Rade Westergren Mathematics Handbook

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1 Rade Westergren Mathematics Handbook

2 Springer-Verlag Berlin Heidelberg GmbH

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

4 Lennart Rade Bertil Westergren Lennart Rade, Berti! Westergren and Studentlitteratur, Box 141, SE Lund, Sweden 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 ISBN (ebook) DOI / ISBN 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 / o- Printed on acid-free paper

5 Contents Preface 7 1 Fundamentals. Discrete Mathematics Logic Set Theory Binary Relations and Functions Algebraic Structures Graph Theory Codes 37 2 Algebra Basic Algebra of Real Numbers Number Theory Complex Numbers Algebraic Equations 63 3 Geometry and Trigonometry Plane Figures Solids Spherical Trigonometry Geometrical Vectors Plane Analytic Geometry Analytic Geometry in Space 83 4 Linear Algebra Matrices Determinants Systems of Linear Equations Linear Coordinate Transformations Eigenvalues. Diagonalization Quadratic Forms Linear Spaces Linear Mappings Tensors Complex matrices The Elementary Functions A Survey of the Elementary Functions Polynomials and Rational Functions 116 3

6 5.3 Logarithmic, Exponential, Power and Hyperbolic Functions Trigonometric and Inverse Trigonometric Functions Differential Calculus (one variable) Some Basic Concepts Limits and Continuity Derivatives Monotonicity. Extremes of Functions Integral Calculus Indefinite Integrals Definite Integrals Applications of Differential and Integral Calculus Tables of Indefinite Integrals Tables of Definite Integrals Sequences and Series Sequences of Numbers Sequences of Functions Series of Constant Terms Series of Functions Taylor Series Special Sums and Series Ordinary Differential Equations (ODE) Differential Equations of the First Order Differential Equations of the Second Order Linear Differential Equations Autonomous systems General Concepts and Results Linear Difference Equations Multidimensional Calculus The Space Rn Surfaces. Tangent Planes Limits and Continuity Partial Derivatives Extremes of Functions Functionsf Rn ~ Rm (Rn ~Rn) Double Integrals Triple Integrals Partial Differential Equations Vector Analysis Curves Vector Fields 244 4

7 11.3 Line Integrals Surface Integrals Orthogonal Series and Special Functions Orthogonal Systems Orthogonal Polynomials Bernoulli and Euler Polynomials Bessel Functions Functions Defined by Transcendental Integrals Step and Impulse Functions Functional Analysis Lebesgue Integrals Generalized functions (Distributions) Transforms Trigonometric Fourier Series Fourier Transforms Discrete Fourier Transforms The z-transform Laplace Transforms Dynamical Systems (Filters) Hankel and Hilbert transforms Wavelets Complex Analysis Functions of a Complex Variable Complex Integration Power Series Expansions Zeros and Singularities Conformal Mappings Optimization Calculus of Variations Linear Optimization Non-linear Optimization Dynamic Optimization Numerical Analysis and Programming Approximations and Errors Numerical Solution of Equations Interpolation Numerical Integration and Differentiation Numerical Solutions of Differential Equations Numerical summation 404 5

8 17 Probability Theory Basic Probability Theory Probability Distributions Stochastic Processes Algorithms for Calculation of Probability Distributions Simulation Queueing Systems Reliability Tables Statistics Descriptive Statistics Point Estimation Confidence Intervals Tables for Confidence Intervals Tests of Significance Linear Models Distribution-free Methods Statistical Quality Control Factorial Experiments Analysis of life time (failure time) data Statistical glossary Miscellaneous Glossary of functions 527 Glossary of symbols 528 Index 531 6

9 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

10 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 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 ). 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

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