HANDBOOK OF MATHEMATICS

HANDBOOK OF MATHEMATICS

CONTENTS

1 Arithmetics 1
1.1 Elementary Rules for Calculations . 1
1.1.1 Numbers . 1
1.1.2 Methods for Proof . 4
1.1.3 Sums and Products . 6
1.1.4 Powers, Roots, and Logarithms . 7
1.1.5 Algebraic Expressions . 10
1.1.6 Integral Rational Expressions . 11
1.1.7 Rational Expressions . 14
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

1.3 Business Mathematics . 21
1.3.1 Calculation of Interest or Percentage . 21
1.3.2 Calculation of Compound Interest . 22
1.3.3 Amortization Calculus . 23
1.3.4 Annuity Calculations . 25
1.3.5 Depreciation . 26

1.4 Inequalities . 28
1.4.1 Pure Inequalities . 28
1.4.2 Special Inequalities . 30
1.4.3 Solution of Linear and Quadratic Inequalities . 33

1.5 Complex Numbers . 34
1.5.1 Imaginary and Complex Numbers . 34
1.5.2 Geometric Representation . 34
1.5.3 Calculation with Complex Numbers . 36

1.6 Algebraic and Transcendental Equations . 38
1.6.1 Transforming Algebraic Equations to Normal Form . 38
1.6.2 Equations of Degree at Most Four . 39
1.6.3 Equations of Degree n . 43
1.6.4 Reducing Transcendental Equations to Algebraic Equations . 45
Functions 48
2.1 Notion of Functions . 48
2.1.1 Definition of a Function . 48
2.1.2 Methods for Defining a Real Function . 49
2.1.3 Certain Types of Functions . 50
2.1.4 Limits of Functions . 53
2.1.5 Continuity of a Function . 58

2.2 Elementary Functions . 62
2.2.1 Algebraic Functions . 62
2.2.2 Transcendental Functions . 62
2.2.3 Composite Functions . 63

2.3 Polynomials . 63
2.3.1 Linear Function . 63
2.3.2 Quadratic Polynomial . 64
2.3.3 Cubic Polynomials . 64
2.3.4 Polynomials of n-th Degree . 65
2.3.5 Parabola of n-th Degree . 66

2.4 Rational Functions . 66
2.4.1 Special Fractional Linear Function (Inverse Proportionality) . .66
2.4.2 Linear Fractional Function . 66
2.4.3 Curves of Third Degree, Type I . 67
2.4.4 Curves of Third Degree, Type II . 67
2.4.5 Curves of Third Degree, Type III . 68
2.4.6 Reciprocal Powers . 70

2.5 Irrational Functions . 71
2.5.1 Square Root of a Linear Binomial . 71
2.5.2 Square Root of a Quadratic Polynomial . 71
2.5.3 Power Function . 71

2.6 Exponential Functions and Logarithmic Functions . 72
2.6.1 Exponential Functions . 72
2.6.2 Logarithmic Functions . 73
2.6.3 Error Curve . 73
2.6.4 Exponential Sum . 74
2.6.5 Generalized Error Function . 74
2.6.6 Product of Power and Exponential Functions . 75

2.7 Trigonometric Functions (Functions of Angles) . 76
2.7.1 Basic Notions . 76
2.7.2 Important Formulas for Trigonometric Functions . 81
2.7.3 Description of Oscillations . 84

2.8 Cyclometric or Inverse Trigonometric Functions . 85
2.8.1 Definition of the Inverse Trigonometric Functions . 85
2.8.2 Reduction to the Principal Value . 86
2.8.3 Relations Between the Principal Values . 87
2.8.4 Formulas for Negative Arguments . 87
2.8.5 Sum and Difference of arcsin x and arcsin y . 88
2.8.6 Sum and Difference of arccos x and arccos y . 88
2.8.7 Sum and Difference of arctan x and arctan y . 88
2.8.8 Special Relations for arcsin x, arccos x, arctan x . 88

2.9 Hyperbolic Functions . 89
2.9.1 Definition of Hyperbolic Functions . 89
2.9.2 Graphical Representation of the Hyperbolic Functions . 89
2.9.3 Important Formulas for the Hyperbolic Functions . 91

2.10 Area Functions . 93
2.10.1 Definitions . 93

2.11 Curves of Order Three (Cubic Curves) . 95
2.11.1 Semicubic Parabola . 95
2.11.2 Witch of Agnesi . 95
2.11.3 Cartesian Folium (Folium of Descartes) . 96
2.11.4 Cissoid . 96
2.11.5 Strophoide . 97

2.12 Curves of Order Four (Quartics) . 97
2.12.1 Conchoid of Nicomedes . 97
2.12.2 General Conchoid . 98
2.12.3 Pascal’s Lima¸con . 98
2.12.4 Cardioid . 99
2.12.5 Cassinian Curve . 100
2.12.6 Lemniscate . 101

2.13 Cycloids . 101
2.13.1 Common (Standard) Cycloid . 101
2.13.2 Prolate and Curtate Cycloids or Trochoids . 102
2.13.3 Epicycloid . 102
2.13.4 Hypocycloid and Astroid . 103
2.13.5 Prolate and Curtate Epicycloid and Hypocycloid . 104

2.14 Spirals . 105
2.14.1 Archimedean Spiral . 105
2.14.2 Hyperbolic Spiral . 105
2.14.3 Logarithmic Spiral . 106
2.14.4 Evolvent of the Circle . 106
2.14.5 Clothoid . 107

2.15 Various Other Curves . 107
2.15.1 Catenary Curve . 107
2.15.2 Tractrix . 108

2.16 Determination of Empirical Curves . 108
2.16.1 Procedure . 108

2.17 Scales and Graph Paper . 115
2.17.1 Scales . 115
2.17.2 Graph Paper . 116
2.17.2.4 Remark . 118

2.18 Functions of Several Variables . 118
2.18.1 Definition and Representation . 118
2.18.2 Different Domains in the Plane . 119
2.18.3 Limits . 123
2.18.4 Continuity . 124
2.18.5 Properties of Continuous Functions . 124

2.19 Nomography . 125
2.19.1 Nomograms . 125
2.19.2 Net Charts . 125
2.19.3 Alignment Charts . 126
2.19.4 Net Charts for More Than Three Variables . 128

3 Geometry 129
3.1 Plane Geometry . 129
3.1.1 Basic Notations . 129
3.1.2 Geometrical Definition of Circular and Hyperbolic Functions . 131
3.1.3 Plane Triangles . 132
3.1.4 Plane Quadrangles . 135
3.1.5 Polygons in the Plane . 138
3.1.6 The Circle and Related Shapes . 139

3.2 Plane Trigonometry . 142
3.2.1 Triangles . 142
3.2.2 Geodesic Applications . 144

3.3 Stereometry . 151
3.3.1 Lines and Planes in Space . 151
3.3.2 Edge, Corner, Solid Angle . 152
3.3.3 Polyeder or Polyhedron . 153
3.3.4 Solids Bounded by Curved Surfaces. 156

3.4 Spherical Trigonometry. 160
3.4.1 Basic Concepts of Geometry on the Sphere. 160
3.4.2 Basic Properties of Spherical Triangles . 165
3.4.3 Calculation of Spherical Triangles. 169
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