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The Handbook of Mathematics and Computational Science Article · January 1998 DOI: 10.1007/978-1-4612-5317-4
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Handbook of Mathematics and Computational Science With 545 Illustrations
Springer
Contents
Introduction 1 Numerical computation (arithmetics and numerics) 1.1 Sets 1.1.1 Representation of sets 1.1.2 Operations on sets 1.1.3 Laws of the algebra of sets 1.1.4 Mapping and function 1.2 Number systems 1.2.1 Decimal number system 1.2.2 Other number systems 1.2.3 Computer representation . . . ." 1.2.4 Homer's scheme for the representation of numbers 1.3 Natural numbers ' 1.3.1 Mathematical induction 1.3.2 Vectors and fields, indexing 1.3.3 Calculating with natural numbers 1.4 Integers 1.5 Rational numbers (fractional numbers) 1.5.1 Decimal fractions 1.5.2 Fractions 1.5.3 Calculating with fractions 1.6 Calculating with quotients 1.6.1 Proportion 1.6.2 Rule of three 1.7 Mathematics of finance 1.7.1 Calculations of percentage 1.7.2 Interest and compound interest 1.7.3 Amortization
v 1 1 1 2 4 4 4 5 6 6 7 7 8 8 9 11 11 11 13 13 14 14 15 15 16 16 17
viii
Contents
1.8 1.9 1.10 1.11
1.12
2
1.7.4 Annuities 1.7.5 Depreciation Irrational numbers Real numbers Complex numbers 1.10.1 Field of complex numbers Calculating with real numbers 1.11.1 Sign and absolute value 1.11.2 Ordering relations 1.11.3 Intervals 1.11.4 Rounding and truncating 1.11.5 Calculating with intervals 1.11.6 Brackets 1.11.7 Addition and subtraction 1.11.8 Summation sign 1.11.9 Multiplication and division 1.11.10 Product sign 1.11.11 Powers and roots 1.11.12 Exponentiation and logarithms Binomial theorem 1.12.1 Binomial formulas 1.12.2 Binomial coefficients 1.12.3 Pascal's triangle 1.12.4 Properties of binomial coefficients 1.12.5 Expansion of powers of sums
Equations and inequalities (algebra) 2.1 Fundamental algebraic laws 2.1.1 Nomenclature 2.1.2 Group 2.1.3 Ring 2.1.4 Field 2.1.5 Vector space . 2.1.6 Algebra . . . . ' . 2.2 Equations with one unknown . 2.2.1 Elementary equivalence transformations 2.2.2 Overview of the different kinds of equations 2.3 Linear equations 2.3.1 Ordinary linear equations 2.3.2 Linear equations in fractional form 2.3.3 Linear equations in irrational form 2.4 Quadratic equations 2.4.1 Quadratic equations in fractional form 2.4.2 Quadratic equations in irrational form 2.5 Cubic equations 2.6 Quartic equations 2.6.1 General quartic equations 2.6.2 Biquadratic equations 2.6.3 Symmetric quartic equations 2.7 Equations of arbitrary degree 2.7.1 Polynomial division
18 19 20 20 20 21 22 22 23 23 24 25 25 26 27 28 29 30 32 33 33 34 34 35 36 37 37 37 39 39 39 40 40 41 41 41 42 42 42 43 43 44 44 44 46 46 46 46 47 47
Contents 2.8 2.9
ix
Fractional rational equations Irrational equations 2.9.1 Radical equations 2.9.2 Power equations Transcendental equations 2.10.1 Exponential equations 2.10.2 Logarithmic equations 2.10.3 Trigonometric (goniometric) equations Equations with absolute values 2.11.1 Equations with one absolute value 2.11.2 Equations with several absolute values Inequalities 2.12.1 Equivalence transformations for inequalities Numerical solution of equations 2.13.1 Graphical solution 2.13.2 Nesting of intervals 2.13.3 Secant methods and method of false position 2.13.4 Newton's method 2.13.5 Successive approximation
48 48 48 49 49 49 50 51 51 51 52 53 53 54 54 54 55 56 57
Geometry and trigonometry in the plane 3.1 Point curves 3.2 Basic constructions 3.2.1 Construction of the midpoint of a segment 3.2.2 Construction of the bisector of an angle 3.2.3 Construction of perpendiculars 3.2.4 To drop a perpendicular 3.2.5 Construction of parallels at a given distance 3.2.6 Parallels through a given point 3.3 Angles 3.3.1 Specification of angles 3.3.2 Types of angles 3.3.3 Angles between two parallels . .-. 3.4 Similarity and intercept theorems 3.4.1 Intercept theorems * 3.4.2 Division of a segment 3.4.3 Mean values 3.4.4 Golden Section 3.5 Triangles 3.5.1 Congruence theorems 3.5.2 Similarity of triangles 3.5.3 Construction of triangles 3.5.4 Calculation of a right triangle 3.5.5 Calculation of an arbitrary triangle 3.5.6 Relations between angles and sides of a triangle 3.5.7 Altitude 3.5.8 Angle-bisectors 3.5.9 Medians 3.5.10 Mid-perpendiculars, incircle, circumcircle, excircle 3.5.11 Area of a triangle 3.5.12 Generalized Pythagorean theorem
59 60 60 60 61 61 61 61 62 62 62 63 64 64 64 65 66 66 67 67 68 68 70 70 72 73 74 74 75 76 76
2.10
2.11
2.12 2.13
Contents
3.6
3.7
3.8
3.5.13 Angular relations 3.5.14 Sine theorem 3.5.15 Cosine theorem 3.5.16 Tangent theorem 3.5.17 Half-angle theorems 3.5.18 Mollweide's formulas 3.5.19 Theorems of sides 3.5.20 Isosceles triangle 3.5.21 Equilateral triangle 3.5.22 Right triangle 3.5.23 Theorem of Thales 3.5.24 Pythagorean theorem 3.5.25 Theorem of Euclid 3.5.26 Altitude theorem Quadrilaterals 3.6.1 General quadrilateral 3.6.2 Trapezoid 3.6.3 Parallelogram 3.6.4 Rhombus 3.6.5 Rectangle 3.6.6 Square 3.6.7 Quadrilateral of chords 3.6.8 Quadrilateral of tangents 3.6.9 Kite Regular n-gons (polygons) 3.7.1 General regular n-gons 3.7.2 Particular regular n-gons (polygons) Circular objects 3.8.1 Circle 3.8.2 Circular areas 3.8.3 Annulus, circular ring 3.8.4 Sector of a circle 3.8.5 Sector of an annulus . \ 3.8.6 Segment of a circle 3.8.7 Ellipse -
Solid geometry 4.1 General theorems . . 4.1.1 Cavalieri's theorem 4.1.2 Simpson's rule 4.1.3 Guldin's rules 4.2 Prism 4.2.1 Oblique prism 4.2.2 Right prism 4.2.3 Cuboid 4.2.4 Cube 4.2.5 Obliquely truncated n-sided prism 4.3 Pyramid 4.3.1 Tetrahedron 4.3.2 Frustum of a pyramid 4.4 Regular polyhedron
76 76 77 77 77 77 78 78 79 80 81 81 81 81 82 82 82 83 83 84 84 85 86 86 86 87 87 89 89 90 91 91 92 92 93 95 95 95 95 96 96 96 97 97 97 98 98 98 99 99
Contents 4.4.1 Euler's theorem for polyhedrons 4.4.2 Tetrahedron 4.4.3 Cube (hexahedron) 4.4.4 Octahedron 4.4.5 Dodecahedron 4.4.6 Icosahedron 4.5 Other solids 4.5.1 Prismoid, prismatoid 4.5.2 Wedge 4.5.3 Obelisk 4.6 Cylinder 4.6.1 General cylinder 4.6.2 Right circular cylinder 4.6.3 Obliquely cut circular cylinder 4.6.4 Segment of a cylinder 4.6.5 Hollow cylinder (tube) 4.7 Cone 4.7.1 Right circular cone 4.7.2 Frustum of a right circular cone 4.8 Sphere 4.8.1 Solid sphere 4.8.2 Hollow sphere 4.8.3 Spherical sector 4.8.4 Spherical segment (spherical cap) 4.8.5 Spherical zone (spherical layer) 4.8.6 Spherical wedge 4.9 Spherical geometry 4.9.1 General spherical triangle (Euler's triangle) 4.9.2 Right-angled spherical triangle 4.9.3 Oblique spherical triangle 4.10 Solids of rotation 4.10.1 Ellipsoid 4.10.2 Paraboloid of revolution 4.10.3 Hyperboloid of revolution 4.10.4 Barrel 4.10.5 Torus 4.11 Fractal geometry 4.11.1 Scaling invariance and self-similarity 4.11.2 Construction of self-similar objects 4.11.3 Hausdorff dimension 4.11.4 Cantor set 4.11.5 Koch's curve 4.11.6 Koch's snowflake 4.11.7 Sierpinski gasket 4.11.8 Box-counting algorithm 5 Functions 5.1 Sequences, series, and functions 5.1.1 Sequences and series 5.1.2 Properties of sequences, limits 5.1.3 Functions
xi 99 99 100 100 101 101 102 102 102 102 102 103 103 103 104 104 104 105 105 106 106 106 106 107 107 108 108 108 109 110 Ill Ill 112 112 112 113 113 113 113 113 114 114 115 115 116 117 117 117 119 120
xii
Contents
5.2
5.3
5.1.4 Classification of functions 5.1.5 Limit and continuity Discussion of curves 5.2.1 Domain of definition 5.2.2 Symmetry 5.2.3 Behavior at infinity 5.2.4 Gaps of definition and points of discontinuity 5.2.5 Zeros 5.2.6 Behavior of sign 5.2.7 Behavior of slope, extremes 5.2.8 Curvature 5.2.9 Point of inflection Basic properties of functions
122 123 124 124 124 125 126 127 127 128 129 129 130
Simple functions
137
5.4 5.5 5.6 5.7 5.8
137 139 143 147 150
Constant function Step function Absolute value function Delta function Integer-part function, fractional-part function
Integral rational functions
155
5.9 5.10 5.11 5.12 5.13 5.14
155 158 162 166 170 174 174 175 176 179 180 181 187
Linear function—straight line Quadratic function — parabola Cubic equation Power function of higher degree Polynomials of higher degree Representation of polynomials and particular polynomials 5.14.1 Representation by sums and products 5.14.2 Taylor series . . 5.14.3 Homer's scheme 5.14.4 Newton's interpolation polynomial 5.14.5 Lagrange polynomials 5.14.6 Bezier polynomials and splines 5.14.7 Particular polynomials
Fractional rational functions 5.15 5.16 5.17 5.18
Hyperbola 189 Reciprocal quadratic function 192 Power functions with a negative exponent 196 Quotient of two polynomials 200 5.18.1 Polynomial division and partial fraction decomposition . . . 203 5.18.2 Pade's approximation 205
Irrational algebraic functions 5.19 5.20 5.21
189
Square-root function Root function Power functions with fractional exponents
209 209 212 ...216
Contents
5.22
Roots of rational functions
xiii
219
Transcendental functions
228
5.23 5.24
Logarithmic function Expansion function
228 233
5.25
Exponential functions of powers
239
Hyperbolic functions
245
5.26 Hyperbolic sine and cosine functions 5.27 Hyperbolic tangent and cotangent function 5.28 Hyperbolic secant and hyperbolic cosecant functions Area hyperbolic functions
247 252 258 263
5.29 5.30 5.31
264 267 271
Area hyperbolic sine and hyperbolic cosine Area-hyperbolic tangent and hyperbolic cotangent Area-hyperbolic secant and hyperbolic cosecant
Trigonometric functions
274
5.32
278 287 292 294 300
5.33 5.34
Sine and cosine functions 5.32.1 Superpositions of oscillations 5.32.2 Periodic functions Tangent and cotangent functions Secant and cosecant
Inverse trigonometric functions
306
5.35 5.36 5.37
307 311 315
Inverse sine and cosine functions Inverse tangent and cotangent functions Inverse secant and cosecant functions
Plane curves
319
5.38
Algebraic curves of the n-th order 5.38.1 Curves of the second order 5.38.2 Curves of the third order 5.38.3 Curves of the fourth and higher order Cycloidal curves Spirals : Other curves
319 319 321 323 324 327 328
Vector analysis 6.1 Vector algebra 6.1.1 Vector and scalar 6.1.2 Particular vectors 6.1.3 Multiplication of a vector by a scalar 6.1.4 Vector addition 6.1.5 Vector subtraction 6.1.6 Calculating laws 6.1.7 Linear dependence/independence of vectors
331 331 331 332 332 333 333 333 334
5.39 5.40 5.41
xiv
Contents
6.2
6.3 6.4
7
6.1.8 Basis Scalar product or inner product 6.2.1 Calculating laws 6.2.2 Properties and applications of the scalar product 6.2.3 Schmidt's orthonormalization method 6.2.4 Direction cosine 6.2.5 Application hypercubes of vector analysis Vector product of two vectors 6.3.1 Properties of the vector product Multiple products of vectors 6.4.1 Scalar triple product
Coordinate systems 7.1 Coordinate systems in two dimensions 7.1.1 Cartesian coordinates 7.1.2 Polar coordinates 7.1.3 Conversions between two-dimensional coordinate systems . 7.2 Two-dimensional coordinate transformation 7.2.1 Parallel displacement (translation) 7.2.2 Rotation 7.2.3 Reflection 7.2.4 Scaling 7.3 Coordinate systems in three dimensions 7.3.1 Cartesian coordinates 7.3.2 Cylindrical coordinates 7.3.3 Spherical coordinates 7.3.4 Conversions between three-dimensional coordinate systems . 7.4 Coordinate transformation in three dimensions 7.4.1 Parallel displacement (translation) 7.4.2 Rotation 7.5 Application in computer graphics 7.6 Transformations 7.6.1 Object representation and object description 7.6.2 Homogeneous coordinates 7.6.3 Two-dimensional translations with homogeneous coordinates 7.6.4 Two-dimensional scaling with homogeneous coordinates . . 7.6.5 Three-dimensional translation with homogeneous coordinates 7.6.6 Three-dimensional scaling with homogeneous coordinates . 7.6.7 Three-dimensional rotation of points with homogeneous coordinates 7.6.8 Positioning of an object in space 7.6.9 Rotation of objects about an arbitrary axis in space 7.6.10 Animation 7.6.11 Reflections 7.6.12 Transformation of coordinate systems 7.6.13 Translation of a coordinate system 7.6.14 Rotation of a coordinate system about a principal axis . . . . 7.7 Projections 7.7.1 Fundamental principles
335 338 339 339 341 341 342 343 344 345 345 349 349 349 350 350 350 351 352 353 353 354 354 354 355 355 356 356 357 357 358 358 359 360 360 361 361 362 363 364 366 366 367 367 368 370 370
Contents
7.8
7.7.2 Parallel projection 7.7.3 Central projection 7.7.4 General formulation of projections Window/viewport transformation
8 Analytic geometry 8.1 Elements of the plane 8.1.1 Distance between two points 8.1.2 Division of a segment 8.1.3 Area of a triangle 8.1.4 Equation of a curve 8.2 Straight line 8.2.1 Forms of straight-line equations 8.2.2 Hessian normal form 8.2.3 Point of intersection of straight lines 8.2.4 Angle between straight lines 8.2.5 Parallel and perpendicular straight lines 8.3 Circle ,. 8.3.1 Equations of a circle 8.3.2 Circle and straight line 8.3.3 Intersection of two circles 8.3.4 Equation of the tangent to a circle 8.4 Ellipse 8.4.1 Equations of the ellipse 8.4.2 Focal properties of the ellipse 8.4.3 Diameters of the ellipse 8.4.4 Tangent and normal to the ellipse 8.4.5 Curvature of the ellipse 8.4.6 Areas and circumference of the ellipse 8.5 Parabola 8.5.1 Equations of the parabola 8.5.2 Focal properties of the parabola 8.5.3 Diameters of the parabola 8.5.4 Tangent and normal of the parabola 8.5.5 Curvature of a parabola 8.5.6 Areas and arc lengths of the parabola 8.5.7 Parabola and straight line 8.6 Hyperbola 8.6.1 Equations of the hyperbola 8.6.2 Focal properties of the hyperbola 8.6.3 Tangent and normal to the hyperbola 8.6.4 Conjugate hyperbolas and diameter 8.6.5 Curvature of a hyperbola 8.6.6 Areas of hyperbola 8.6.7 Hyperbola and straight line 8.7 General equation of conies 8.7.1 Form of conies 8.7.2 Transformation to principal axes 8.7.3 Geometric construction (conic section) 8.7.4 Directrix property 8.7.5 Polar equation
xv
370 373 374 376
,
377 377 377 377 378 378 378 379 380 381 381 382 382 382 383 383 384 384 384 385 385 385 386 386 387 387 388 388 388 389 389 389 390 390 391 392 392 392 393 393 393 394 394 395 395 396
xvi
Contents 8.8
8.9
8.10
8.11
8.12
9
Elements in space 8.8.1 Distance between two points 8.8.2 Division of a segment 8.8.3 Volume of a tetrahedron Straight lines in space 8.9.1 Parametric representation of a straight line 8.9.2 Point of intersection of two straight lines 8.9.3 Angle of intersection between two intersecting straight lines 8.9.4 Foot of a perpendicular (perpendicular line) 8.9.5 Distance between a point and a straight line 8.9.6 Distance between two lines Planes in space 8.10.1 Parametric representation of the plane 8.10.2 Coordinate representation of the plane 8.10.3 Hessian normal form of the plane 8.10.4 Conversions 8.10.5 Distance between a point and a plane 8.10.6 Point of intersection of a line and a plane 8.10.7 Angle of intersection between two intersecting planes . . . . 8.10.8 Foot of the perpendicular (perpendicular line) 8.10.9 Reflection 8.10.10 Distance between two parallel planes 8.10.11 Cut set of two planes Plane of the second order in normal form 8.11.1 Ellipsoid 8.11.2 Hyperboloid 8.11.3 Cone 8.11.4 Paraboloid 8.11.5 Cylinder General plane of the second order 8.12.1 General equation 8.12.2 Transformation to principal axes 8.12.3 Shape of a surface of the second order
Matrices, determinants, and systems of linear equations 9.1 Matrices 9.1.1 Row and column vectors 9.2 Special matrices 9.2.1 Transposed, conjugate, and adjoint matrices 9.2.2 Square matrices 9.2.3 Triangular matrices 9.2.4 Diagonal matrices 9.3 Operations with matrices 9.3.1 Addition and subtraction of matrices 9.3.2 Multiplication of a matrix by a scalar factor c 9.3.3 Multiplication of vectors, scalar product 9.3.4 Multiplication of a matrix by a vector 9.3.5 Multiplication of matrices 9.3.6 Calculating rules of matrix multiplication 9.3.7 Multiplication by a diagonal matrix
396 396 396 396 397 397 397 398 398 398 399 399 399 399 400 400 401 401 401 401 402 402 402 403 403 403 404 404 405 406 406 406 407 409 409 411 412 412 412 414 415 418 418 418 419 421 421 422 424
Contents
xvii
9.3.8 Matrix multiplication according to Falk's scheme 424 9.3.9 Checking of row and column sums 425 9.4 Determinants 426 9.4.1 Two-row determinants 427 9.4.2 General computational rules for determinants 427 9.4.3 Zero value of the determinant 429 9.4.4 Three-row determinants 430 9.4.5 Determinants of higher (n-th) order 432 9.4.6 Calculation of n-row determinants 433 9.4.7 Regular and inverse matrix 434 9.4.8 Calculation of the inverse matrix in terms of determinants . . 435 9.4.9 Rank of a matrix 436 9.4.10 Determination of the rank by means of minor determinants . 437 9.5 Systems of linear equations 437 9.5.1 Systems of two equations with two unknowns 439 9.6 Numerical solution methods 441 9.6.1 Gaussian algorithm for systems of linear equations 441 9.6.2 Forward elimination 441 9.6.3 Pivoting 443 9.6.4 Backsubstitution 444 9.6.5 LU-decomposition 445 9.6.6 Solvability of (m x n) systems of equations 448 9.6.7 Gauss-Jordan method for matrix inversion 450 9.6.8 Calculation of the inverse matrix A" 1 452 9.7 Iterative solution of systems of linear equations 454 9.7.1 Total-step methods (Jacobi) 456 9.7.2 Single-step methods (Gauss-Seidel) 456 9.7.3 Criteria of convergence for iterative methods 457 9.7.4 Storage of the coefficient matrix 458 9.8 Table of solution methods 459 9.9 Eigenvalue equations 461 9.10 Tensors 463 9.10.1 Algebraic operations with tensors 465 10 Boolean algebra-application in switching algebra 10.1 Basic notions 10.1.1 Propositions and truth values 10.1.2 Proposition variables 10.2 Boolean connectives 10.2.1 Negation: not 10.2.2 Conjunction: and 10.2.3 Disjunction (inclusive): or 10.2.4 Calculating rules 10.3 Boolean functions 10.3.1 Operator basis 10.4 Normal forms 10.4.1 Disjunctive normal forms 10.4.2 Conjunctive normal form 10.4.3 Representation of functions by normal forms 10.5 Karnaugh-Veitch diagrams 10.5.1 Producing a KV-diagram
'
467 467 467 468 468 469 469 469 470 471 472 472 472 473 473 475 476
xviii
Contents
10.6 10.7 -
10.5.2 Entering a function in a KV-diagram 10.5.3 Minimization with the help of KV-diagrams Minimization according to Quine and McCluskey Multi-valued logic and fuzzy logic 10.7.1 Multi-valued logic 10.7.2 Fuzzy logic
476 477 478 481 481 481
11 Graphs and Algorithms 11.1 Graphs 11.1.1 Basic definitions 11.1.2 Representation of graphs 11.1.3 Trees 11.2 Matchings 11.3 Networks 11.3.1 Flows in networks 11.3.2 Eulerian line and Hamiltonian circuit
483 483 483 485 485 486 487 487 487
12 Differential calculus 12.1 Derivative of a function 12.1.1 Differential 12.1.2 Differentiability 12.2 Differentiation rules 12.2.1 Derivatives of elementary functions 12.2.2 Derivatives of trigonometric functions 12.2.3 Derivatives of hyperbolic functions 12.2.4 Constant rule 12.2.5 Factor rule 12.2.6 Power rule 12.2.7 Sum rule 12.2.8 Product rule 12.2.9 Quotient rule 12.2.10 Chain rule 12.2.11 Logarithmic differentiation of functions 12.2.12 Differentiation of functions in parametric representation . . . 12.2.13 Differentiation of functions in polar coordinates 12.2.14 Differentiation of an implicit function 12.2.15 Differentiation of the inverse function 12.2.16 Table of differentiation rules 12.3 Mean value theorems 12.3.1 Rolle's theorem 12.3.2 Mean value theorem of differential calculus 12.3.3 Extended mean value theorem of differential calculus . . . . 12.4 Higher derivatives 12.4.1 Slope, extremes 12.4.2 Curvature 12.4.3 Point of inflection 12.5 Approximation method of differentiation 12.5.1 Graphical differentiation 12.5.2 Numerical differentiation 12.6 Differentiation of functions with several variables 12.6.1 Partial derivative
489 489 490 491 492 492 492 492 493 493 493 493 493 494 494 495 495 496 496 497 498 499 499 499 500 500 502 503 503 504 504 505 506 506
Contents
12.6.2 Total differential 12.6.3 Extremes of functions in two dimensions 12.6.4 Extremes with constraints 12.7 Application of differential calculus 12.7.1 Calculation of indefinite expressions 12.7.2 Discussion of curves 12.7.3 Extreme value problems 12.7.4 Calculus of errors 12.7.5 Determination of zeros according to Newton's method
...
xix
508 508 509 510 510 511 512 513 514
13 Differential geometry 13.1 Plane curves 13.1.1 Representation of curves 13.1.2 Differentiation by implicit representation 13.1.3 Differentiation by parametric representation 13.1.4 Differentiation by polar coordinates 13.1.5 Differential of arc of a curve 13.1.6 Tangent, normal 13.1.7 Curvature of a curve 13.1.8 Evolutes and evolvents 13.1.9 Points of inflection, vertices 13.1.10 Singular points 13.1.11 Asymptotes 13.1.12 Envelope of a family of curves 13.2 Space curves 13.2.1 Representation of space curves 13.2.2 Moving trihedral 13.2.3 Curvature 13.2.4 Torsion of a curve 13.2.5 Frenet formulas 13.3 Surfaces 13.3.1 Representation of a surface 13.3.2 Tangent plane and normal to the surface 13.3.3 Singular points of the surface
517 517 517 517 518 518 518 519 520 522 522 522 523 524 524 524 525 527 527 528 528 528 529 530
14 Infinite series 14.1 Series 14.2 Criteria of convergence 14.2.1 Special number series 14.3 Taylor and MacLaurin series 14.3.1 Taylor's formula 14.3.2 Taylor series 14.4 Power series 14.4.1 Test of convergence for power series 14.4.2 Properties of convergent power series 14.4.3 Inversion of power series 14.5 Special expansions of series and products 14.5.1 Binomial series 14.5.2 Special binomial series 14.5.3 Series of exponential functions 14.5.4 Series of logarithmic functions
531 531 532 535 535 535 536 537 537 538 540 540 540 540 541 542
xx
Contents 14.5.5 14.5.6 14.5.7 14.5.8 14.5.9 14.5.10
Series of trigonometric functions Series of inverse trigonometric functions Series of hyperbolic functions Series of area hyperbolic functions Partial fraction expansions Infinite products
15 Integral calculus 15.1 Definition and integrability 15.1.1 Primitive 15.1.2 Definite and indefinite integrals 15.1.3 Geometrical interpretation 15.1.4 Rules for integrability 15.1.5 Improper integrals 15.2' Integration rules 15.2.1 Rules for indefinite integrals 15.2.2 Rules for definite integrals 15.2.3 Table of integration rules 15.2.4 Integrals of some elementary functions 15.3 Integration methods 15.3.1 Integration by substitution 15.3.2 Integration by parts 15.3.3 Integration by partial fraction decomposition 15.3.4 Integration by series expansion 15.4 Numerical integration 15.4.1 Rectangular rule 15.4.2 Trapezoidal rule 15.4.3 Simpson's rule 15.4.4 Romberg integration 15.4.5 Gaussian quadrature 15.4.6 Table of numerical integration methods 15.5 Mean value theorem of integral calculus 15.6 Line, surface, and volume integrals 15.6.1 Arc length (rectification) 15.6.2 Area 15.6.3 Solid of rotation (solid of revolution) 15.7 Functions in parametric representation 15.7.1 Arc length in parametric representation 15.7.2 Sector formula 15.7.3 Solids of rotation in parametric representation 15.8 Multiple integrals and their applications 15.8.1 Definition of multiple integrals 15.8.2 Calculation of areas 15.8.3 Center of mass of arcs 15.8.4 Moment of inertia of an area 15.8.5 Center of mass of areas 15.8.6 Moment of inertia of planes 15.8.7 Center of mass of a body o 15.8.8 Moment of inertia of a body 15.8.9 Center of mass of rotational solids 15.8.10 Moment of inertia of rotational solids
542 543 544 544 544 545 547 547 547 548 549 550 551 552 552 553 554 555 557 557 560 562 565 567 567 568 568 569 570 572 574 574 574 575 576 577 577 578 578 579 579 580 581 581 582 582 582 583 583 583
Contents 15.9 Technical 15.9.1 15.9.2 15.9.3 15.9.4 15.9.5
applications of integral calculus Static moment, center of mass Mass moment of inertia Statics Calculation of work Mean values
xxi 584 584 585 588 588 589
16 Vector analysis 591 16.1 Fields 591 16.1.1 Symmetries of fields 592 16.2 Differentiation and integration of vectors 594 16.2.1 Scale factors in general orthogonal coordinates 596 16.2.2 Differential operators 597 16.3 Gradient and potential 598 16.4 Directional derivative and vector gradient 600 16.5 Divergence and Gaussian integral theorem 601 16.6 Rotation and Stokes's theorem 604 16.7 Laplace operator and Green's formulas 607 16.7.1 Combinations of div, rot, and grad; calculation of fields . . . 609 16.8 Summary 610 17 Complex variables and functions 17.1 Complex numbers 17.1.1 Imaginary numbers 17.1.2 Algebraic representation of complex numbers 17.1.3 Cartesian representation of complex numbers 17.1.4 Conjugate complex numbers 17.1.5 Absolute value of a complex number 17.1.6 Trigonometric representation of complex numbers 17.1.7 Exponential representation of complex numbers 17.1.8 Transformation from Cartesian to trigonometric representation 17.1.9 Riemann sphere 17.2 Elementary arithmetical operations with complex numbers 17.2.1 Addition and subtraction of complex numbers 17.2.2 Multiplication and division of complex numbers 17.2.3 Exponentiation in the complex domain 17.2.4 Taking the root in the complex domain 17.3 Elementary functions of a complex variable 17.3.1 Sequences in the complex domain 17.3.2 Series in the complex domain 17.3.3 Exponential function in the complex domain 17.3.4 Natural logarithm in the complex domain 17.3.5 General power in the complex domain 17.3.6 Trigonometric functions in the complex domain 17.3.7 Hyperbolic functions in the complex domain 17.3.8 Inverse trigonometric, inverse hyperbolic functions in the complex domain 17.4 Applications of complex functions 17.4.1 Representation of oscillations in the complex plane 17.4.2 Superposition of oscillations of equal frequency
613 613 613 614 614 615 615 616 616 617 618 619 619 619 622 623 623 624 625 626 626 627 627 629 630 631 631 632
xxii
Contents
17.5
17.6
17.4.3 Loci 17.4.4 Inversion of loci Differentiation of functions of a complex variable 17.5.1 Definition of the derivative in the complex domain 17.5.2 Differentiation rales in the complex domain 17.5.3 Cauchy-Riemann differentiability conditions 17.5.4 Conformal mapping Integration in the complex plane 17.6.1 Complex curvilinear integrals 17.6.2 Cauchy's integral theorem 17.6.3 Primitive functions in the complex domain 17.6.4 Cauchy's integral formulas 17.6.5 Taylor series of an analytic function 17.6.6 Laurent series 17.6.7 Classification of singular points 17.6.8 Residue theorem 17.6.9 Inverse Laplace transformation
18 Differential equations 18.1 General definitions 18.2 Geometric interpretation 18.3 Solution methods for first-order differential equations 18.3.1 Separation of variables 18.3.2 Substitution 18.3.3 Exact differential equations 18.3.4 Integrating factor 18.4 Linear differential equations of the first order 18.4.1 Variation of the constants 18.4.2 General solution 18.4.3 Determination of a particular solution 18.4.4 Linear differential equations of the first order with constant coefficients 18.5 Some specific equations 18.5.1 Bernoulli differential equation 18.5.2 Riccati differential equation 18.6 Differential equations of the second order 18.6.1 Simple special cases 18.7 Linear differential equations of the second order 18.7.1 Homogeneous linear differential equation of the second order 18.7.2 Inhomogeneous linear differential equations of the second order 18.7.3 Reduction of special differential equations of the second order to differential equations of the first order 18.7.4 Linear differential equations of the second order with constant coefficients 18.8 Differential equations ofthew-th order 18.9 Systems of coupled differential equations of the first order 18.10 Systems of linear homogeneous differential equations with constant coefficients 18.11 Partial differential equations
633 634 635 635 636 637 637 639 639 640 641 641 642 643 643 644 645 647 647 649 650 650 651 651 651 652 652 653 653 653 654 654 654 655 655 656 657 657 659 659 662 668 670 672
Contents
xxiii
18.11.1 Solution by separation 673 18.12 Numerical integration of differential equations 676 18.12.1 Euler method 676 18.12.2 Heun method 677 18.12.3 Modified Euler method 679 18.12.4~ Runge-Kutta methods 679 18.12.5 Runge-Kutta method for systems of differential equations . . 685 18.12.6 Difference method for the solution of partial differential equations 685 18.12.7 Method of finite elements 688 19 Fourier transformation 19.1 Fourier series 19.1.1 Introduction 19.1.2 Definition and coefficients 19.1.3 Condition of convergence 19.1.4 Extended interval 19.1.5 Symmetries 19.1.6 Fourier series in complex and spectral representation 19.1.7 Formulas for the calculation of Fourier series 19.1.8 Fourier expansion of simple periodic functions 19.1.9 Fourier series (table) 19.2 Fourier integrals 19.2.1 Introduction 19.2.2 Definition and coefficients 19.2.3 Conditions for convergence 19.2.4 Complex representation, Fourier sine and cosine transformation 19.2.5 Symmetries 19.2.6 Convolution and some calculating rales 19.3 Discrete Fourier transformation (DFT) 19.3.1 Definition and coefficients 19.3.2 Shannon scanning theorem 19.3.3 Discrete sine and cosine transformation 19.3.4 Fast Fourier transformation (FFT) 19.3.5 Particular pairs of Fourier transforms 19.3.6 Fourier transforms (table) 19.3.7 Particular Fourier sine transforms 19.3.8 Particular Fourier cosine transforms 19.4 Wavelet transformation 19.4.1 Signals 19.4.2 Linear signal analysis 19.4.3 Symmetry transformations 19.4.4 Time-frequency analysis and Gabor transformation 19.4.5 Wavelet transformation 19.4.6 Discrete wavelet-transformation 20 Laplace and z transformations 20.1 Introduction 20.2 Definition of the Laplace transformation 20.3 Transformation theorems
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691 691 691 691 693 694 696 698 699 699 705 707 707 707 708 708 710 710 712 712 713 714 715 720 720 722 723 724 724 725 726 727 728 732 735 735 736 737
xxiv 20.4
20.5
20.6
Contents Partial fraction separation 745 20.4.1 Partial fraction separation with simple real zeros 745 20.4.2 Partial fraction decomposition with multiple real zeros . . . 746 20.4.3 Partial fraction decomposition with complex zeros 747 Linear differential equations with constant coefficients 748 20.5.1 Laplace transformation: linear differential equation of the first order with constant coefficients 749 20.5.2 Laplace transformation: linear differential equation of the second order with constant coefficients 751 20.5.3 Example: linear differential equations 753 20.5.4 Laplace transforms (table) 756 z transformation 764 20.6.1 Definition of the z transformation 764 20.6.2 Convergence conditions for the z transformation 766 20.6.3 Inversion of the z transformation 767 20.6.4 Calculating rales 767 20.6.5 Calculating rales for the z transformation 770 20.6.6 Table of z transforms 770
21 Probability theory and mathematical statistics 21.1 Combinatorics 21.2 Random events 21.2.1 Basic notions 21.2.2 Event relations and event operations 21.2.3 Structural representation of events 21.3 Probability of events 21.3.1 Properties of probabilities 21.3.2 Methods to calculate probabilities 21.3.3 Conditional probabilities 21.3.4 Calculating with probabilities 21.4 Random variables and their distributions 21.4.1 Individual probability, density function and distribution function x 21.4.2 Parameters of distributions 21.4.3 Special discrete distribution 21.4.4 Special continuous distributions 21.5 Limit theorems 21.5.1 Laws of large numbers 21.5.2 Limit theorems 21.6 Multidimensional random variables 21.6.1 Distribution functions of two-dimensional random variables . 21.6.2 Two-dimensional discrete random variables 21.6.3 Two-dimensional continuous random variables 21.6.4 Independence of random variables 21.6.5 Parameters of two-dimensional random variables 21.6.6 Two-dimensional normal distribution 21.7 Basics of mathematical statistics 21.7.1 Description of measurements . . 21.7.2 Types of error 21.8 Parameters for describing distributions of measured values 21.8.1 Position parameter, means of series of measurements . . . .
773 773 774 774 775 777 778 778 778 779 779 781 782 783 785 793 800 800 801 802 802 803 804 805 806 807 808. 809 810 812 812
Contents 21.8.2 Dispersion parameter 21.9 Special distributions 21.9.1 Frequency distributions 21.9.2 Distribution of random sample functions 21.10 Analysis by means of random sampling (theory of testing and estimating) 21.10.1 Estimation methods 21.10.2 Construction principles for estimators 21.10.3 Method of moments 21.10.4 Maximum likelihood method 21.10.5 Method of least squares 21.10.6 x 2 -minimum method 21.10.7 Method of quantiles, percentiles 21.10.8 Interval estimation 21.10.9 Interval bounds for normal distribution 21.10.10 Prediction and confidence interval bounds for binomial and hypergeometric distributions 21.10.11 Interval bounds for a Poisson distribution 21.10.12 Determination of sample sizes n 21.10.13 Test methods 21.10.14 Parameter tests 21.10.15 Parameter tests for a normal distribution 21.10.16 Hypotheses about the mean value of arbitrary distributions 21.10.17 Hypotheses about p of binomial and hypergeometric distributions 21.10.18 Tests of goodness of fit 21.10.19 Application: acceptance/rejection test 21.11 Reliability 21.12 Computation of adjustment, regression 21.12.1 Linear regression, least squares method 21.12.2 Regression of the n-th order 22 Fuzzy logic 22.1 Fuzzy sets 22.2 Fuzzy concept 22.3 Functional graphs for the modeling of fuzzy sets 22.4 Combination of fuzzy sets 22.4.1 Elementary operations 22.4.2 Calculating rales for fuzzy sets 22.4.3 Rules for families of fuzzy sets 22.4.4 t norm and t conorm 22.4.5 Non-parametrized operators: t norms and s norms (t conorms) 22.4.6 Parametrized t and s norms 22.4.7 Compensatory operators 22.5 Fuzzy relations 22.6 Fuzzy inference 22.7 Denazification methods 22.8 Example: erect pendulum 22.9 Fuzzy realizations
xxv 814 815 815 816 820 821 823 823 824 824 825 825 826 828 829 830 830 831 834 834 836 837 837 838 839 841 843 844 847 847 848 849 852 852 855 856 856 858 859 860 861 863 864 866 870
xxvi
Contents
23 Neural networks 23.1 Function and structure 23.1.1 Function 23.1.2 Structure 23.2 Implementation of the neuron model ~ 23.2.1 Time-independent systems 23.2.2 Time-dependent systems 23.2.3 Application 23.3 Supervised learning 23.3.1 Principle of supervised learning 23.3.2 Standard backpropagation 23.3.3 Backpropagation through time 23.3.4 Improved learning methods 23.3.5 Hopfield network 23.4 Unsupervised learning 23.4.1 Principle of unsupervised learning 23.4.2 Kohonen model
871 871 871 872 873 873 873 874 874 874 876 877 878 879 881 881 881
24 Computers 24.1 Operating systems 24.1.1 Introduction to MS-DOS 24.1.2 Introduction to UNIX 24.2 High-level programming languages 24.2.1 Program structures 24.2.2 Object-oriented programming (OOP)
883 883 885 886 889 890 892
Introduction to PASCAL
893
24.3 24.4
894 894 895 895 895 895 896 897 898 899 900 900 901 902 903 904 905 905 906 906 908 909 909
24.5
24.6
24.7 24.8
Basic structure Variables and types 24.4.1 Integers 24.4.2 Real numbers . . . 24.4.3 Boolean values 24.4.4 ARRAYS \ 24.4.5 Characters and character strings 24.4.6 RECORD ' 24.4.7 Pointers 24.4.8 Self-defined types Statements 24.5.1 Assignments and expressions 24.5.2 Input and output 24.5.3 Compound statements 24.5.4 Conditional statements I F and CASE 24.5.5 Loops FOR, WHILE, and REPEAT Procedures and functions 24.6.1 Procedures 24.6.2 Functions 24.6.3 Local and global variables, parameter passing Recursion Basic algorithms 24.8.1 Dynamic data structures
Contents
24.9
24.8.2 Search 24.8.3 Sorting Computer graphics 24.9.1 Basic functions
Introduction to C 24.9.2 24.9.3 24.9.4 24.9.5
xxvii 910 911 913 913 914
Basic structures Operators Data structures Loops and branches
914 916 918 921
Introduction to C++
924
24.9.6 24.9.7 24.9.8 24.9.9 24.9.10 24.9.11 24.9.12 24.9.13 24.9.14 24.9.15
924 924 924 925 926 926 926 927 928 929
Variables and constants Overloading of functions Overloading of operators Classes Instantiation of classes f r i e n d functions Operators as member functions Constructors Derived classes (inheritance) Class libraries
Introduction to FORTRAN 24.9.16 24.9.17 24.9.18 24.9.19 24.9.20 24.9.21
Program structure Data structures Type conversion Operators Loops and branches Subprograms
Computer algebra 24.9.22 24.9.23 24.9.24 24.9.25 24.9.26 24.9.27 24.9.28 24.9.29 24.9.30
Structural elements of Mathematica Structural elements of Maple Algebraic expressions Equations and systems of equations Linear algebra Differential and integral calculus Programming Fitting curves and interpolation with Mathematica Graphics
930 930 930 931 933 933 934 937 937 940 942 943 944 945 947 948 949
25 Tables of integrals 951 25.1 Integrals of rational functions 951 25.1.1 Integrals with P - ax + b, a^0 951 25.1.2 Integrals with x" 1 /(ax + fc)\ P = ax + b,a ^ 0, F ^ 0 . . 952 25.1.3 Integrals with 1/(xn(ax + b)m), P = ax + b b ^ 0 . . . 953 25.1.4 Integrals with ax + b and ex + d c ^ 0 955
xxviii
Contents 25.1.5 25.1.6 25.1.7
25.2
25.3
25.4
Index
View
Integrals with a + x and b + x a =£ b 955 Integrals with P = ax2 + bx + c (a / 0) 956 Integrals with xn/(ax2 + bx + c)m, P = ax2 + bx+c a / 0 956 25.1.8 Integrals with l/(xn(ax2 + bx + c)m), P = ax2 + bx + c c/0 957 25.1.9 Integrals with P = a 2 ± x 2 958 25.1.10 Integrals with l / ( a 2 ± x 2 ) " , P = a 2 ± x 2 a / 0 . . . . 9 5 8 2 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 8 2 5 . 1 . 1 2 Integrals with 1/ (xn{a2 ±x2)m) P = a2±x2 a / 0 . . 960 3 3 25.1.13 Integrals with P = a ± x a / 0 961 25.1.14 Integrals with a4 + xA (a > 0) 962' 25.1.15 Integrals with a4 - x 4 (a > 0) 962 Integrals of irrational functions 963 25.2.1 Integrals with x 1 / 2 and P = ax + b a,b^0 963 25.2.2 Integrals with (ax + b)l/2 P = ax + b a / 0 964 25.2.3 Integrals with (ax + b)l/2 and (ex + d ) 1 / 2 , a, c / 0 . . . . 966 25.2.4 Integrals with R = (a2 + x2)1'2 a / 0 966 25.2.5 Integrals with S = (x 2 - a2)y'2 a # 0 968 25.2.6 Integrals with T = {a2- x2)x'2 a / 0 970 25.2.7 Integrals with (ax 2 + bx + c)l/2 X = ax2 + bx + c a / 0 972 Integrals of transcendental functions 973 25.3.1 Integrals with exponential functions 973 25.3.2 Integrals with logarithmic functions (x > 0) 975 25.3.3 Integrals with hyperbolic functions (a / 0) 977 25.3.4 Integrals with inverse hyperbolic functions 979 25.3.5 Integrals with sine and cosine functions (a / 0) 979 25.3.6 Integrals with sine and cosine functions (a / 0) 984 25.3.7 Integrals with tangent or cotangent functions (a / 0) . . . 989 25.3.8 Integrals with inverse trigonometric functions (a / 0) . . . 990 Definite integrals 992 25.4.1 Definite integrals with algebraic functions 992 25.4.2 Definite integrals with exponential functions 992 25.4.3 Definite integrals with logarithmic functions 994 25.4.4 Definite integrals with trigonometric functions 995 999
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