John W. Harris Horst Stocker

Handbook of Mathematics andComputational Science

With 545 Illustrations

Springer

Contents

Introduction v

1 Numerical computation (arithmetics and numerics) 11.1 Sets 1

1.1.1 Representation of sets 11.1.2 Operations on sets 21.1.3 Laws of the algebra of sets 41.1.4 Mapping and function 4

1.2 Number systems 41.2.1 Decimal number system 51.2.2 Other number systems 61.2.3 Computer representation . . . ." 61.2.4 Homer's scheme for the representation of numbers 7

1.3 Natural numbers ' 71.3.1 Mathematical induction 81.3.2 Vectors and fields, indexing 81.3.3 Calculating with natural numbers 9

1.4 Integers 111.5 Rational numbers (fractional numbers) 11

1.5.1 Decimal fractions 111.5.2 Fractions 131.5.3 Calculating with fractions 13

1.6 Calculating with quotients 141.6.1 Proportion 141.6.2 Rule of three 15

1.7 Mathematics of finance 151.7.1 Calculations of percentage 161.7.2 Interest and compound interest 161.7.3 Amortization 17

viii Contents

1.7.4 Annuities 181.7.5 Depreciation 19

1.8 Irrational numbers 201.9 Real numbers 201.10 Complex numbers 20

1.10.1 Field of complex numbers 211.11 Calculating with real numbers 22

1.11.1 Sign and absolute value 221.11.2 Ordering relations 231.11.3 Intervals 231.11.4 Rounding and truncating 241.11.5 Calculating with intervals 251.11.6 Brackets 251.11.7 Addition and subtraction 261.11.8 Summation sign 271.11.9 Multiplication and division 281.11.10 Product sign 291.11.11 Powers and roots 301.11.12 Exponentiation and logarithms 32

1.12 Binomial theorem 331.12.1 Binomial formulas 331.12.2 Binomial coefficients 341.12.3 Pascal's triangle 341.12.4 Properties of binomial coefficients 351.12.5 Expansion of powers of sums 36

2 Equations and inequalities (algebra) 372.1 Fundamental algebraic laws 37

2.1.1 Nomenclature 372.1.2 Group 392.1.3 Ring 392.1.4 Field 392.1.5 Vector space . 402.1.6 Algebra . . . . ' . 40

2.2 Equations with one unknown . 412.2.1 Elementary equivalence transformations 412.2.2 Overview of the different kinds of equations 41

2.3 Linear equations 422.3.1 Ordinary linear equations 422.3.2 Linear equations in fractional form 422.3.3 Linear equations in irrational form 43

2.4 Quadratic equations 432.4.1 Quadratic equations in fractional form 442.4.2 Quadratic equations in irrational form 44

2.5 Cubic equations 442.6 Quartic equations 46

2.6.1 General quartic equations 462.6.2 Biquadratic equations 462.6.3 Symmetric quartic equations 46

2.7 Equations of arbitrary degree 472.7.1 Polynomial division 47

Contents ix

2.8 Fractional rational equations 482.9 Irrational equations 48

2.9.1 Radical equations 482.9.2 Power equations 49

2.10 Transcendental equations 492.10.1 Exponential equations 492.10.2 Logarithmic equations 502.10.3 Trigonometric (goniometric) equations 51

2.11 Equations with absolute values 512.11.1 Equations with one absolute value 512.11.2 Equations with several absolute values 52

2.12 Inequalities 532.12.1 Equivalence transformations for inequalities 53

2.13 Numerical solution of equations 542.13.1 Graphical solution 542.13.2 Nesting of intervals 542.13.3 Secant methods and method of false position 552.13.4 Newton's method 562.13.5 Successive approximation 57

Geometry and trigonometry in the plane 593.1 Point curves 603.2 Basic constructions 60

3.2.1 Construction of the midpoint of a segment 603.2.2 Construction of the bisector of an angle 613.2.3 Construction of perpendiculars 613.2.4 To drop a perpendicular 613.2.5 Construction of parallels at a given distance 613.2.6 Parallels through a given point 62

3.3 Angles 623.3.1 Specification of angles 623.3.2 Types of angles 633.3.3 Angles between two parallels . .-. 64

3.4 Similarity and intercept theorems 643.4.1 Intercept theorems * 643.4.2 Division of a segment 653.4.3 Mean values 663.4.4 Golden Section 66

3.5 Triangles 673.5.1 Congruence theorems 673.5.2 Similarity of triangles 683.5.3 Construction of triangles 683.5.4 Calculation of a right triangle 703.5.5 Calculation of an arbitrary triangle 703.5.6 Relations between angles and sides of a triangle 723.5.7 Altitude 733.5.8 Angle-bisectors 743.5.9 Medians 743.5.10 Mid-perpendiculars, incircle, circumcircle, excircle 753.5.11 Area of a triangle 763.5.12 Generalized Pythagorean theorem 76

Contents

3.5.13 Angular relations 763.5.14 Sine theorem 763.5.15 Cosine theorem 773.5.16 Tangent theorem 773.5.17 Half-angle theorems 773.5.18 Mollweide's formulas 773.5.19 Theorems of sides 783.5.20 Isosceles triangle 783.5.21 Equilateral triangle 793.5.22 Right triangle 803.5.23 Theorem of Thales 813.5.24 Pythagorean theorem 813.5.25 Theorem of Euclid 813.5.26 Altitude theorem 81

3.6 Quadrilaterals 823.6.1 General quadrilateral 823.6.2 Trapezoid 823.6.3 Parallelogram 833.6.4 Rhombus 833.6.5 Rectangle 843.6.6 Square 843.6.7 Quadrilateral of chords 853.6.8 Quadrilateral of tangents 863.6.9 Kite 86

3.7 Regular n-gons (polygons) 863.7.1 General regular n-gons 873.7.2 Particular regular n-gons (polygons) 87

3.8 Circular objects 893.8.1 Circle 893.8.2 Circular areas 903.8.3 Annulus, circular ring 913.8.4 Sector of a circle 913.8.5 Sector of an annulus . \ 923.8.6 Segment of a circle 923.8.7 Ellipse - 93

Solid geometry 954.1 General theorems . . 95

4.1.1 Cavalieri's theorem 954.1.2 Simpson's rule 954.1.3 Guldin's rules 96

4.2 Prism 964.2.1 Oblique prism 964.2.2 Right prism 974.2.3 Cuboid 974.2.4 Cube 974.2.5 Obliquely truncated n-sided prism 98

4.3 Pyramid 984.3.1 Tetrahedron 984.3.2 Frustum of a pyramid 99

4.4 Regular polyhedron 99

Contents xi

4.4.1 Euler's theorem for polyhedrons 994.4.2 Tetrahedron 994.4.3 Cube (hexahedron) 1004.4.4 Octahedron 1004.4.5 Dodecahedron 1014.4.6 Icosahedron 101

4.5 Other solids 1024.5.1 Prismoid, prismatoid 1024.5.2 Wedge 1024.5.3 Obelisk 102

4.6 Cylinder 1024.6.1 General cylinder 1034.6.2 Right circular cylinder 1034.6.3 Obliquely cut circular cylinder 1034.6.4 Segment of a cylinder 1044.6.5 Hollow cylinder (tube) 104

4.7 Cone 1044.7.1 Right circular cone 1054.7.2 Frustum of a right circular cone 105

4.8 Sphere 1064.8.1 Solid sphere 1064.8.2 Hollow sphere 1064.8.3 Spherical sector 1064.8.4 Spherical segment (spherical cap) 1074.8.5 Spherical zone (spherical layer) 1074.8.6 Spherical wedge 108

4.9 Spherical geometry 1084.9.1 General spherical triangle (Euler's triangle) 1084.9.2 Right-angled spherical triangle 1094.9.3 Oblique spherical triangle 110

4.10 Solids of rotation I l l4.10.1 Ellipsoid I l l4.10.2 Paraboloid of revolution 1124.10.3 Hyperboloid of revolution 1124.10.4 Barrel 1124.10.5 Torus 113

4.11 Fractal geometry 1134.11.1 Scaling invariance and self-similarity 1134.11.2 Construction of self-similar objects 1134.11.3 Hausdorff dimension 1134.11.4 Cantor set 1144.11.5 Koch's curve 1144.11.6 Koch's snowflake 1154.11.7 Sierpinski gasket 1154.11.8 Box-counting algorithm 116

5 Functions 1175.1 Sequences, series, and functions 117

5.1.1 Sequences and series 1175.1.2 Properties of sequences, limits 1195.1.3 Functions 120

xii Contents

5.1.4 Classification of functions 1225.1.5 Limit and continuity 123

5.2 Discussion of curves 1245.2.1 Domain of definition 1245.2.2 Symmetry 1245.2.3 Behavior at infinity 1255.2.4 Gaps of definition and points of discontinuity 1265.2.5 Zeros 1275.2.6 Behavior of sign 1275.2.7 Behavior of slope, extremes 1285.2.8 Curvature 1295.2.9 Point of inflection 129

5.3 Basic properties of functions 130

Simple functions 137

5.4 Constant function 1375.5 Step function 1395.6 Absolute value function 1435.7 Delta function 1475.8 Integer-part function, fractional-part function 150

Integral rational functions 155

5.9 Linear function—straight line 1555.10 Quadratic function — parabola 1585.11 Cubic equation 1625.12 Power function of higher degree 1665.13 Polynomials of higher degree 1705.14 Representation of polynomials and particular polynomials 174

5.14.1 Representation by sums and products 1745.14.2 Taylor series . . 1755.14.3 Homer's scheme 1765.14.4 Newton's interpolation polynomial 1795.14.5 Lagrange polynomials 1805.14.6 Bezier polynomials and splines 1815.14.7 Particular polynomials 187

Fractional rational functions 189

5.15 Hyperbola 1895.16 Reciprocal quadratic function 1925.17 Power functions with a negative exponent 1965.18 Quotient of two polynomials 200

5.18.1 Polynomial division and partial fraction decomposition . . . 2035.18.2 Pade's approximation 205

Irrational algebraic functions 209

5.19 Square-root function 2095.20 Root function 2125.21 Power functions with fractional exponents . . . 2 1 6

Contents xiii

5.22 Roots of rational functions 219

Transcendental functions 228

5.23 Logarithmic function 2285.24 Expansion function 233

5.25 Exponential functions of powers 239

Hyperbolic functions 245

5.26 Hyperbolic sine and cosine functions 2475.27 Hyperbolic tangent and cotangent function 2525.28 Hyperbolic secant and hyperbolic cosecant functions 258Area hyperbolic functions 263

5.29 Area hyperbolic sine and hyperbolic cosine 2645.30 Area-hyperbolic tangent and hyperbolic cotangent 2675.31 Area-hyperbolic secant and hyperbolic cosecant 271

Trigonometric functions 274

5.32 Sine and cosine functions 2785.32.1 Superpositions of oscillations 2875.32.2 Periodic functions 292

5.33 Tangent and cotangent functions 2945.34 Secant and cosecant 300

Inverse trigonometric functions 306

5.35 Inverse sine and cosine functions 3075.36 Inverse tangent and cotangent functions 3115.37 Inverse secant and cosecant functions 315

Plane curves 319

5.38 Algebraic curves of the n-th order 3195.38.1 Curves of the second order 3195.38.2 Curves of the third order 3215.38.3 Curves of the fourth and higher order 323

5.39 Cycloidal curves 3245.40 Spirals : 3275.41 Other curves 328

Vector analysis 3316.1 Vector algebra 331

6.1.1 Vector and scalar 3316.1.2 Particular vectors 3326.1.3 Multiplication of a vector by a scalar 3326.1.4 Vector addition 3336.1.5 Vector subtraction 3336.1.6 Calculating laws 3336.1.7 Linear dependence/independence of vectors 334

xiv Contents

6.1.8 Basis 3356.2 Scalar product or inner product 338

6.2.1 Calculating laws 3396.2.2 Properties and applications of the scalar product 3396.2.3 Schmidt's orthonormalization method 3416.2.4 Direction cosine 3416.2.5 Application hypercubes of vector analysis 342

6.3 Vector product of two vectors 3436.3.1 Properties of the vector product 344

6.4 Multiple products of vectors 3456.4.1 Scalar triple product 345

7 Coordinate systems 3497.1 Coordinate systems in two dimensions 349

7.1.1 Cartesian coordinates 3497.1.2 Polar coordinates 3507.1.3 Conversions between two-dimensional coordinate systems . 350

7.2 Two-dimensional coordinate transformation 3507.2.1 Parallel displacement (translation) 3517.2.2 Rotation 3527.2.3 Reflection 3537.2.4 Scaling 353

7.3 Coordinate systems in three dimensions 3547.3.1 Cartesian coordinates 3547.3.2 Cylindrical coordinates 3547.3.3 Spherical coordinates 3557.3.4 Conversions between three-dimensional coordinate systems . 355

7.4 Coordinate transformation in three dimensions 3567.4.1 Parallel displacement (translation) 3567.4.2 Rotation 357

7.5 Application in computer graphics 3577.6 Transformations 358

7.6.1 Object representation and object description 3587.6.2 Homogeneous coordinates 3597.6.3 Two-dimensional translations with homogeneous

coordinates 3607.6.4 Two-dimensional scaling with homogeneous coordinates . . 3607.6.5 Three-dimensional translation with homogeneous

coordinates 3617.6.6 Three-dimensional scaling with homogeneous coordinates . 3617.6.7 Three-dimensional rotation of points with homogeneous

coordinates 3627.6.8 Positioning of an object in space 3637.6.9 Rotation of objects about an arbitrary axis in space 3647.6.10 Animation 3667.6.11 Reflections 3667.6.12 Transformation of coordinate systems 3677.6.13 Translation of a coordinate system 3677.6.14 Rotation of a coordinate system about a principal axis . . . . 368

7.7 Projections 3707.7.1 Fundamental principles 370

Contents xv

7.7.2 Parallel projection 3707.7.3 Central projection 3737.7.4 General formulation of projections 374

7.8 Window/viewport transformation 376

8 Analytic geometry 3778.1 Elements of the plane 377

8.1.1 Distance between two points 3778.1.2 Division of a segment 3778.1.3 Area of a triangle 3788.1.4 Equation of a curve 378

8.2 Straight line 3788.2.1 Forms of straight-line equations 3798.2.2 Hessian normal form 3808.2.3 Point of intersection of straight lines 3818.2.4 Angle between straight lines 3818.2.5 Parallel and perpendicular straight lines 382

8.3 Circle ,. 3828.3.1 Equations of a circle 3828.3.2 Circle and straight line 3838.3.3 Intersection of two circles 3838.3.4 Equation of the tangent to a circle 384

8.4 Ellipse 3848.4.1 Equations of the ellipse 3848.4.2 Focal properties of the ellipse 3858.4.3 Diameters of the ellipse 3858.4.4 Tangent and normal to the ellipse 3858.4.5 Curvature of the ellipse 3868.4.6 Areas and circumference of the ellipse 386

8.5 Parabola 3878.5.1 Equations of the parabola 3878.5.2 Focal properties of the parabola 3888.5.3 Diameters of the parabola 3888.5.4 Tangent and normal of the parabola , 3888.5.5 Curvature of a parabola 3898.5.6 Areas and arc lengths of the parabola 3898.5.7 Parabola and straight line 389

8.6 Hyperbola 3908.6.1 Equations of the hyperbola 3908.6.2 Focal properties of the hyperbola 3918.6.3 Tangent and normal to the hyperbola 3928.6.4 Conjugate hyperbolas and diameter 3928.6.5 Curvature of a hyperbola 3928.6.6 Areas of hyperbola 3938.6.7 Hyperbola and straight line 393

8.7 General equation of conies 3938.7.1 Form of conies 3948.7.2 Transformation to principal axes 3948.7.3 Geometric construction (conic section) 3958.7.4 Directrix property 3958.7.5 Polar equation 396

xvi Contents

8.8 Elements in space 3968.8.1 Distance between two points 3968.8.2 Division of a segment 3968.8.3 Volume of a tetrahedron 396

8.9 Straight lines in space 3978.9.1 Parametric representation of a straight line 3978.9.2 Point of intersection of two straight lines 3978.9.3 Angle of intersection between two intersecting straight

lines 3988.9.4 Foot of a perpendicular (perpendicular line) 3988.9.5 Distance between a point and a straight line 3988.9.6 Distance between two lines 399

8.10 Planes in space 3998.10.1 Parametric representation of the plane 3998.10.2 Coordinate representation of the plane 3998.10.3 Hessian normal form of the plane 4008.10.4 Conversions 4008.10.5 Distance between a point and a plane 4018.10.6 Point of intersection of a line and a plane 4018.10.7 Angle of intersection between two intersecting planes . . . . 4018.10.8 Foot of the perpendicular (perpendicular line) 4018.10.9 Reflection 4028.10.10 Distance between two parallel planes 4028.10.11 Cut set of two planes 402

8.11 Plane of the second order in normal form 4038.11.1 Ellipsoid 4038.11.2 Hyperboloid 4038.11.3 Cone 4048.11.4 Paraboloid 4048.11.5 Cylinder 405

8.12 General plane of the second order 4068.12.1 General equation 4068.12.2 Transformation to principal axes 4068.12.3 Shape of a surface of the second order 407

9 Matrices, determinants, and systems of linear equations 4099.1 Matrices 409

9.1.1 Row and column vectors 4119.2 Special matrices 412

9.2.1 Transposed, conjugate, and adjoint matrices 4129.2.2 Square matrices 4129.2.3 Triangular matrices 4149.2.4 Diagonal matrices 415

9.3 Operations with matrices 4189.3.1 Addition and subtraction of matrices 4189.3.2 Multiplication of a matrix by a scalar factor c 4189.3.3 Multiplication of vectors, scalar product 4199.3.4 Multiplication of a matrix by a vector 4219.3.5 Multiplication of matrices 4219.3.6 Calculating rules of matrix multiplication 4229.3.7 Multiplication by a diagonal matrix 424

Contents xvii

9.3.8 Matrix multiplication according to Falk's scheme 4249.3.9 Checking of row and column sums 425

9.4 Determinants 4269.4.1 Two-row determinants 4279.4.2 General computational rules for determinants 4279.4.3 Zero value of the determinant 4299.4.4 Three-row determinants 4309.4.5 Determinants of higher (n-th) order 4329.4.6 Calculation of n-row determinants 4339.4.7 Regular and inverse matrix 4349.4.8 Calculation of the inverse matrix in terms of determinants . . 4359.4.9 Rank of a matrix 4369.4.10 Determination of the rank by means of minor determinants . 437

9.5 Systems of linear equations 4379.5.1 Systems of two equations with two unknowns 439

9.6 Numerical solution methods 4419.6.1 Gaussian algorithm for systems of linear equations 4419.6.2 Forward elimination 4419.6.3 Pivoting 4439.6.4 Backsubstitution 4449.6.5 LU-decomposition 4459.6.6 Solvability of (m x n) systems of equations 4489.6.7 Gauss-Jordan method for matrix inversion 4509.6.8 Calculation of the inverse matrix A"1 452

9.7 Iterative solution of systems of linear equations 4549.7.1 Total-step methods (Jacobi) 4569.7.2 Single-step methods (Gauss-Seidel) 4569.7.3 Criteria of convergence for iterative methods 4579.7.4 Storage of the coefficient matrix 458

9.8 Table of solution methods 4599.9 Eigenvalue equations 4619.10 Tensors 463

9.10.1 Algebraic operations with tensors 465

10 Boolean algebra-application in switching algebra 46710.1 Basic notions 467

10.1.1 Propositions and truth values 46710.1.2 Proposition variables ' 468

10.2 Boolean connectives 46810.2.1 Negation: not 46910.2.2 Conjunction: and 46910.2.3 Disjunction (inclusive): or 46910.2.4 Calculating rules 470

10.3 Boolean functions 47110.3.1 Operator basis 472

10.4 Normal forms 47210.4.1 Disjunctive normal forms 47210.4.2 Conjunctive normal form 47310.4.3 Representation of functions by normal forms 473

10.5 Karnaugh-Veitch diagrams 47510.5.1 Producing a KV-diagram 476

xviii Contents

10.5.2 Entering a function in a KV-diagram 47610.5.3 Minimization with the help of KV-diagrams 477

10.6 Minimization according to Quine and McCluskey 47810.7 Multi-valued logic and fuzzy logic 481

10.7.1 Multi-valued logic 481- 10.7.2 Fuzzy logic 481

11 Graphs and Algorithms 48311.1 Graphs 483

11.1.1 Basic definitions 48311.1.2 Representation of graphs 48511.1.3 Trees 485

11.2 Matchings 48611.3 Networks 487

11.3.1 Flows in networks 48711.3.2 Eulerian line and Hamiltonian circuit 487

12 Differential calculus 48912.1 Derivative of a function 489

12.1.1 Differential 49012.1.2 Differentiability 491

12.2 Differentiation rules 49212.2.1 Derivatives of elementary functions 49212.2.2 Derivatives of trigonometric functions 49212.2.3 Derivatives of hyperbolic functions 49212.2.4 Constant rule 49312.2.5 Factor rule 49312.2.6 Power rule 49312.2.7 Sum rule 49312.2.8 Product rule 49312.2.9 Quotient rule 49412.2.10 Chain rule 49412.2.11 Logarithmic differentiation of functions 49512.2.12 Differentiation of functions in parametric representation . . . 49512.2.13 Differentiation of functions in polar coordinates 49612.2.14 Differentiation of an implicit function 49612.2.15 Differentiation of the inverse function 49712.2.16 Table of differentiation rules 498

12.3 Mean value theorems 49912.3.1 Rolle's theorem 49912.3.2 Mean value theorem of differential calculus 49912.3.3 Extended mean value theorem of differential calculus . . . . 500

12.4 Higher derivatives 50012.4.1 Slope, extremes 50212.4.2 Curvature 50312.4.3 Point of inflection 503

12.5 Approximation method of differentiation 50412.5.1 Graphical differentiation 50412.5.2 Numerical differentiation 505

12.6 Differentiation of functions with several variables 50612.6.1 Partial derivative 506

Contents xix

12.6.2 Total differential 50812.6.3 Extremes of functions in two dimensions 50812.6.4 Extremes with constraints 509

12.7 Application of differential calculus 51012.7.1 Calculation of indefinite expressions 51012.7.2 Discussion of curves 51112.7.3 Extreme value problems 51212.7.4 Calculus of errors 51312.7.5 Determination of zeros according to Newton's method . . . 514

13 Differential geometry 51713.1 Plane curves 517

13.1.1 Representation of curves 51713.1.2 Differentiation by implicit representation 51713.1.3 Differentiation by parametric representation 51813.1.4 Differentiation by polar coordinates 51813.1.5 Differential of arc of a curve 51813.1.6 Tangent, normal 51913.1.7 Curvature of a curve 52013.1.8 Evolutes and evolvents 52213.1.9 Points of inflection, vertices 52213.1.10 Singular points 52213.1.11 Asymptotes 52313.1.12 Envelope of a family of curves 524

13.2 Space curves 52413.2.1 Representation of space curves 52413.2.2 Moving trihedral 52513.2.3 Curvature 52713.2.4 Torsion of a curve 52713.2.5 Frenet formulas 528

13.3 Surfaces 52813.3.1 Representation of a surface 52813.3.2 Tangent plane and normal to the surface 52913.3.3 Singular points of the surface 530

14 Infinite series 53114.1 Series 53114.2 Criteria of convergence 532

14.2.1 Special number series 53514.3 Taylor and MacLaurin series 535

14.3.1 Taylor's formula 53514.3.2 Taylor series 536

14.4 Power series 53714.4.1 Test of convergence for power series 53714.4.2 Properties of convergent power series 53814.4.3 Inversion of power series 540

14.5 Special expansions of series and products 54014.5.1 Binomial series 54014.5.2 Special binomial series 54014.5.3 Series of exponential functions 54114.5.4 Series of logarithmic functions 542

xx Contents

14.5.5 Series of trigonometric functions 54214.5.6 Series of inverse trigonometric functions 54314.5.7 Series of hyperbolic functions 54414.5.8 Series of area hyperbolic functions 54414.5.9 Partial fraction expansions 54414.5.10 Infinite products 545

15 Integral calculus 54715.1 Definition and integrability 547

15.1.1 Primitive 54715.1.2 Definite and indefinite integrals 54815.1.3 Geometrical interpretation 54915.1.4 Rules for integrability 55015.1.5 Improper integrals 551

15.2' Integration rules 55215.2.1 Rules for indefinite integrals 55215.2.2 Rules for definite integrals 55315.2.3 Table of integration rules 55415.2.4 Integrals of some elementary functions 555

15.3 Integration methods 55715.3.1 Integration by substitution 55715.3.2 Integration by parts 56015.3.3 Integration by partial fraction decomposition 56215.3.4 Integration by series expansion 565

15.4 Numerical integration 56715.4.1 Rectangular rule 56715.4.2 Trapezoidal rule 56815.4.3 Simpson's rule 56815.4.4 Romberg integration 56915.4.5 Gaussian quadrature 57015.4.6 Table of numerical integration methods 572

15.5 Mean value theorem of integral calculus 57415.6 Line, surface, and volume integrals 574

15.6.1 Arc length (rectification) 57415.6.2 Area 57515.6.3 Solid of rotation (solid of revolution) 576

15.7 Functions in parametric representation 57715.7.1 Arc length in parametric representation 57715.7.2 Sector formula 57815.7.3 Solids of rotation in parametric representation 578

15.8 Multiple integrals and their applications 57915.8.1 Definition of multiple integrals 57915.8.2 Calculation of areas 58015.8.3 Center of mass of arcs 58115.8.4 Moment of inertia of an area 58115.8.5 Center of mass of areas 58215.8.6 Moment of inertia of planes 58215.8.7 Center of mass of a body 582

o 15.8.8 Moment of inertia of a body 58315.8.9 Center of mass of rotational solids 58315.8.10 Moment of inertia of rotational solids 583

Contents xxi

15.9 Technical applications of integral calculus 58415.9.1 Static moment, center of mass 58415.9.2 Mass moment of inertia 58515.9.3 Statics 58815.9.4 Calculation of work 58815.9.5 Mean values 589

16 Vector analysis 59116.1 Fields 591

16.1.1 Symmetries of fields 59216.2 Differentiation and integration of vectors 594

16.2.1 Scale factors in general orthogonal coordinates 59616.2.2 Differential operators 597

16.3 Gradient and potential 59816.4 Directional derivative and vector gradient 60016.5 Divergence and Gaussian integral theorem 60116.6 Rotation and Stokes's theorem 60416.7 Laplace operator and Green's formulas 607

16.7.1 Combinations of div, rot, and grad; calculation of fields . . . 60916.8 Summary 610

17 Complex variables and functions 61317.1 Complex numbers 613

17.1.1 Imaginary numbers 61317.1.2 Algebraic representation of complex numbers 61417.1.3 Cartesian representation of complex numbers 61417.1.4 Conjugate complex numbers 61517.1.5 Absolute value of a complex number 61517.1.6 Trigonometric representation of complex numbers 61617.1.7 Exponential representation of complex numbers 61617.1.8 Transformation from Cartesian to trigonometric

representation 61717.1.9 Riemann sphere 618

17.2 Elementary arithmetical operations with complex numbers 61917.2.1 Addition and subtraction of complex numbers 61917.2.2 Multiplication and division of complex numbers 61917.2.3 Exponentiation in the complex domain 62217.2.4 Taking the root in the complex domain 623

17.3 Elementary functions of a complex variable 62317.3.1 Sequences in the complex domain 62417.3.2 Series in the complex domain 62517.3.3 Exponential function in the complex domain 62617.3.4 Natural logarithm in the complex domain 62617.3.5 General power in the complex domain 62717.3.6 Trigonometric functions in the complex domain 62717.3.7 Hyperbolic functions in the complex domain 62917.3.8 Inverse trigonometric, inverse hyperbolic functions in the

complex domain 63017.4 Applications of complex functions 631

17.4.1 Representation of oscillations in the complex plane 63117.4.2 Superposition of oscillations of equal frequency 632

xxii Contents

17.4.3 Loci 63317.4.4 Inversion of loci 634

17.5 Differentiation of functions of a complex variable 63517.5.1 Definition of the derivative in the complex domain 63517.5.2 Differentiation rales in the complex domain 63617.5.3 Cauchy-Riemann differentiability conditions 63717.5.4 Conformal mapping 637

17.6 Integration in the complex plane 63917.6.1 Complex curvilinear integrals 63917.6.2 Cauchy's integral theorem 64017.6.3 Primitive functions in the complex domain 64117.6.4 Cauchy's integral formulas 64117.6.5 Taylor series of an analytic function 64217.6.6 Laurent series 64317.6.7 Classification of singular points 64317.6.8 Residue theorem 64417.6.9 Inverse Laplace transformation 645

18 Differential equations 64718.1 General definitions 64718.2 Geometric interpretation 64918.3 Solution methods for first-order differential equations 650

18.3.1 Separation of variables 65018.3.2 Substitution 65118.3.3 Exact differential equations 65118.3.4 Integrating factor 651

18.4 Linear differential equations of the first order 65218.4.1 Variation of the constants 65218.4.2 General solution 65318.4.3 Determination of a particular solution 65318.4.4 Linear differential equations of the first order with constant

coefficients 65318.5 Some specific equations 654

18.5.1 Bernoulli differential equation 65418.5.2 Riccati differential equation 654

18.6 Differential equations of the second order 65518.6.1 Simple special cases 655

18.7 Linear differential equations of the second order 65618.7.1 Homogeneous linear differential equation of the

second order 65718.7.2 Inhomogeneous linear differential equations of the second

order 65718.7.3 Reduction of special differential equations of the second

order to differential equations of the first order 65918.7.4 Linear differential equations of the second order with

constant coefficients 65918.8 Differential equations ofthew-th order 66218.9 Systems of coupled differential equations of the first order 66818.10 Systems of linear homogeneous differential equations with constant

coefficients 67018.11 Partial differential equations 672

Contents xxiii

18.11.1 Solution by separation 67318.12 Numerical integration of differential equations 676

18.12.1 Euler method 67618.12.2 Heun method 67718.12.3 Modified Euler method 67918.12.4~ Runge-Kutta methods 67918.12.5 Runge-Kutta method for systems of differential equations . . 68518.12.6 Difference method for the solution of partial differential

equations 68518.12.7 Method of finite elements 688

19 Fourier transformation 69119.1 Fourier series 691

19.1.1 Introduction 69119.1.2 Definition and coefficients 69119.1.3 Condition of convergence 69319.1.4 Extended interval 69419.1.5 Symmetries 69619.1.6 Fourier series in complex and spectral representation . . . . 69819.1.7 Formulas for the calculation of Fourier series 69919.1.8 Fourier expansion of simple periodic functions 69919.1.9 Fourier series (table) 705

19.2 Fourier integrals 70719.2.1 Introduction 70719.2.2 Definition and coefficients 70719.2.3 Conditions for convergence 70819.2.4 Complex representation, Fourier sine and cosine

transformation 70819.2.5 Symmetries 71019.2.6 Convolution and some calculating rales 710

19.3 Discrete Fourier transformation (DFT) 71219.3.1 Definition and coefficients 71219.3.2 Shannon scanning theorem 71319.3.3 Discrete sine and cosine transformation 71419.3.4 Fast Fourier transformation (FFT) 71519.3.5 Particular pairs of Fourier transforms 72019.3.6 Fourier transforms (table) 72019.3.7 Particular Fourier sine transforms 72219.3.8 Particular Fourier cosine transforms 723

19.4 Wavelet transformation 72419.4.1 Signals 72419.4.2 Linear signal analysis 72519.4.3 Symmetry transformations 72619.4.4 Time-frequency analysis and Gabor transformation 72719.4.5 Wavelet transformation 72819.4.6 Discrete wavelet-transformation 732

20 Laplace and z transformations 73520.1 Introduction 73520.2 Definition of the Laplace transformation 73620.3 Transformation theorems 737

xxiv Contents

20.4 Partial fraction separation 74520.4.1 Partial fraction separation with simple real zeros 74520.4.2 Partial fraction decomposition with multiple real zeros . . . 74620.4.3 Partial fraction decomposition with complex zeros 747

20.5 Linear differential equations with constant coefficients 74820.5.1 Laplace transformation: linear differential equation of the

first order with constant coefficients 74920.5.2 Laplace transformation: linear differential equation of the

second order with constant coefficients 75120.5.3 Example: linear differential equations 75320.5.4 Laplace transforms (table) 756

20.6 z transformation 76420.6.1 Definition of the z transformation 76420.6.2 Convergence conditions for the z transformation 76620.6.3 Inversion of the z transformation 76720.6.4 Calculating rales 76720.6.5 Calculating rales for the z transformation 77020.6.6 Table of z transforms 770

21 Probability theory and mathematical statistics 77321.1 Combinatorics 77321.2 Random events 774

21.2.1 Basic notions 77421.2.2 Event relations and event operations 77521.2.3 Structural representation of events 777

21.3 Probability of events 77821.3.1 Properties of probabilities 77821.3.2 Methods to calculate probabilities 77821.3.3 Conditional probabilities 77921.3.4 Calculating with probabilities 779

21.4 Random variables and their distributions 78121.4.1 Individual probability, density function and distribution

function x 78221.4.2 Parameters of distributions 78321.4.3 Special discrete distribution 78521.4.4 Special continuous distributions 793

21.5 Limit theorems 80021.5.1 Laws of large numbers 80021.5.2 Limit theorems 801

21.6 Multidimensional random variables 80221.6.1 Distribution functions of two-dimensional random variables . 80221.6.2 Two-dimensional discrete random variables 80321.6.3 Two-dimensional continuous random variables 80421.6.4 Independence of random variables 80521.6.5 Parameters of two-dimensional random variables 80621.6.6 Two-dimensional normal distribution 807

21.7 Basics of mathematical statistics 808.21.7.1 Description of measurements . . 80921.7.2 Types of error 810

21.8 Parameters for describing distributions of measured values 81221.8.1 Position parameter, means of series of measurements . . . . 812

Contents xxv

21.8.2 Dispersion parameter 81421.9 Special distributions 815

21.9.1 Frequency distributions 81521.9.2 Distribution of random sample functions 816

21.10 Analysis by means of random sampling (theory of testingand estimating) 82021.10.1 Estimation methods 82121.10.2 Construction principles for estimators 82321.10.3 Method of moments 82321.10.4 Maximum likelihood method 82421.10.5 Method of least squares 82421.10.6 x2-minimum method 82521.10.7 Method of quantiles, percentiles 82521.10.8 Interval estimation 82621.10.9 Interval bounds for normal distribution 82821.10.10 Prediction and confidence interval bounds for binomial

and hypergeometric distributions 82921.10.11 Interval bounds for a Poisson distribution 83021.10.12 Determination of sample sizes n 83021.10.13 Test methods 83121.10.14 Parameter tests 83421.10.15 Parameter tests for a normal distribution 83421.10.16 Hypotheses about the mean value of arbitrary

distributions 83621.10.17 Hypotheses about p of binomial and hypergeometric

distributions 83721.10.18 Tests of goodness of fit 83721.10.19 Application: acceptance/rejection test 838

21.11 Reliability 83921.12 Computation of adjustment, regression 841

21.12.1 Linear regression, least squares method 84321.12.2 Regression of the n-th order 844

22 Fuzzy logic 84722.1 Fuzzy sets 84722.2 Fuzzy concept 84822.3 Functional graphs for the modeling of fuzzy sets 84922.4 Combination of fuzzy sets 852

22.4.1 Elementary operations 85222.4.2 Calculating rales for fuzzy sets 85522.4.3 Rules for families of fuzzy sets 85622.4.4 t norm and t conorm 85622.4.5 Non-parametrized operators: t norms and s norms

(t conorms) 85822.4.6 Parametrized t and s norms 85922.4.7 Compensatory operators 860

22.5 Fuzzy relations 86122.6 Fuzzy inference 86322.7 Denazification methods 86422.8 Example: erect pendulum 86622.9 Fuzzy realizations 870

xxvi Contents

23 Neural networks 87123.1 Function and structure 871

23.1.1 Function 87123.1.2 Structure 872

23.2 Implementation of the neuron model 873~ 23.2.1 Time-independent systems 873

23.2.2 Time-dependent systems 87323.2.3 Application 874

23.3 Supervised learning 87423.3.1 Principle of supervised learning 87423.3.2 Standard backpropagation 87623.3.3 Backpropagation through time 87723.3.4 Improved learning methods 87823.3.5 Hopfield network 879

23.4 Unsupervised learning 88123.4.1 Principle of unsupervised learning 88123.4.2 Kohonen model 881

24 Computers 88324.1 Operating systems 883

24.1.1 Introduction to MS-DOS 88524.1.2 Introduction to UNIX 886

24.2 High-level programming languages 88924.2.1 Program structures 89024.2.2 Object-oriented programming (OOP) 892

Introduction to PASCAL 893

24.3 Basic structure 89424.4 Variables and types 894

24.4.1 Integers 89524.4.2 Real numbers . . . 89524.4.3 Boolean values 89524.4.4 ARRAYS \ 89524.4.5 Characters and character strings 89624.4.6 RECORD ' 89724.4.7 Pointers 89824.4.8 Self-defined types 899

24.5 Statements 90024.5.1 Assignments and expressions 90024.5.2 Input and output 90124.5.3 Compound statements 90224.5.4 Conditional statements IF and CASE 90324.5.5 Loops FOR, WHILE, and REPEAT 904

24.6 Procedures and functions 90524.6.1 Procedures 90524.6.2 Functions 90624.6.3 Local and global variables, parameter passing 906

24.7 Recursion 90824.8 Basic algorithms 909

24.8.1 Dynamic data structures 909

Contents xxvii

24.8.2 Search 91024.8.3 Sorting 911

24.9 Computer graphics 91324.9.1 Basic functions 913

Introduction to C 914

24.9.2 Basic structures 91424.9.3 Operators 91624.9.4 Data structures 91824.9.5 Loops and branches 921

Introduction to C++ 924

24.9.6 Variables and constants 92424.9.7 Overloading of functions 92424.9.8 Overloading of operators 92424.9.9 Classes 92524.9.10 Instantiation of classes 92624.9.11 f r i e n d functions 92624.9.12 Operators as member functions 92624.9.13 Constructors 92724.9.14 Derived classes (inheritance) 92824.9.15 Class libraries 929

Introduction to FORTRAN 930

24.9.16 Program structure 93024.9.17 Data structures 93024.9.18 Type conversion 93124.9.19 Operators 93324.9.20 Loops and branches 93324.9.21 Subprograms 934

Computer algebra 937

24.9.22 Structural elements of Mathematica 93724.9.23 Structural elements of Maple 94024.9.24 Algebraic expressions 94224.9.25 Equations and systems of equations 94324.9.26 Linear algebra 94424.9.27 Differential and integral calculus 94524.9.28 Programming 94724.9.29 Fitting curves and interpolation with Mathematica 94824.9.30 Graphics 949

25 Tables of integrals 95125.1 Integrals of rational functions 951

25.1.1 Integrals with P - ax + b, a^0 95125.1.2 Integrals with x"1/(ax + fc)\ P = ax + b,a ^ 0, F ^ 0 . . 95225.1.3 Integrals with 1/(xn(ax + b)m), P = ax + b b ^ 0 . . . 95325.1.4 Integrals with ax + b and ex + d c ^ 0 955

xxviii Contents

25.1.5 Integrals with a + x and b + x a =£ b 95525.1.6 Integrals with P = ax2 + bx + c (a / 0) 95625.1.7 Integrals with xn/(ax2 + bx + c)m, P = ax2 + bx+c

a / 0 95625.1.8 Integrals with l/(xn(ax2 + bx + c)m), P = ax2 + bx + c

c / 0 9 5 72 5 . 1 . 9 I n t e g r a l s w i t h P = a 2 ± x 2 9 5 82 5 . 1 . 1 0 I n t e g r a l s w i t h l / ( a 2 ± x 2 ) " , P = a 2 ± x 2 a / 0 . . . . 9 5 82 5 . 1 . 1 1 I n t e g r a l s w i t h x " / {a2 ± x 2 ) m , P = a 2 ± x 2 a / 0 . . . 9 5 82 5 . 1 . 1 2 Integrals with 1/ (xn{a2 ±x2)m) P = a2±x2 a / 0 . . 96025.1.13 Integrals with P = a 3 ± x3 a / 0 96125.1.14 Integrals with a4 + xA (a > 0) 962'25.1.15 Integrals with a4 - x 4 (a > 0) 962

25.2 Integrals of irrational functions 96325.2.1 Integrals with x 1 / 2 and P = ax + b a,b^0 96325.2.2 Integrals with (ax + b)l/2 P = ax + b a / 0 96425.2.3 Integrals with (ax + b)l/2 and (ex + d ) 1 / 2 , a, c / 0 . . . . 96625.2.4 Integrals with R = (a2 + x2)1'2 a / 0 96625.2.5 Integrals with S = (x2 - a2)y'2 a # 0 96825.2.6 Integrals with T = {a2- x2)x'2 a / 0 97025.2.7 Integrals with (ax2 + bx + c)l/2

X = ax2 + bx + c a / 0 97225.3 Integrals of transcendental functions 973

25.3.1 Integrals with exponential functions 97325.3.2 Integrals with logarithmic functions (x > 0) 97525.3.3 Integrals with hyperbolic functions (a / 0) 97725.3.4 Integrals with inverse hyperbolic functions 97925.3.5 Integrals with sine and cosine functions (a / 0) 97925.3.6 Integrals with sine and cosine functions (a / 0) 98425.3.7 Integrals with tangent or cotangent functions (a / 0) . . . 98925.3.8 Integrals with inverse trigonometric functions (a / 0) . . . 990

25.4 Definite integrals 99225.4.1 Definite integrals with algebraic functions 99225.4.2 Definite integrals with exponential functions 99225.4.3 Definite integrals with logarithmic functions 99425.4.4 Definite integrals with trigonometric functions 995

Index 999