras ......................... 339
5.7 Boolean Algebras and Switch Algebra......................... 340
5.7.1 Definition.................................... 340
5.7.2 Duality Principle ................................ 341
5.7.3 Finite Boolean Algebras ............................ 341
5.7.4 Boolean Algebras as Orderings......................... 341
5.7.5 Boolean Functions, Boolean Expressions.................... 342
5.7.6 Normal Forms.................................. 343
5.7.7 Switch Algebra ................................. 344
5.8 Algorithms of Graph Theory.............................. 346
5.8.1 Basic Notions and Notation........................... 346
5.8.2 Traverse of Undirected Graphs......................... 349
5.8.2.1 Edge Sequences or Paths....................... 349
5.8.2.2 Euler Trails.............................. 350
5.8.2.3 Hamiltonian Cycles.......................... 351
5.8.3 Trees and Spanning Trees............................ 352
5.8.3.1 Trees.................................. 352!
5.8.3.2 Spanning Trees............................ 353
5.8.4 Matchings.................................... 354
5.8.5 Planar Graphs.................................. 355
5.8.6 Paths in Directed Graphs............................ 355
5.8.7 Transport Networks............................... 356
5.9 Fuzzy Logic....................................... 358
5.9.1 Basic Notions of Fuzzy Logic.......................... 358
5.9.1.1 Interpretation of Fuzzy Sets..................... 358
5.9.1.2 Membership Functions on the Real Line............... 359
5.9.1.3 Fuzzy Sets............................... 3gj
5.9.2 Aggregation of Fuzzy Sets ........................... 353
5.9.2.1 Concepts for Aggregation of Fuzzy Sets............... 353
5.9.2.2 Practical Aggregator Operations of Fuzzy Sets........... 354
5.9.2.3 Compensatory Operators....................... 3gg
5.9.2.4 Extension Principle.......................... 3gg
5.9.2.5 Fuzzy Complement.......................... 3gg
Contents XIX
5.9.3 Fuzzy-Valued Relations............................. 367
5.9.3.1 Fuzzy Relations............................ 367
5.9.3.2 Fuzzy Product Relation R o S.................... 369
5.9.4 Fuzzy Inference (Approximate Reasoning)................... 370
5.9.5 Defuzzification Methods ............................ 371
5.9.6 Knowledge-Based Fuzzy Systems........................ 372
5.9.6.1 Method of Mamdani ......................... 372
5.9.6.2 Method of Sugeno........................... 373
5.9.6.3 Cognitive Systems .......................... 373
5.9.6.4 Knowledge-Based Interpolation Systems .............. 375
Differentiation 377
6.1 Differentiation of Functions of One Variable...................... 377
6.1.1 Differential Quotient .............................. 377
6.1.2 Rules of Differentiation for Functions of One Variable............. 378
6.1.2.1 Derivatives of the Elementary Functions............... 378
6.1.2.2 Basic Rules of Differentiation .................... 378
6.1.3 Derivatives of Higher Order........................... 383
6.1.3.1 Definition of Derivatives of Higher Order .............. 383
6.1.3.2 Derivatives of Higher Order of some Elementary Functions..... 383
6.1.3.3 Leibniz s Formula........................... 383
6.1.3.4 Higher Derivatives of Functions Given in Parametric Form..... 385
6.1.3.5 Derivatives of Higher Order of the Inverse Function......... 385
6.1.4 Fundamental Theorems of Differential Calculus................ 386
6.1.4.1 Monotonicity............................. 386
6.1.4.2 Fermat s Theorem .......................... 386
6.1.4.3 Rolle s Theorem............................ 386
6.1.4.4 Mean Value Theorem of Differential Calculus............ 387
6.1.4.5 Taylor s Theorem of Functions of One Variable........... 387
6.1.4.6 Generalized Mean Value Theorem of Differential Calculus (Cauchy s
Theorem)............................... 388
6.1.5 Determination of the Extreme Values and Inflection Points.......... 388
6.1.5.1 Maxima and Minima......................... 388
6.1.5.2 Necessary Conditions for the Existence of a Relative Extreme Value 388
6.1.5.3 Relative Extreme Values of a Differentiable, Explicit Function . . . 389
6.1.5.4 Determination of Absolute Extrema................. 390
6.1.5.5 Determination of the Extrema of Implicit Functions ........ 390
6.2 Differentiation of Functions of Several Variables.................... 390
6.2.1 Partial Derivatives ............................... 390
6.2.1.1 Partial Derivative of a Function................... 390
6.2.1.2 Geometrical Meaning for Functions of Two Variables........ 391
6.2.1.3 Differentials of x and f(x)...................... 391
6.2.1.4 Basic Properties of the Differential.................. 392
6.2.1.5 Partial Differential.......................... 392
6.2.2 Total Differential and Differentials of Higher Order.............. 392
6.2.2.1 Notion of Total Differential of a Function of Several Variables (Com-
plete Differential) ........................... 392
6.2.2.2 Derivatives and Differentials of Higher Order............ 393
6.2.2.3 Taylor s Theorem for Functions of Several Variables ........ 394
6.2.3 Rules of Differentiation for Functions of Several Variables.......... 395
6.2.3.1 Differentiation of Composite Functions............... 395
6.2.3.2 Differentiation of Implicit Functions................. 396
XX Contents
I
6.2.4 Substitution of Variables in Differential Expressions and Coordinate Transfor-
mations ..................................... 397
6.2.4.1 Function of One Variable....................... 397
6.2.4.2 Function of Two Variables...................... 398
6.2.5 Extreme Values of Functions of Several Variables............... 399
6.2.5.1 Definition............................... 399
6.2.5.2 Geometrie Representation...................... 399
6.2.5.3 Determination of Extreme Values of Functions of Two Variables . . 400
6.2.5.4 Determination of the Extreme Values of a Function of n Variables . 400
6.2.5.5 Solution of Approximation Problems ................ 401
6.2.5.6 Extreme Value Problem with Side Conditions............ 401
7 Infinite Series 402
7.1 Sequences of Numbers.................................. 402
7.1.1 Properties of Sequences of Numbers...................... 402
7.1.1.1 Definition of Sequence of Numbers.................. 402
7.1.1.2 Monotone Sequences of Numbers .................. 402
7.1.1.3 Bounded Sequences.......................... 402
7.1.2 Limits of Sequences of Numbers........................ 403
7.2 Number Series...................................... 404
7.2.1 General Convergence Theorems........................ 404
7.2.1.1 Convergence and Divergence of Infinite Series............ 404
7.2.1.2 General Theorems about the Convergence of Series......... 404
7.2.2 Convergence Criteria for Series with Positive Terms.............. 405
7.2.2.1 Comparison Criterion......................... 405
7.2.2.2 D Alembert s Ratio Test....................... 405
7.2.2.3 Root Test of Cauchy......................... 406
7.2.2.4 Integral Test of Cauchy........................ 406
7.2.3 Absolute and Conditional Convergence .................... 407
7.2.3.1 Definition............................... 407
7.2.3.2 Properties of Absolutely Convergent Series............. 407
7.2.3.3 Alternating Series........................... 408
7.2.4 Some Special Series............................... 408
7.2.4.1 The Values of Some Important Number Series ........... 408
7.2.4.2 Bernoulli and Euler Numbers .................... 410
7.2.5 Estimation of the Remainder.......................... 411
7.2.5.1 Estimation with Majorant...................... 411
7.2.5.2 Alternating Convergent Series.................... 412
7.2.5.3 Special Series............................. 412
7.3 Function Series ..................................... 412
7.3.1 Definitions.................................... 412
7.3.2 Uniform Convergence.............................. 412
7.3.2.1 Definition, Weierstrass Theorem................... 412
7.3.2.2 Properties of Uniformly Convergent Series ............. 413
7.3.3 Power series................................... 414
7.3.3.1 Definition, Convergence....................... 444:
7.3.3.2 Calculations with Power Series.................... 444
7.3.3.3 Taylor Series Expansion, Maclaurin Series.............. 445
7.3.4 Approximation Formulas............................ 44g
7.3.5 Asymptotic Power Series............................ 447
7.3.5.1 Asymptotic Behavior......................... 447
7.3.5.2 Asymptotic Power Series....................... 41§ ,
Contents XXI
7.4 Fourier Series...................................... 418
7.4.1 Trigonometrie Sum and Fourier Series..................... 418
7.4.1.1 Basic Notions............................. 418
7.4.1.2 Most Important Properties of the Fourier Series .......... 419
7.4.2 Determination of Coefficients for Symmetric Functions............ 420
7.4.2.1 Different Kinds of Symmetries.................... 420
7.4.2.2 Forms of the Expansion into a Fourier Series ............ 421
7.4.3 Determination of the Fourier Coefficients with Numerical Methods ..... 422
7.4.4 Fourier Series and Fourier Integrals ...................... 422
7.4.5 Remarks on the Table of Some Fourier Expansions.............. 423
Integral Calculus 425
8.1 Indefinite Integrals ................................... 425
8.1.1 Primitive Function or Antiderivative...................... 425
8.1.1.1 Indefinite Integrals .......................... 426
8.1.1.2 Integrals of Elementary Functions.................. 426
8.1.2 Rules of Integration............................... 427
8.1.3 Integration of Rational Functions ....................... 430
8.1.3.1 Integrals of Integer Rational Functions (Polynomials) ....... 430
8.1.3.2 Integrals of Fractional Rational Functions.............. 430
8.1.3.3 Four Cases of Partial Fraction Decomposition............ 430
8.1.4 Integration of Irrational Functions....................... 433
8.1.4.1 Substitution to Reduce to Integration of Rational Functions .... 433
8.1.4.2 Integration of Binomial Integrands.................. 434
8.1.4.3 Elliptic Integrals ........................... 435
8.1.5 Integration of Trigonometric Functions .................... 436
8.1.5.1 Substitution.............................. 436
8.1.5.2 Simplified Methods.......................... 436
8.1.6 Integration of Further Transcendental Functions............... 437
8.1.6.1 Integrals with Exponential Functions................ 437
8.1.6.2 Integrals with Hyperbolic Functions................. 438
8.1.6.3 Application of Integration by Parts................. 438
8.1.6.4 Integrals of Transcendental Functions................ 438
8.2 Definite Integrals.................................... 438
8.2.1 Basic Notions. Rules and Theorems...................... 438
8.2.1.1 Definition and Existence of the Definite Integral .......... 438
8.2.1.2 Properties of Definite Integrals.................... 439
8.2.1.3 Further Theorems about the Limits of Integration ......... 441
8.2.1.4 Evaluation of the Definite Integral.................. 443
8.2.2 Application of Definite Integrals........................ 445
8.2.2.1 General Principles for Application of the Definite Integral..... 445
8.2.2.2 Applications in Geometry ...................... 446
8.2.2.3 Applications in Mechanics and Physics ............... 449
8.2.3 Improper Integrals, Stieltjes and Lebesgue Integrals ............. 451
8.2.3.1 Generalization of the Notion of the Integral............. 451
8.2.3.2 Integrals with Infinite Integration Limits.............. 452
8.2.3.3 Integrals with Unbounded Integrand................. 454
8.2.4 Parametric Integrals .............................. 457
8.2.4.1 Definition of Parametric Integrals.................. 457
8.2.4.2 Differentiation Under the Symbol of Integration .......... 457
8.2.4.3 Integration Under the Symbol of Integration ............ 457
8.2.5 Integration by Series Expansion, Special Non-Elementary Functions..... 458
XXII Contents
I
8.3 Line Integrals...................................... 460
8.3.1 Line Integrals of the First Type......................... 461
8.3.1.1 Definitions............................... 461
8.3.1.2 Existence Theorem.......................... 461
8.3.1.3 Evaluation of the Line Integral of the First Type.......... 461
8.3.1.4 Application of the Line Integral of the First Type.......... 462
8.3.2 Line Integrals of the Second Type....................... 462
8.3.2.1 Definitions............................... 462
8.3.2.2 Existence Theorem.......................... 464
8.3.2.3 Calculation of the Line Integral of the Second Type......... 464
8.3.3 Line Integrals of General Type......................... 465
8.3.3.1 Definition............................... 465
8.3.3.2 Properties of the Line Integral of General Type........... 465
8.3.3.3 Integral Along a Closed Curve.................... 466
8.3.4 Independence of the Line Integral of the Path of Integration......... 466
8.3.4.1 Two-Dimensional Case........................ 466
8.3.4.2 Existence of a Primitive Function.................. 467
8.3.4.3 Three-Dimensional Case....................... 467
8.3.4.4 Determination of the Primitive Function .............. 467
8.3.4.5 Zero-Valued Integral Along a Closed Curve............. 468
8.4 Multiple Integrals.................................... 469
8.4.1 Double Integrals................................. 469
8.4.1.1 Notion of the Double Integral .................... 469
8.4.1.2 Evaluation of the Double Integral.................. 470
8.4.1.3 Applications of the Double Integral ................. 472
8.4.2 Triple Integrals ................................. 474
8.4.2.1 Notion of the Triple Integral..................... 474
8.4.2.2 Evaluation of the Triple Integral................... 474
8.4.2.3 Applications of the Triple Integral.................. 477
8.5 Surface Integrals..................................... 477
8.5.1 Surface Integral of the First Type....................... 477
8.5.1.1 Notion of the Surface Integral of the First Type........... 478
8.5.1.2 Evaluation of the Surface Integral of the First Type........ 479
8.5.1.3 Applications of the Surface Integral of the First Type....... 480
8.5.2 Surface Integral of the Second Type...................... 481
8.5.2.1 Notion of the Surface Integral of the Second Type......... 481
8.5.2.2 Evaluation of Surface Integrals of the Second Type......... 482
8.5.2.3 An Application of the Surface Integral................ 484
I Differential Equations 485
9.1 Ordinary Differential Equations ............................ 485
9.1.1 First-Order Differential Equations....................... 486
9.1.1.1 Existence Theorems, Direction Field................. 486
9.1.1.2 Important Solution Methods..................... 487
9.1.1.3 Implicit Differential Equations.................... 490
9.1.1.4 Singular Integrals and Singular Points................ 491
9.1.1.5 Approximation Methods for Solution of First-Order Differential Equa-
tions .................................. 4g4
9.1.2 Differential Equations of Higher Order and Systems of Differential Equations 495
9.1.2.1 Basic Results............................. 405
9.1.2.2 Lowering the Order.......................... 407
9.1.2.3 Linear n-th Order Differential Equations.............. 400
Contents XXIII
9.1.2.4 Solution of Linear Differential Equations with Constant Coefficients 500
9.1.2.5 Systems of Linear Differential Equations with Constant Coefficients 503
9.1.2.6 Linear Second-Order Differential Equations............. 505
9.1.3 Boundary Value Problems ........................... 512
9.1.3.1 Problem Formulation......................... 512
9.1.3.2 Fundamental Properties of Eigenfunctions and Eigenvalues .... 513
9.1.3.3 Expansion in Eigenfunctions..................... 514
9.1.3.4 Singular Cases............................. 514
9.2 Partial Differential Equations.............................. 515
9.2.1 First-Order Partial Differential Equations................... 515
9.2.1.1 Linear First-Order Partial Differential Equations.......... 515
9.2.1.2 Non-Linear First-Order Partial Differential Equations....... 517
9.2.2 Linear Second-Order Partial Differential Equations.............. 520
9.2.2.1 Classification and Properties of Second-Order Differential Equations
with Two Independent Variables................... 520
9.2.2.2 Classification and Properties of Linear Second-Order Differential Equa-
tions with more than two Independent Variables.......... 521
9.2.2.3 Integration Methods for Linear Second-Order Partial Differential Equa-
tions .................................. 522
9.2.3 Some further Partial Differential Equations From Natural Sciences and Engi-
neering ...................................... 532
9.2.3.1 Formulation of the Problem and the Boundary Conditions..... 532
9.2.3.2 Wave Equation............................ 534
9.2.3.3 Heat Conduction and Diffusion Equation for Homogeneous Media . 535
9.2.3.4 Potential Equation.......................... 536
9.2.3.5 Schrödinger s Equation........................ 536
9.2.4 Non-Linear Partial Differential Equations: Solitons, Periodic Patterns and Chaos 544
9.2.4.1 Formulation of the Physical-Mathematical Problem ........ 544
9.2.4.2 Korteweg de Vries Equation (KdV)................. 546
9.2.4.3 Non-Linear Schrödinger Equation (NLS) .............. 547
9.2.4.4 Sine-Gordon Equation (SG)..................... 547
9.2.4.5 Further Non-linear Evolution Equations with Soliton Solutions . . 549
10 Calculus of Variations 550
10.1 Defining the Problem.................................. 550
10.2 Historical Problems................................... 551
10.2.1 Isoperimetric Problem ............................. 551
10.2.2 Brachistochrone Problem............................ 551
10.3 Variational Problems of One Variable ......................... 551
10.3.1 Simple Variational Problems and Extremal Curves.............. 551
10.3.2 Euler Differential Equation of the Variational Calculus............ 552
10.3.3 Variational Problems with Side Conditions.................. 553
10.3.4 Variational Problems with Higher-Order Derivatives............. 554
10.3.5 Variational Problem with Several Unknown Functions ............ 555
10.3.6 Variational Problems using Parametric Representation............ 555
10.4 Variational Problems with Functions of Several Variables............... 556
10.4.1 Simple Variational Problem........................... 556
10.4.2 More General Variational Problems...................... 558
10.5 Numerical Solution of Variational Problems...................... 558
10.6 Supplementary Problems................................ 559
10.6.1 First and Second Variation........................... 559
10.6.2 Application in Physics ............................. 560
I
XXIV Contents
I
11 Linear Integral Equations 561
11.1 Introduction and Classification............................. 561
11.2 Fredholm Integral Equations of the Second Kind................... 562
11.2.1 Integral Equations with Degenerate Kernel.................. 562
11.2.2 Successive Approximation Method, Neumann Series............. 565
11.2.3 Fredholm Solution Method, Fredholm Theorems ............... 567
11.2.3.1 Fredholm Solution Method...................... 567
11.2.3.2 Fredholm Theorems.......................... 569
11.2.4 Numerical Methods for Fredholm Integral Equations of the Second Kind . . 570
11.2.4.1 Approximation of the Integral.................... 57Q.
11.2.4.2 Kernel Approximation........................ 572
11.2.4.3 Collocation Method.......................... 574,
11.3 Fredholm Integral Equations of the First Kind..................... 575.
11.3.1 Integral Equations with Degenerate Kernels.................. 575{
11.3.2 Analytic Basis.................................. 576Ì
11.3.3 Reduction of an Integral Equation into a Linear System of Equations .... 578:
11.3.4 Solution of the Homogeneous Integral Equation of the First Kind...... 579:
11.3.5 Construction of Two Special Orthonormal Systems for a Given Kernel . . . 580-
11.3.6 Iteration Method ................................ 582
11.4 Volterra Integral Equations............................... 583
11.4.1 Theoretical Foundations............................ 583
11.4.2 Solution by Differentiation........................... 584
11.4.3 Solution of the Volterra Integral Equation of the Second Kind by Neumann
Series....................................... 585
11.4.4 Convolution Type Volterra Integral Equations ................ 585
11.4.5 Numerical Methods for Volterra Integral Equations of the Second Kind . . . 586
11.5 Singular Integral Equations............................... 588,1
11.5.1 Abel Integral Equation............................. 588
11.5.2 Singular Integral Equation with Cauchy Kernel................ 589
11.5.2.1 Formulation of the Problem..................... 589
11.5.2.2 Existence of a Solution........................ 590
11.5.2.3 Properties of Cauchy Type Integrals................. 590
11.5.2.4 The Hilbert Boundary Value Problem................ 591
11.5.2.5 Solution of the Hilbert Boundary Value Problem (in short: Hilbert
Problem) ............................... 591
11.5.2.6 Solution of the Characteristic Integral Equation.......... 592
12 Functional Analysis 594
12.1 Vector Spaces...................................... 594,
12.1.1 Notion of a Vector Space............................ 594;*
12.1.2 Linear and Affine Linear Subsets ........................ 595;
12.1.3 Linearly Independent Elements......................... 596!
12.1.4 Convex Subsets and the Convex Hull...................... 597
12.1.4.1 Convex Sets.............................. 597;
12.1.4.2 Cones................................. 597:
12.1.5 Linear Operators and Functionals....................... 598Î
12.1.5.1 Mappings............................... 598)
12.1.5.2 Homomorphism and Endomorphism................. 598j
12.1.5.3 Isomorphic Vector Spaces ...................... 599.
12.1.6 Complexification of Real Vector Spaces.................... 599
12.1.7 Ordered Vector Spaces............................. 599.
12.1.7.1 Cone and Partial Ordering...................... 599:
Contents XXV
12.1.7.2 Order Bounded Sets ......................... 600
12.1.7.3 Positive Operators .......................... 600
12.1.7.4 Vector Lattices ............................ 600
12.2 Metrie Spaces...................................... 602
12.2.1 Notion of a Metrie Space............................ 602
12.2.1.1 Balls, Neighborhoods and Open Sets................. 603
12.2.1.2 Convergence of Sequences in Metrie Spaces............. 604
12.2.1.3 Closed Sets and Closure ....................... 604
12.2.1.4 Dense Subsets and Separable Metrie Spaces............. 605
12.2.2 Complete Metrie Spaces ............................ 605
12.2.2.1 Cauchy Sequences........................... 605
12.2.2.2 Complete Metrie Spaces....................... 606
12.2.2.3 Some Fundamental Theorems in Complete Metrie Spaces..... 606
12.2.2.4 Some Applications of the Contraction Mapping Principle ..... 606
12.2.2.5 Completion of a Metric Space.................... 608
12.2.3 Continuous Operators.............................. 608
12.3 Normed Spaces..................................... 609
12.3.1 Notion of a Normed Space ........................... 609
12.3.1.1 Axioms of a Normed Space...................... 609
12.3.1.2 Some Properties of Normed Spaces ................. 610
12.3.2 Banach Spaces.................................. 610
12.3.2.1 Series in Normed Spaces....................... 610
12.3.2.2 Examples of Banach Spaces..................... 610
12.3.2.3 Sobolev Spaces ............................ 611
12.3.3 Ordered Normed Spaces ............................ 611
12.3.4 Normed Algebras................................ 612
12.4 Hilbert Spaces...................................... 613
12.4.1 Notion of a Hilbert Space............................ 613
12.4.1.1 Scalar Product ............................ 613
12.4.1.2 Unitary Spaces and Some of their Properties ............ 613
12.4.1.3 Hilbert Space............................. 613
12.4.2 Orthogonality.................................. 614
12.4.2.1 Properties of Orthogonality..................... 614
12.4.2.2 Orthogonal Systems ......................... 614
12.4.3 Fourier Series in Hilbert Spaces......................... 615
12.4.3.1 Best Approximation ......................... 615
12.4.3.2 Parseval Equation, Riesz-Fischer Theorem............. 616
12.4.4 Existence of a Basis, Isomorphic Hilbert Spaces................ 616
12.5 Continuous Linear Operators and Functionals..................... 617
12.5.1 Boundedness, Norm and Continuity of Linear Operators........... 617
12.5.1.1 Boundedness and the Norm of Linear Operators .......... 617
12.5.1.2 The Space of Linear Continuous Operators............. 617
12.5.1.3 Convergence of Operator Sequences................. 618
12.5.2 Linear Continuous Operators in Banach Spaces................ 618
12.5.3 Elements of the Spectral Theory of Linear Operators............. 620
12.5.3.1 Resolvent Set and the Resolvent of an Operator............ 620
12.5.3.2 Spectrum of an Operator....................... 620
12.5.4 Continuous Linear Functionals......................... 621
12.5.4.1 Definition............................... 621
12.5.4.2 Continuous Linear Functionals in Hilbert Spaces, Riesz Representa-
tion Theorem ............................. 622
12.5.4.3 Continuous Linear Functionals in Lp ................ 622
XXVI Contents
I
12.5.5 Extension of a Linear Functional........................ 622
12.5.6 Separation of Convex Sets ........................... 623
12.5.7 Second Adjoint Space and Reflexive Spaces.................. 624
12.6 Adjoint Operators in Normed Spaces.......................... 624
12.6.1 Adjoint of a Bounded Operator......................... 624
12.6.2 Adjoint Operator of an Unbounded Operator................. 625
12.6.3 Self-Adjoint Operators............................. 625
12.6.3.1 Positive Definite Operators...................... 626
12.6.3.2 Projectors in a Hilbert Space..................... 626
12.7 Compact Sets and Compact Operators......................... 626
12.7.1 Compact Subsets of a Normed Space...................... 626
12.7.2 Compact Operators............................... 626
12.7.2.1 Definition of Compact Operator................... 626
12.7.2.2 Properties of Linear Compact Operators .............. 627
12.7.2.3 Weak Convergence of Elements ................... 627
12.7.3 Fredholm Alternative.............................. 627
12.7.4 Compact Operators in Hilbert Space...................... 628
12.7.5 Compact Self-Adjoint Operators........................ 628
12.8 Non-Linear Operators.................................. 629
12.8.1 Examples of Non-Linear Operators....................... 629
12.8.2 Differentiability of Non-Linear Operators................... 630
12.8.3 Newton s Method................................ 630
12.8.4 Schauder s Fixed-Point Theorem........................ 631
12.8.5 Leray-Schauder Theory............................. 631
12.8.6 Positive Non-Linear Operators......................... 631
12.8.7 Monotone Operators in Banach Spaces .................... 632
12.9 Measure and Lebesgue Integral............................. 633
12.9.1 Sigma Algebra and Measures.......................... 633
12.9.2 Measurable Functions.............................. 634
12.9.2.1 Measurable Function......................... 634
12.9.2.2 Properties of the Class of Measurable Functions.......... 634
12.9.3 Integration.................................... 635
12.9.3.1 Definition of the Integral....................... 635
12.9.3.2 Some Properties of the Integral ................... 635
12.9.3.3 Convergence Theorems........................ 636
12.9.4 V Spaces.................................... 637
12.9.5 Distributions .................................. 637
12.9.5.1 Formula of Partial Integration.................... 637
12.9.5.2 Generalized Derivative........................ 638
12.9.5.3 Distributions............................. 638
12.9.5.4 Derivative of a Distribution ..................... 639
13 Vector Analysis and Vector Fields 640
13.1 Basic Notions of the Theory of Vector Fields...................... 640
13.1.1 Vector Functions of a Scalar Variable ..................... 640
13.1.1.1 Definitions............................... 640
13.1.1.2 Derivative of a Vector Function ................... 640
13.1.1.3 Rules of Differentiation for Vectors ................. 640
13.1.1.4 Taylor Expansion for Vector Functions ............... 641
13.1.2 Scalar Fields................................... 641
13.1.2.1 Scalar Field or Scalar Point Function ................ 641
13.1.2.2 Important Special Cases of Scalar Fields .............. 64]
Contents XXVII
13.1.2.3 Coordinate Definition of a Field................... 642
13.1.2.4 Level Surfaces and Level Lines of a Field .............. 642
13.1.3 Vector Fields................................... 643
13.1.3.1 Vector Field or Vector Point Function................ 643
13.1.3.2 Important Cases of Vector Fields.................. 643
13.1.3.3 Coordinate Representation of Vector Fields............. 644
13.1.3.4 Transformation of Coordinate Systems ............... 645
13.1.3.5 Vector Lines.............................. 646
13.2 Differential Operators of Space............................. 647
13.2.1 Directional and Space Derivatives....................... 647
13.2.1.1 Directional Derivative of a Scalar Field............... 647
13.2.1.2 Directional Derivative of a Vector Field............... 648
13.2.1.3 Volume Derivative .......................... 648
13.2.2 Gradient of a Scalar Field............................ 648
13.2.2.1 Definition of the Gradient ...................... 648
13.2.2.2 Gradient and Volume Derivative................... 649
13.2.2.3 Gradient and Directional Derivative................. 649
13.2.2.4 Further Properties of the Gradient.................. 649
13.2.2.5 Gradient of the Scalar Field in Different Coordinates........ 649
13.2.2.6 Rules of Calculations......................... 650
13.2.3 Vector Gradient................................. 650
13.2.4 Divergence of Vector Fields........................... 651
13.2.4.1 Definition of Divergence....................... 651
13.2.4.2 Divergence in Different Coordinates................. 651
13.2.4.3 Rules for Evaluation of the Divergence ............... 651
13.2.4.4 Divergence of a Central Field..................... 652
13.2.5 Rotation of Vector Fields............................ 652
13.2.5.1 Definitions of the Rotation...................... 652
13.2.5.2 Rotation in Different Coordinates.................. 653
13.2.5.3 Rules for Evaluating the Rotation.................. 653
13.2.5.4 Rotation of a Potential Field..................... 654
13.2.6 Nabla Operator, Laplace Operator....................... 654
13.2.6.1 Nabla Operator............................ 654
13.2.6.2 Rules for Calculations with the Nabla Operator........... 654
13.2.6.3 Vector Gradient............................ 655
13.2.6.4 Nabla Operator Applied Twice.................... 655
13.2.6.5 Laplace Operator........................... 655
13.2.7 Review of Spatial Differential Operations................... 656
13.2.7.1 Fundamental Relations and Results (see Table 13.2)........ 656
13.2.7.2 Rules of Calculation for Spatial Differential Operators....... 656
13.2.7.3 Expressions of Vector Analysis in Cartesian, Cylindrical, and Spher-
ical Coordinates ............................ 657
13.3 Integration in Vector Fields............................... 658
13.3.1 Line Integral and Potential in Vector Fields.................. 658
13.3.1.1 Line Integral in Vector Fields..................... 658
13.3.1.2 Interpretation of the Line Integral in Mechanics........... 659
13.3.1.3 Properties of the Line Integral.................... 659
13.3.1.4 Line Integral in Cartesian Coordinates ............... 659
13.3.1.5 Integral Along a Closed Curve in a Vector Field........... 660
13.3.1.6 Conservative or Potential Field................... 660
13.3.2 Surface Integrals................................. 661
13.3.2.1 Vector of a Plane Sheet........................ 661
XXVIII Contents
13.3.2.2 Evaluation of the Surface Integral.................. 661
13.3.2.3 Surface Integrals and Flow of Fields................. 662
13.3.2.4 Surface Integrals in Cartesian Coordinates as Surface Integral of Sec-
ond Type ............................... 662
13.3.3 Integral Theorems................................ 663
13.3.3.1 Integral Theorem and Integral Formula of Gauss.......... 663
13.3.3.2 Integral Theorem of Stokes...................... 664
13.3.3.3 Integral Theorems of Green ..................... 664 !
13.4 Evaluation of Fields................................... 665 j
13.4.1 Pure Source Fields ............................... 665 |
13.4.2 Pure Rotation Field or Zero-Divergence Field................. 666
13.4.3 Vector Fields with Point-Like Sources..................... 666
13.43.1 Coulomb Field of a Point-Like Charge................ 666
13.4.3.2 Gravitational Field of a Point Mass ................. 667!
13.4.4 Superposition of Fields............................. 667:
13.4.4.1 Discrete Source Distribution..................... 667!
13.4.4.2 Continuous Source Distribution................... 667:
13.4.4.3 Conclusion .............................. 667
13.5 Differential Equations of Vector Field Theory..................... 667
13.5.1 Laplace Differential Equation.......................... 667!
13.5.2 Poisson Differential Equation.......................... 668
14 Function Theory 669
14.1 Functions of Complex Variables ............................ 669;
14.1.1 Continuity, Differentiability........................... 669
141.1.1 Definition of a Complex Function .................. 669
14.1.1.2 Limit of a Complex Function..................... 669,
14.1.1.3 Continuous Complex Functions ................... 669
14.1.1.4 Differentiability of a Complex Function............... 669
14.1.2 Analytic Functions ............................... 670
14.1.2.1 Definition of Analytic Functions................... 670
14.1.2.2 Examples of Analytic Functions................... 670
14.1.2.3 Properties of Analytic Functions................... 670
14.1.2.4 Singular Points............................ 671
14.1.3 Conformai Mapping............................... 672
14.1.3.1 Notion and Properties of Conformai Mappings........... 672
14.1.3.2 Simplest Conformai Mappings.................... 673
14.1.3.3 The Schwarz Reflection Principle .................. 679
14.1.3.4 Complex Potential.......................... 679
14.1.3.5 Superposition Principle........................ 681
14.1.3.6 Arbitrary Mappings of the Complex Plane............. 682:
14.2 Integration in the Complex Plane ........................... 683.
14.2.1 Definite and Indefinite Integral......................... 683:
14.2.1.1 Definition of the Integral in the Complex Plane........... 683
14.2.1.2 Properties and Evaluation of Complex Integrals.......... 684,
14.2.2 Cauchy Integral Theorem............................ 686;
14.2.2.1 Cauchy Integral Theorem for Simply Connected Domains..... 686;
142.2.2 Cauchy Integral Theorem for Multiply Connected Domains .... 686!
14.2.3 Cauchy Integral Formulas............................ 687,
14.2.3.1 Analytic Function on the Interior of a Domain........... 687
14.2.3.2 Analytic Function on the Exterior of a Domain........... 687!
14.3 Power Series Expansion of Analytic Functions..................... 687Ï
Contents XXIX
14.3.1 Convergence of Series with Complex Terms.................. 687
14.3.1.1 Convergence of a Number Sequence with Complex Terms ..... 687
14.3.1.2 Convergence of an Infinite Series with Complex Terms....... 688
14.3.1.3 Power Series with Complex Terms.................. 688
14.3.2 Taylor Series................................... 689
14.3.3 Principle of Analytic Continuation....................... 689
14.3.4 Laurent Expansion ............................... 690
14.3.5 Isolated Singular Points and the Residue Theorem.............. 690
14.3.5.1 Isolated Singular Points ....................... 690
14.3.5.2 Meromorphic Functions ....................... 691
14.3.5.3 Elliptic Functions........................... 691
14.3.5.4 Residue................................ 691
14.3.5.5 Residue Theorem........................... 692
14.4 Evaluation of Real Integrals by Complex Integrals.................. 692
14.4.1 Application of Cauchy Integral Formulas................... 692
14.4.2 Application of the Residue Theorem...................... 693
14.4.3 Application of the Jordan Lemma....................... 693
14.4.3.1 Jordan Lemma ............................ 693
14.4.3.2 Examples of the Jordan Lemma................... 694
14.5 Algebraic and Elementary Transcendental Functions................. 696
14.5.1 Algebraic Functions............................... 696
14.5.2 Elementary Transcendental Functions..................... 696
14.5.3 Description of Curves in Complex Form.................... 699
14.6 Elliptic Functions.................................... 700
14.6.1 Relation to Elliptic Integrals.......................... 700
14.6.2 Jacobian Functions............................... 701
14.6.3 Theta Function................................. 703
14.6.4 Weierstrass Functions.............................. 703
15 Integral Transformations 705
15.1 Notion of Integral Transformation........................... 705
15.1.1 General Definition of Integral Transformations................ 705
15.1.2 Special Integral Transformations........................ 705
15.1.3 Inverse Transformations ............................ 705
15.1.4 Linearity of Integral Transformations..................... 705
15.1.5 Integral Transformations for Functions of Several Variables ......... 707
15.1.6 Applications of Integral Transformations................... 707
15.2 Laplace Transformation................................. 708
15.2.1 Properties of the Laplace Transformation................... 708
15.2.1.1 Laplace Transformation, Original and Image Space......... 708
15.2.1.2 Rules for the Evaluation of the Laplace Transformation ...... 709
15.2.1.3 Transforms of Special Functions................... 712
15.2.1.4 Dirac 5 Function and Distributions................. 715
15.2.2 Inverse Transformation into the Original Space................ 716
15.2.2.1 Inverse Transformation with the Help of Tables........... 716
15.2.2.2 Partial Fraction Decomposition................... 716
15.2.2.3 Series Expansion ........................... 717
15.2.2.4 Inverse Integral............................ 718
15.2.3 Solution of Differential Equations using Laplace Transformation....... 719
15.2.3.1 Ordinary Linear Differential Equations with Constant Coefficients 719
15.2.3.2 Ordinary Linear Differential Equations with Coefficients Depending
on the Variable............................ 720
XXX Contents
15.2.3.3 Partial Differential Equations .................... 721
15.3 Fourier Transformation................................. 722
15.3.1 Properties of the Fourier Transformation ................... 722
15.3.1.1 Fourier Integral............................ 722
15.3.1.2 Fourier Transformation and Inverse Transformation........ 723
15.3.1.3 Rules of Calculation with the Fourier Transformation....... 725
15.3.1.4 Transforms of Special Functions................... 728
15.3.2 Solution of Differential Equations using the Fourier Transformation..... 729
15.3.2.1 Ordinary Linear Differential Equations ............... 729
15.3.2.2 Partial Differential Equations.................... 730
15.4 Z-Transformation.................................... 731
15.4.1 Properties of the Z-Transformation....................... 732
15.4.1.1 Discrete Functions.......................... 732!
15.4.1.2 Definition of the Z-Transformation.................. 732
15.4.1.3 Rules of Calculations......................... 733:
15.4.1.4 Relation to the Laplace Transformation............... 734:
15.4.1.5 Inverse of the Z-Transformation................... 735,
15.4.2 Applications of the Z-Transformation..................... 736;
15.4.2.1 General Solution of Linear Difference Equations.......... 736
15.4.2.2 Second-Order Difference Equations (Initial Value Problem) .... 737
15.4.2.3 Second-Order Difference Equations (Boundary Value Problem) . . 738
15.5 Wavelet Transformation................................. 738
15.5.1 Signals...................................... 738
15.5.2 Wavelets..................................... 739
15.5.3 Wavelet Transformation ............................ 739
15.5.4 Discrete Wavelet Transformation........................ 741
15.5.4.1 Fast Wavelet Transformation..................... 741
15.5.4.2 Discrete Haar Wavelet Transformation ............... 741
15.5.5 Gabor Transformation ............................. 741
15.6 Walsh Functions..................................... 742
15.6.1 Step Functions.................................. 742
15.6.2 Walsh Systems.................................. 742
16 Probability Theory and Mathematical Statistics 743
16.1 Combinatorics...................................... 743
16.1.1 Permutations .................................. 743
16.1.2 Combinations.................................. 743
16.1.3 Arrangements.................................. 744
16.1.4 Collection of the Formulas of Combinatorics (see Table 16.1)......... 745
16.2 Probability Theory ................................... 745
16.2.1 Event, Frequency and Probability....................... 745
16.2.1.1 Events................................. 745
16.2.1.2 Frequencies and Probabilities .................... 746;
16.2.1.3 Conditional Probability, Bayes Theorem .............. 748:
16.2.2 Random Variables, Distribution Functions .................. 749
16.2.2.1 Random Variable........................... 749;
16.2.2.2 Distribution Function......................... 74g;
16.2.2.3 Expected Value and Variance, Chebyshev Inequality........ 754
16.2.2.4 Multidimensional Random Variable................. 752
16.2.3 Discrete Distributions.............................. 752
16.2.3.1 Binomial Distribution........................ 753
16.2.3.2 Hypergeometric Distribution..................... 754
Contents XXXI
16.2.3.3 Poisson Distribution ......................... 755
16.2.4 Continuous Distributions............................ 756
16.2.4.1 Normal Distribution ......................... 756
16.2.4.2 Standard Normal Distribution, Gaussian Error Function...... 757
16.2.4.3 Logarithmic Normal Distribution.................. 757
16.2.4.4 Exponential Distribution....................... 758
16.2.4.5 Weibull Distribution ......................... 759
16.2.4.6 x2 (Chi-Square) Distribution..................... 760
16.2.4.7 Fisher F Distribution......................... 761
16.2.4.8 Student t Distribution ........................ 761
16.2.5 Law of Large Numbers, Limit Theorems.................... 762
16.2.6 Stochastic Processes and Stochastic Chains.................. 763
16.2.6.1 Basic Notions, Markov Chains.................... 763
16.2.6.2 Poisson Process............................ 766
16.3 Mathematical Statistics................................. 767
16.3.1 Statistic Function or Sample Function..................... 767
16.3.1.1 Population, Sample, Random Vector................. 767
16.3.1.2 Statistic Function or Sample Function................ 768
16.3.2 Descriptive Statistics.............................. 770
16.3.2.1 Statistical Summarization and Analysis of Given Data....... 770
16.3.2.2 Statistical Parameters ........................ 771
16.3.3 Important Tests................................. 772
16.3.3.1 Goodness of Fit Test for a Normal Distribution........... 772
16.3.3.2 Distribution of the Sample Mean................... 774
16.3.3.3 Confidence Limits for the Mean................... 775
16.3.3.4 Confidence Interval for the Variance................. 776
16.3.3.5 Structure of Hypothesis Testing................... 777
16.3.4 Correlation and Regression........................... 777
16.3.4.1 Linear Correlation of two Measurable Characters.......... 777
16.3.4.2 Linear Regression for two Measurable Characters.......... 778
16.3.4.3 Multidimensional Regression..................... 779
16.3.5 Monte Carlo Methods.............................. 781
16.3.5.1 Simulation............................... 781
16.3.5.2 Random Numbers........................... 781
16.3.5.3 Example of a Monte Carlo Simulation................ 782
16.3.5.4 Application of the Monte Carlo Method in Numerical Mathematics 783
16.3.5.5 Further Applications of the Monte Carlo Method.......... 785
16.4 Calculus of Errors.................................... 785
16.4.1 Measurement Error and its Distribution.................... 785
16.4.1.1 Qualitative Characterization of Measurement Errors........ 785
16.4.1.2 Density Function of the Measurement Error............. 786
16.4.1.3 Quantitative Characterization of the Measurement Error ..... 788
16.4.1.4 Determining the Result of a Measurement with Bounds on the Error 790
16.4.1.5 Error Estimation for Direct Measurements with the Same Accuracy 791
16.4.1.6 Error Estimation for Direct Measurements with Different Accuracy 791
16.4.2 Error Propagation and Error Analysis..................... 792
16.4.2.1 Gauss Error Propagation Law.................... 792
16.4.2.2 Error Analysis............................. 794
XXXII Contents
I
17 Dynamical Systems and Chaos 795
17.1 Ordinary Differential Equations and Mappings.................... 795
17.1.1 Dynamical Systems............................... 795
17.1.1.1 Basic Notions............................. 795,
17.1.1.2 Invariant Sets............................. 797
17.1.2 Qualitative Theory of Ordinary Differential Equations............ 798
17.1.2.1 Existence of Flows, Phase Space Structure ............. 798
17.1.2.2 Linear Differential Equations..................... 799
17.1.2.3 Stability Theory ........................... 801
17.1.2.4 Invariant Manifolds.......................... 804 ¦
17.1.2.5 Poincaré Mapping........................... 806
17.1.2.6 Topological Equivalence of Differential Equations.......... 808;
17.1.3 Discrete Dynamical Systems.......................... 809:
17.1.3.1 Steady States, Periodic Orbits and Limit Sets............ 809 i
17.1.3.2 Invariant Manifolds.......................... 810?
17.1.3.3 Topological Conjugacy of Discrete Systems............. 811
17.1.4 Structural Stability (Robustness) ....................... 811:
17.1.4.1 Structurally Stable Differential Equations.............. 811;
17.1.4.2 Structurally Stable Discrete Systems ................ 812
17.1.4.3 Generic Properties.......................... 812
17.2 Quantitative Description of Attractors......................... 814
17.2.1 Probability Measures on Attractors...................... 814
17.2.1.1 Invariant Measure........................... 814
17.2.1.2 Elements of Ergodic Theory..................... 815
17.2.2 Entropies .................................... 817
17.2.2.1 Topological Entropy......................... 817
17.2.2.2 Metric Entropy............................ 817
17.2.3 Lyapunov Exponents.............................. 818
17.2.4 Dimensions ................................... 820
17.2.4.1 Metric Dimensions.......................... 820
17.2.4.2 Dimensions Defined by Invariant Measures............. 822
17.2.4.3 Local Hausdorff Dimension According to Douady and Oesterlé . . 824
17.2.4.4 Examples of Attractors........................ 825
17.2.5 Strange Attractors and Chaos......................... 826
17.2.6 Chaos in One-Dimensional Mappings..................... 827
17.3 Bifurcation Theory and Routes to Chaos........................ 827
17.3.1 Bifurcations in Morse-Smale Systems..................... 827
17.3.1.1 Local Bifurcations in Neighborhoods of Steady States....... 828
17.3.1.2 Local Bifurcations in a Neighborhood of a Periodic Orbit..... 833
17.3.1.3 Global Bifurcation.......................... 836
17.3.2 Transitions to Chaos .............................. 837
17.3.2.1 Cascade of Period Doublings..................... 837
17.3.2.2 Intermittency............................. 837
17.3.2.3 Global Homoclinic Bifurcations................... 838
17.3.2.4 Destruction of a Torus........................ 839
18 Optimization 844
18.1 Linear Programming .................................. 844
18.1.1 Formulation of the Problem and Geometrical Representation ........ 844
18.1.1.1 The Form of a Linear Programming Problem............ 844
18.1.1.2 Examples and Graphical Solutions.................. 845
18.1.2 Basic Notions of Linear Programming, Normal Form............. 847
Contents XXXIII
18.1.2.1 Extreme Points and Basis ...................... 847
18.1.2.2 Normal Form of the Linear Programming Problem......... 848
18.1.3 Simplex Method................................. 849
18.1.3.1 Simplex Tableau ........................... 849
18.1.3.2 Transition to the New Simplex Tableau............... 850
18.1.3.3 Determination of an Initial Simplex Tableau ............ 852
18.1.3.4 Revised Simplex Method....................... 853
18.1.3.5 Duality in Linear Programming................... 854
18.1.4 Special Linear Programming Problems..................... 855
18.1.4.1 Transportation Problem....................... 855
18.1.4.2 Assignment Problem......................... 858
18.1.4.3 Distribution Problem......................... 858
18.1.4.4 Travelling Salesman.......................... 859
18.1.4.5 Scheduling Problem.......................... 859
18.2 Non-linear Optimization................................ 859
18.2.1 Formulation of the Problem, Theoretical Basis................ 859
18.2.1.1 Formulation of the Problem..................... 859
18.2.1.2 Optimality Conditions........................ 860
18.2.1.3 Duality in Optimization ....................... 861
18.2.2 Special Non-linear Optimization Problems................... 861
18.2.2.1 Convex Optimization......................... 861
18.2.2.2 Quadratic Optimization ....................... 862
18.2.3 Solution Methods for Quadratic Optimization Problems........... 863
18.2.3.1 Wolfe s Method............................ 863
18.2.3.2 Hildreth-d Esopo Method...................... 865
18.2.4 Numerical Search Procedures.......................... 865
18.2.4.1 One-Dimensional Search....................... 865
18.2.4.2 Minimum Search in n-Dimensional Euclidean Vector Space .... 866
18.2.5 Methods for Unconstrained Problems..................... 866
18.2.5.1 Method of Steepest Descent..................... 867
18.2.5.2 Application of the Newton Method................. 867
18.2.5.3 Conjugate Gradient Methods .................... 867
18.2.5.4 Method of Davidon, Fletcher and Powell (DFP) .......... 868
18.2.6 Gradient Methods for Problems with Inequality Type Constraints) ..... 868
18.2.6.1 Method of Feasible Directions.................... 869
18.2.6.2 Gradient Projection Method..................... 870
18.2.7 Penalty Function and Barrier Methods..................... 872
18.2.7.1 Penalty Function Method ...................... 872
18.2.7.2 Barrier Method............................ 873
18.2.8 Cutting Plane Methods............................. 874
18.3 Discrete Dynamic Programming............................ 875
18.3.1 Discrete Dynamic Decision Models....................... 875
18.3.1.1 n-Stage Decision Processes...................... 875
18.3.1.2 Dynamic Programming Problem................... 875
18.3.2 Examples of Discrete Decision Models..................... 876
18.3.2.1 Purchasing Problem ......................... 876
18.3.2.2 Knapsack Problem.......................... 876
18.3.3 Bellman Functional Equations......................... 876
18.3.3.1 Properties of the Cost Function................... 876
18.3.3.2 Formulation of the Functional Equations.............. 877
18.3.4 Bellman Optimality Principle ......................... 878
18.3.5 Bellman Functional Equation Method..................... 878
XXXIV Contents
18.3.5.1 Determination of Minimal Costs................... 878
18.3.5.2 Determination of the Optimal Policy ................ 878
18.3.6 Examples of Applications of the Functional Equation Method........ 879
18.3.6.1 Optimal Purchasing Policy...................... 879
18.3.6.2 Knapsack Problem . ......................... 880
19 Numerical Analysis 881
19.1 Numerical Solution of Non-Linear Equations in a Single Unknown.......... 881
19.1.1 Iteration Method ................................ 881
19.1.1.1 Ordinary Iteration Method...................... 881
19.1.1.2 Newton s Method........................... 882
19.1.1.3 Regula Falsi.............................. 883
19.1.2 Solution of Polynomial Equations....................... 884
19.1.2.1 Horner s Scheme ........................... 884
19.1.2.2 Positions of the Roots......................... 885
19.1.2.3 Numerical Methods.......................... 886
19.2 Numerical Solution of Equation Systems........................ 887
19.2.1 Systems of Linear Equations.......................... 887
19.2.1.1 Triangular Decomposition of a Matrix................ 887
19.2.1.2 Cholesky s Method for a Symmetric Coefficient Matrix....... 890
19.2.1.3 Orthogonalization Method...................... 890
19.2.1.4 Iteration Methods........................... 892
19.2.2 Non-Linear Equation Systems......................... 893
19.2.2.1 Ordinary Iteration Method...................... 893
19.2.2.2 Newton s Method........................... 894
19.2.2.3 Derivative-Free Gauss-Newton Method............... 894
19.3 Numerical Integration.................................. 895
19.3.1 General Quadrature Formulas......................... 895
19.3.2 Interpolation Quadratures........................... 896
19.3.2.1 Rectangular Formula......................... 896
19.3.2.2 Trapezoidal Formula......................... 896
19.3.2.3 Simpson s Formula.......................... 897
19.3.2.4 Hermite s Trapezoidal Formula ................... 897
19.3.3 Quadrature Formulas of Gauss......................... 897
19.3.3.1 Gauss Quadrature Formulas..................... 897
19.3.3.2 Lobatto s Quadrature Formulas................... 898
19.3.4 Method of Romberg............................... 898
19.3.4.1 Algorithm of the Romberg Method.................. 898
19.3.4.2 Extrapolation Principle........................ 899
19.4 Approximate Integration of Ordinary Differential Equations............. 901
19.4.1 Initial Value Problems ............................. 901
19.4.1.1 Euler Polygonal Method....................... 901
19.4.1.2 Runge-Kutta Methods........................ 901
19.4.1.3 Multi-Step Methods ......................... 902
19.4.1.4 Predictor-Corrector Method..................... 903
19.4.1.5 Convergence, Consistency, Stability................. 904
19.4.2 Boundary Value Problems........................... 905
19.4.2.1 Difference Method .......................... 905
19.4.2.2 Approximation by Using Given Functions.............. 9Q6
19.42.3 Shooting Method........................... 907
19.5 Approximate Integration of Partial Differential Equations.............. 908
19.5.1 Difference Method................................ 908
Contents XXXV
19.5.2 Approximation by Given Functions ...................... 909
19.5.3 Finite Element Method (FEM)......................... 910
19.6 Approximation, Computation of Adjustment, Harmonie Analysis.......... 914
19.6.1 Polynomial Interpolation............................ 914
19.6.1.1 Newton s Interpolation Formula................... 914
19.6.1.2 Lagrange s Interpolation Formula.................. 915
19.6.1.3 Aitken-Neville Interpolation..................... 915
19.6.2 Approximation in Mean............................. 916
19.6.2.1 Continuous Problems, Normal Equations.............. 916
19.6.2.2 Discrete Problems, Normal Equations, Householder s Method ... 918
19.6.2.3 Multidimensional Problems ..................... 919
19.6.2.4 Non-Linear Least Squares Problems................. 919
19.6.3 Chebyshev Approximation........................... 920
19.6.3.1 Problem Definition and the Alternating Point Theorem ...... 920
19.6.3.2 Properties of the Chebyshev Polynomials.............. 921
19.6.3.3 Remes Algorithm........................... 922
19.6.3.4 Discrete Chebyshev Approximation and Optimization....... 923
19.6.4 Harmonic Analysis ............................... 924
19.6.4.1 Formulas for Trigonometric Interpolation.............. 924
19.6.4.2 Fast Fourier Transformation (FFT)................. 925
19.7 Representation of Curves and Surfaces with Splines.................. 928
19.7.1 Cubic Splines.................................. 928
19.7.1.1 Interpolation Splines......................... 928
19.7.1.2 Smoothing Splines .......................... 929
19.7.2 Bicubic Splines................................. 930
19.7.2.1 Use of Bicubic Splines ........................ 930
19.7.2.2 Bicubic Interpolation Splines..................... 930
19.7.2.3 Bicubic Smoothing Splines...................... 932
19.7.3 Bernstein-Bézier Representation of Curves and Surfaces........... 932
19.7.3.1 Principle of the B-B Curve Representation............. 932
19.7.3.2 B-B Surface Representation..................... 933
19.8 Using the Computer................................... 933
19.8.1 Internal Symbol Representation........................ 933
19.8.1.1 Number Systems ........................... 933
19.8.1.2 Internal Number Representation................... 935
19.8.2 Numerical Problems in Calculations with Computers............. 936
19.8.2.1 Introduction, Error Types...................... 936
19.8.2.2 Normalized Decimal Numbers and Round-Off............ 936
19.8.2.3 Accuracy in Numerical Calculations................. 938
19.8.3 Libraries of Numerical Methods ........................ 941
19.8.3.1 NAG Library............................. 941
19.8.3.2 IMSL Library............................. 942
19.8.3.3 Aachen Library............................ 943
19.8.4 Application of Computer Algebra Systems................... 943
19.8.4.1 Mathematica............................. 943
19.8.4.2 Maple................................. 946
20 Computer Algebra Systems 950
20.1 Introduction....................................... 950
20.1.1 Brief Characterization of Computer Algebra Systems............. 950
20.1.2 Examples of Basic Application Fields..................... 950
20.1.2.1 Manipulation of Formulas...................... 950
XXXVI Contents
20.1.2.2 Numerical Calculations........................ 951
20.1.2.3 Graphical Representations...................... 952
20.1.2.4 Programming in Computer Algebra Systems............ 952
20.1.3 Structure of Computer Algebra Systems.................... 952
20.1.3.1 Basic Structure Elements....................... 952
20.2 Mathematica ...................................... 953
20.2.1 Basic Structure Elements............................ 953
20.2.2 Types of Numbers in Mathematica....................... 954
2Ó.2.2.1 Basic Types of Numbers in Mathematica.............. 954
20.2.2.2 Special Numbers........................... 955
20.2.2.3 Representation and Conversion of Numbers............. 955
20.2.3 Important Operators.............................. 956
20.2.4 Lists....................................... 956
20.2.4.1 Notions................................ 956
20.2.4.2 Nested Lists, Arrays or Tables.................... 957
20.2.4.3 Operations with Lists......................... 957
20.2.4.4 Special Lists.............................. 958
20.2.5 Vectors and Matrices as Lists.......................... 958
20.2.5.1 Creating Appropriate Lists...................... 958
20.2.5.2 Operations with Matrices and Vectors................ 959
20.2.6 Functions .................................... 960
20.2.6.1 Standard Functions.......................... 960
20.2.6.2 Special Functions........................... 960
20.2.6.3 Pure Functions............................ 961
20.2.7 Patterns..................................... 961
20.2.8 Functional Operations ............................. 962
20.2.9 Programming.................................. 963
20.2.10Supplement about Syntax, Information, Messages............... 964
20.2.10.1 Contexts, Attributes......................... 964
20.2.10.2 Information.............................. 964
20.2.10.3 Messages ............................... 965
20.3 Maple.......................................... 965
20.3.1 Basic Structure Elements............................ 965
20.3.1.1 Types and Objects.......................... 965
20.3.1.2 Input and Output........................... 966
20.3.2 Types of Numbers in Maple........................... 967
20.3.2.1 Basic Types of Numbers in Maple.................. 967
20.3.2.2 Special Numbers........................... 967
20.3.2.3 Representation and Conversion of Numbers............. 967
20.3.3 Important Operators in Maple......................... 968
20.3.4 Algebraic Expressions.............................. 969
20.3.5 Sequences and Lists............................... 969
20.3.6 Tables, Arrays, Vectors and Matrices...................... 970
20.3.6.1 Tables and Arrays........................... 970
20.3.6.2 One-Dimensional Arrays....................... 971
20.3.6.3 Two-Dimensional Arrays....................... 971
20.3.6.4 Special Commands for Vectors and Matrices............ 972
20.3.7 Functions and Operators............................ 972
20.3.7.1 Functions............................... 972
20.3.7.2 Operators............................... 973
20.3.7.3 Differential Operators ........................ 973
20.3.7.4 The Functional Operator map .................... 974
Contents XXXVII
20.3.8 Programming in Maple............................. 974
20.3.9 Supplement about Syntax, Information and Help............... 975
20.3.9.1 Using the Maple Library....................... 975
20.3.9.2 Environment Variable ........................ 975
20.3.9.3 Information and Help......................... 975
20.4 Applications of Computer Algebra Systems...................... 975
20.4.1 Manipulation of Algebraic Expressions..................... 976
20.4.1.1 Mathematica............................. 976
20.4.1.2 Maple................................. 978
20.4.2 Solution of Equations and Systems of Equations ............... 981
20.4.2.1 Mathematica............................. 981
20.4.2.2 Maple................................. 983
20.4.3 Elements of Linear Algebra........................... 984
20.4.3.1 Mathematica............................. 984
20.4.3.2 Maple................................. 986
20.4.4 Differential and Integral Calculus ....................... 989
20.4.4.1 Mathematica.............................. 989
20.4.4.2 Maple................................. 992
20.5 Graphics in Computer Algebra Systems........................ 994
20.5.1 Graphics with Mathematica .......................... 995
20.5.1.1 Basic Elements of Graphics...................... 995
20.5.1.2 Graphics Primitives.......................... 995
20.5.1.3 Syntax of Graphical Representation................. 995
20.5.1.4 Graphical Options .......................... 996
20.5.1.5 Two-Dimensional Curves....................... 998
20.5.1.6 Parametric Representation of Curves................ 999
20.5.1.7 Representation of Surfaces and Space Curves............ 1000
20.5.2 Graphics with Maple.............................. 1002
20.5.2.1 Two-Dimensional Graphics...................... 1002
20.5.2.2 Three-Dimensional Graphics..................... 1004
21 Tables 1007
21.1 Frequently Used Constants............................... 1007
21.2 Natural Constants.................................... 1007
21.3 Important Series Expansions.............................. 1009
21.4 Fourier Series...................................... 1014
21.5 Indefinite Integrals................................... 1017
21.5.1 Integral Rational Functions........................... 1017
21.5.1.1 Integrals with X = ax + b...................... 1017
21.5.1.2 Integrals with X = ox2 + bx + c................... 1019
21.5.1.3 Integrals with X = a2±x2...................... 1020
21.5.1.4 Integrals with X =a3±x3...................... 1022
21.5.1.5 Integrals with X = a4 + x4...................... 1023
21.5.1.6 Integrals with X = o4 - x4...................... 1023
21.5.1.7 Some Cases of Partial Fraction Decomposition........... 1023
21.5.2 Integrals of Irrational Functions........................ 1024
21.5.2.1 Integrals with sfi and a2 ± b2x.................... 1024
21.5.2.2 Other Integrals with Jx....................... 1024
21.5.2.3 Integrals with fäx + b........................ 1025
21.5.2.4 Integrals with /ax + b and //i + g................. 1026
21.5.2.5 Integrals with Va2 - x2........................ 1027
XXXVIII Contents
21.5.2.6 Integrals with sjx2 + o?........................1029
21.5.2.7 Integrals with s/x2 - a2........................ 1030
21.5.2.8 Integrals with Vax2 + bx + c..................... 1032
21.5.2.9 Integrals with other Irrational Expressions ............. 1034
21.5.2.10 Recursion Formulas for an Integral with Binomial Differential . . . 1034
21.5.3 Integrals of Trigonometrie Functions...................... 1035
21.5.3.1 Integrals with Sine Function..................... 1035
21.5.3.2 Integrals with Cosine Function.................... 1037
21.5.3.3 Integrals with Sine and Cosine Function............... 1039
21.5.3.4 Integrals with Tangent Function................... 1043
21.5.3.5 Integrals with Cotangent Function.................. 1043
21.5.4 Integrals of other Transcendental Functions.................. 1044
21.5.4.1 Integrals with Hyperbolic Functions................. 1044
21.5.4.2 Integrals with Exponential Functions................ 1045
21.5.4.3 Integrals with Logarithmic Functions................ 1047
21.5.4.4 Integrals with Inverse Trigonometrie Functions........... 1048
21.5.4.5 Integrals with Inverse Hyperbolic Functions............. 1049
21.6 Definite Integrals.................................... 1050
21.6.1 Definite Integrals of Trigonometrie Functions................. 1050
21.6.2 Definite Integrals of Exponential Functions.................. 1051
21.6.3 Definite Integrals of Logarithmic Functions.................. 1052
21.6.4 Definite Integrals of Algebraic Functions.................... 1053
21.7 Elliptic Integrals..................................... 1055
21.7.1 Elliptic Integral of the First Kind F ( p, k), k = sina............. 1055
21.7.2 Elliptic Integral of the Second Kind E(tp,k), fc = sina............ 1055
21.7.3 Complete Elliptic Integral, k =sina...................... 1056
21.8 Gamma Function.................................... 1057
21.9 Bessel Functions (Cylindrical Functions)........................ 1058
21.10 Legendre Polynomials of the First Kind........................ 1060
21.11 Laplace Transformation................................. 1061
21.12 Fourier Transformation................................. 1066
21.12.1 Fourier Cosine Transformation......................... 1066
21.12.2 Fourier Sine Transformation.......................... 1072
21.12.3Fourier Transformation............................. 1077
21.12.4Exponential Fourier Transformation...................... 1079
21.13 Z Transformation.................................... 1080
21.14 Poisson Distribution................................... 1083
21.15 Standard Normal Distribution............................. 1085
21.15.1 Standard Normal Distribution for 0.00 x 1.99 .............. 1085
21.15.2StandardNormalDistributionfor2.00 a: 3.90 .............. 1086
21.16 x2 Distribution..................................... 1087
21.17 Fisher F Distribution.................................. 1088
21.18 Student t Distribution.................................. 1090
21.19 Random Numbers.................................... 1091
22 Bibliography 1092
Index 1103
Mathematic Symbols A
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| language | English German Russian |
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| spelling | Spravočnik po matematike Handbook of mathematics I.N. Bronshtein ... 4. ed. Berlin [u.a.] Springer 2004 XLII, 1153 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematik Mathematics Handbooks, manuals, etc Mathematik (DE-588)4037944-9 gnd rswk-swf Mathematik (DE-588)4037944-9 s DE-604 Bronštejn, Ilʹja N. 1903-1976 Sonstige (DE-588)104989742 oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012823489&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Handbook of mathematics Mathematik Mathematics Handbooks, manuals, etc Mathematik (DE-588)4037944-9 gnd |
| subject_GND | (DE-588)4037944-9 |
| title | Handbook of mathematics |
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| title_auth | Handbook of mathematics |
| title_exact_search | Handbook of mathematics |
| title_full | Handbook of mathematics I.N. Bronshtein ... |
| title_fullStr | Handbook of mathematics I.N. Bronshtein ... |
| title_full_unstemmed | Handbook of mathematics I.N. Bronshtein ... |
| title_short | Handbook of mathematics |
| title_sort | handbook of mathematics |
| topic | Mathematik Mathematics Handbooks, manuals, etc Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Mathematik Mathematics Handbooks, manuals, etc |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012823489&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | UT spravocnikpomatematike AT bronstejnilʹjan handbookofmathematics |