Rade Westergren Mathematics Handbook

Rade Westergren Mathematics Handbook

Springer-Verlag Berlin Heidelberg GmbH

Lennart Rade Bertil Westergren Mathematics Handbook for Science and Engineering Fourth Edition Springer ~ Studentlitteratur

Lennart Rade Bertil Westergren Lennart Rade, Berti! Westergren and Studentlitteratur, Box 141, SE-22100 Lund, Sweden 1998. Mathematics Subject Classification (1991): ooa22 Die Deutsche Bibliothek- CIP-Einheitsaufnahme Rade, Lennart: Mathematics handbook for science and engineering /Lennart Rade; Berti! Westergren. 4.ed. ISBN 978-3-662-03558-0 ISBN 978-3-662-03556-6 (ebook) DOI 10.1007/978-3-662-03556-6 ISBN 978-3-662-03558-o This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1999 Originally published by Springer-Verlag Berlin Heidelberg New York in 1999 Softcover reprint of the hardcover 4th edition 1999 Cover design: Erich Kirchner, Heidelberg SPIN 10690491 40/3143-54 3 2 1 o- Printed on acid-free paper

Contents Preface 7 1 Fundamentals. Discrete Mathematics 9 1.1 Logic 9 1.2 Set Theory 14 1.3 Binary Relations and Functions 17 1.4 Algebraic Structures 21 1.5 Graph Theory 33 1.6 Codes 37 2 Algebra 43 2.1 Basic Algebra of Real Numbers 43 2.2 Number Theory 49 2.3 Complex Numbers 61 2.4 Algebraic Equations 63 3 Geometry and Trigonometry 66 3.1 Plane Figures 66 3.2 Solids 71 3.3 Spherical Trigonometry 75 3.4 Geometrical Vectors 77 3.5 Plane Analytic Geometry 79 3.6 Analytic Geometry in Space 83 4 Linear Algebra 87 4.1 Matrices 87 4.2 Determinants 90 4.3 Systems of Linear Equations 92 4.4 Linear Coordinate Transformations 94 4.5 Eigenvalues. Diagonalization 95 4.6 Quadratic Forms 100 4.7 Linear Spaces 103 4.8 Linear Mappings 105 4.9 Tensors 110 4.10 Complex matrices 111 5 The Elementary Functions 115 5.1 A Survey of the Elementary Functions 115 5.2 Polynomials and Rational Functions 116 3

5.3 Logarithmic, Exponential, Power and Hyperbolic Functions 118 5.4 Trigonometric and Inverse Trigonometric Functions 122 6 Differential Calculus (one variable) 129 6.1 Some Basic Concepts 129 6.2 Limits and Continuity 130 6.3 Derivatives 132 6.4 Monotonicity. Extremes of Functions 135 7 Integral Calculus 137 7.1 Indefinite Integrals 137 7.2 Definite Integrals 142 7.3 Applications of Differential and Integral Calculus 144 7.4 Tables of Indefinite Integrals 149 7.5 Tables of Definite Integrals 174 8 Sequences and Series 179 8.1 Sequences of Numbers 179 8.2 Sequences of Functions 180 8.3 Series of Constant Terms 181 8.4 Series of Functions 183 8.5 Taylor Series 185 8.6 Special Sums and Series 188 9 Ordinary Differential Equations (ODE) 196 9.1 Differential Equations of the First Order 196 9.2 Differential Equations of the Second Order 198 9.3 Linear Differential Equations 201 9.4 Autonomous systems 209 9.5 General Concepts and Results 212 9.6 Linear Difference Equations 214 10 Multidimensional Calculus 217 10.1 The Space Rn 217 10.2 Surfaces. Tangent Planes 218 10.3 Limits and Continuity 219 10.4 Partial Derivatives 220 10.5 Extremes of Functions 223 10.6 Functionsf Rn ~ Rm (Rn ~Rn) 225 10.7 Double Integrals 227 10.8 Triple Integrals 230 10.9 Partial Differential Equations 235 11 Vector Analysis 242 11.1 Curves 242 11.2 Vector Fields 244 4

11.3 Line Integrals 249 11.4 Surface Integrals 252 12 Orthogonal Series and Special Functions 255 12.1 Orthogonal Systems 255 12.2 Orthogonal Polynomials 259 12.3 Bernoulli and Euler Polynomials 265 12.4 Bessel Functions 266 12.5 Functions Defined by Transcendental Integrals 283 12.6 Step and Impulse Functions 293 12.7 Functional Analysis 294 12.8 Lebesgue Integrals 299 12.9 Generalized functions (Distributions) 304 13 Transforms 306 13.1 Trigonometric Fourier Series 306 13.2 Fourier Transforms 311 13.3 Discrete Fourier Transforms 320 13.4 The z-transform 322 13.5 Laplace Transforms 325 13.6 Dynamical Systems (Filters) 333 13.7 Hankel and Hilbert transforms 336 13.8 Wavelets 339 14 Complex Analysis 344 14.1 Functions of a Complex Variable 344 14.2 Complex Integration 347 14.3 Power Series Expansions 349 14.4 Zeros and Singularities 350 14.5 Conformal Mappings 351 15 Optimization 360 15.1 Calculus of Variations 360 15.2 Linear Optimization 366 15.3 Non-linear Optimization 370 15.4 Dynamic Optimization 372 16 Numerical Analysis and Programming 374 16.1 Approximations and Errors 374 16.2 Numerical Solution of Equations 375 16.3 Interpolation 381 16.4 Numerical Integration and Differentiation 387 16.5 Numerical Solutions of Differential Equations 395 16.6 Numerical summation 404 5

17 Probability Theory 407 17.1 Basic Probability Theory 407 17.2 Probability Distributions 417 17.3 Stochastic Processes 422 17.4 Algorithms for Calculation of Probability Distributions 426 17.5 Simulation 428 17.6 Queueing Systems 432 17.7 Reliability 435 17.8 Tables 442 18 Statistics 462 18.1 Descriptive Statistics 462 18.2 Point Estimation 471 18.3 Confidence Intervals 474 18.4 Tables for Confidence Intervals 478 18.5 Tests of Significance 484 18.6 Linear Models 490 18.7 Distribution-free Methods 495 18.8 Statistical Quality Control 501 18.9 Factorial Experiments 505 18.10 Analysis of life time (failure time) data 18.11 Statistical glossary 509 19 Miscellaneous 513 508 Glossary of functions 527 Glossary of symbols 528 Index 531 6

Preface This is the fourth edition of the Mathematics handbook for science and engineering (BETA). Compared to the previous editions a number of additions and corrections have been made. The Mathematics handbook covers basic areas of mathematics, numerical analysis, probability and statistics and various applications. The handbook is intended for students and teachers of mathematics, science and engineering and for professionals working in these areas. The aim of the handbook is to provide useful information in a lucid and accessible form in a moderately large volume. The handbook concentrates on definitions, results, formulas, graphs, figures and tables and emphasizes concepts and methods with applications in technology and science. The Mathematics handbook is organised in 19 chapters starting with basic concepts in discrete mathematics and ending with chapters on probability and statistics and a miscellaneous chapter. Crossreferences and an extensive index help the user to find required information. We have not included numerical tables of functions which are available on most scientific calculators and pocket computers. We have treated one variable and multi variable calculus in different chapters, because students, usually, meet these areas in different courses. In formulating theorems and results sometimes all assumptions are not explicitely stated. We are happy to have been able to draw on the expertise of several of our colleagues. Our thanks are especially due to Johan Karlsson, Jan Petersson, Rolf Pettersson and Thomas Weibull. We also want to thank Christer Borell, Juliusz Brzezinski, Kenneth Eriksson, Carl-Henrik Fant, Kjell Holmaker, Lars Homstrom, Eskil Johnson, Jacques de Mare, Jeffrey Steif and Bo Nilsson for their helpful assistance. Furthermore we want to thank Jan Enger of the Royal Institute of Technology in Stockholm for providing new more exact tables of median ranks (section 18.1) and two-sided tolerance limits for the normal distribution (section 18.4). We also want to thank Seppo Mustonen of Helsinki University in Finland for providing us with an algorithm for the simulation of bivariate normal distributions (section 17.5) and Max Nielsen of Odense Teknikum in Denmark for an improved formula for approximation of the normal distribution function. 7

Some tables and graphs have been copied with permission from publishers, whose courtesy is here acknowledged. We are thus indebted to the American Statistical Association for permission to use the table of Gurland-Tripathis correction factors in section 18.2, the table of the Kolmogorov-Smimov test in section 18.7 and the tables for Bartlett's test and the use of Studentized range in section 18.5. For the last two tables we also have permission from Biometrika Trustees. Furthermore we are indebted to the American Society for Quality Control for permission to use the table for construction of single acceptance sampling control plans in section 18.8 (copyright 1952 American Society for Quality Control) to McGraw-Hill Book Company for permission to use the table on tolerance limits for the normal distributions in section 18.4 (originally published in Eisenhart, et al: Techniques of Statistical Analysis, 1947) and to Pergamon Press for permission to use the graph of the Erlang Loss Formula in section 17.6 ( orginally published in L. Kosten, Stochastic Theory of Service System, 1973). In the fourth edition of this handbook two new sections have been included and one section has been removed. A section on wavelets has been added to chapter 13. This new section has been written by Martin Lindberg of Chalmers University of Technology in Gothenburg, Sweden. A new section on autonomous systems has also been added to chapter 9. The former section 16.7 on programming has been removed. In other chapters a number of changes and corrections have been made. These have mostly been inspired by the German edition of the handbook (Springers Mathematische Formeln 1996, editor and author Peter Vachenauer, Springer and Studentlitteratur ISBN 3-540-62829-0). It is a pleasure for the authors to thank Martin Lindberg and Peter Vachenauer for their valuable contributions to the handbook. We shall be grateful for any suggestions about changes, additions, or deletions, as well as corrections in the Mathematics handbook. It is finally our hope that many users will find the Mathematics handbook a useful guide to the world of mathematics. Lennart Rade, Berti! Westergren 8