Mathematics Handbook for Science and Engineering

564 Pages·2004·16.781 MB·English

by  Lennart Råde, Bertil Westergren (auth.)

Rade . Westergren
Mathematics Handbook
Springer-Verlag Berlin Heidelberg GmbH
Lennart Rade · BertH Westergren
Mathematics
Handbook
for Science and Engineering
Fifth Edition
tJJ
, Springer Studentlitteratur
Lennart Râde
Bertil Westergren
© Lennart Râde, Bertil Westergren and Studentlitteratur, Box 141, SE-22100 Lund, Sweden 2004.
Joint1y Published witl! Student1itteratur, Lund, Sweden.lst and 2nd edition published by: Student1ittera
tur, Lund, Sweden; 3rd edition published by: Birkhauser, Basel, Switzerland; 4tl! edition published by:
Springer
ISBN 978-3-642-05936-0 ISBN 978-3-662-08549-3 (eBook)
DOI 10.1007/978-3-662-08549-3
Distribution rights for Denmark, Finland, Norway, Sweden, Iceland: Student1itteratur, Lund, Sweden.
Bibliographic information published by Die Deutsche Bibliotl!ek
Die Deutsche Bibliothek lists tl!is publication in tl!e Deutsche Nationalbibliografie;
detailed bibliographic data is available in tl!e Internet at <http://dnb.ddb.de>.
ISBN 978-3-642-05936-0
Matl!ematics Subject Classification (2000): 00A22
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springeronline.com
© Springer-Verlag Berlin Heidelberg 2004
Originally published by Springer-Verlag Berlin Heidelberg New York in 2004
Softcover reprint ofthe hardcover 5th edition 2004
Cover design: Erich Kirchner, Heidelberg
Printed on acid-free paper 40/3143 - 543210
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
3.7 Fractals 87
4 Linear Algebra 90
4.1 Matrices 90
4.2 Determinants 93
4.3 Systems of Linear Equations 95
4.4 Linear Coordinate Transformations 97
4.5 Eigenvalues. Diagonalization 98
4.6 Quadratic Forms 103
4.7 Linear Spaces 106
4.8 Linear Mappings 108
4.9 Tensors 114
4.10 Complex matrices 114
3
5 The Elementary Functions 118
5.1 A Survey of the Elementary Functions 118
5.2 Polynomials and Rational Functions 119
5.3 Logarithmic, Exponential, Power and Hyperbolic Functions 121
5.4 Trigonometric and Inverse Trigonometric Functions 125
6 Differential Calculus (one variable) 132
6.1 Some Basic Concepts 132
6.2 Limits and Continuity 133
6.3 Derivatives 136
6.4 Monotonicity. Extremes of Functions 139
7 Integral Calculus 141
7.1 Indefinite Integrals 141
7.2 Definite Integrals 146
7.3 Applications of Differential and Integral Calculus 148
7.4 Table of Indefinite Integral 153
7.5 Tables of Definite Integrals 178
8 Sequences and Series 183
8.1 Sequences of Numbers 183
8.2 Sequences of Functions 184
8.3 Series of Constant Terms 185
8.4 Series of Functions 187
8.5 Taylor Series 189
8.6 Special Sums and Series 192
9 Ordinary Differential Equations (ODE) 200
9.1 Differential Equations of the First Order 200
9.2 Differential Equations of the Second Order 202
9.3 Linear Differential Equations 205
9.4 Autonomous systems 213
9.5 General Concepts and Results 216
9.6 Linear Difference Equations 218
10 Multidimensional Calculus 221
10.1 The Space Rn 221
10.2 Surfaces. Tangent Planes 222
10.3 Limits and Continuity 223
10.4 Partial Derivatives 224
10.5 Extremes of Functions 227
10.6 Functions/: Rn ~ Rm (Rn ~Rn) 229
10.7 Double Integrals 231
4
10.8 Triple Integrals 234
10.9 Partial Differential Equations 239
11 Vector Analysis 246
11.1 Curves 246
11.2 Vector Fields 248
11.3 Line Integrals 253
11.4 Surface Integrals 256
12 Orthogonal Series and Special Functions 259
12.1 Orthogonal Systems 259
12.2 Orthogonal Polynomials 263
12.3 Bernoulli and Euler Polynomials 269
12.4 Bessel Functions 270
12.5 Functions Defined by Transcendental Integrals 287
12.6 Step and Impulse Functions 297
12.7 Functional Analysis 298
12.8 Lebesgue Integrals 303
12.9 Generalized functions (Distributions) 308
13 Transforms 310
13.1 Trigonometric Fourier Series 310
13.2 Fourier Transforms 315
13.3 Discrete Fourier Transforms 325
13.4 The z-transform 327
13.5 Laplace Transforms 330
13.6 Dynamical Systems (Filters) 338
13.7 Hankel and Hilbert transforms 341
13.8 Wavelets 344
14 Complex Analysis 349
14.1 Functions of a Complex Variable 349
14.2 Complex Integration 352
14.3 Power Series Expansions 354
14.4 Zeros and Singularities 355
14.5 Conformal Mappings 356
15 Optimization 365
15.1 Calculus of Variations 365
15.2 Linear Optimization 371
15.3 Integer and Combinatorial Optimization 379
15.4 Nonlinear Optimization 383
15.5 Dynamic Optimization 389
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16 Numerical Analysis 391
16.1 Approximations and Errors 391
16.2 Numerical Solution of Equations 392
16.3 Perturbation analysis 397
16.4 Interpolation 398
16.5 Numerical Integration and Differentiation 404
16.6 Numerical Solutions of Differential Equations 412
16.7 Numerical summation 421
17 Probability Theory 424
17.1 Basic Probability Theory 424
17.2 Probability Distributions 434
17.3 Stochastic Processes 439
17.4 Algorithms for Calculation of Probability Distributions 443
17.5 Simulation 445
17.6 Queueing Systems 449
17.7 Reliability 452
17.8 Tables 459
18 Statistics 479
18.1 Descriptive Statistics 479
18.2 Point Estimation 488
18.3 Confidence Intervals 491
18.4 Tables for Confidence Intervals 495
18.5 Tests of Significance 501
18.6 Linear Models 507
18.7 Distribution-free Methods 512
18.8 Statistical Quality Control 518
18.9 Factorial Experiments 522
18.10 Analysis oflife time (failure time) data 525
18.11 Statistical glossary 526
19 Miscellaneous 530
Glossary of functions 544
Glossary of symbols 545
Index 547
6
Preface
This is the fifth 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 profession
als working in these areas. The aim of the handbook is to provide useful informa
tion in a lucid and accessible form in a moderately large volume. The handbook
concentrates on definitions, results, formulas, graphs, figures and tables and emp
hasizes 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 multivariable calculus in different chapters, because students, usu
ally, 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 collea
gues. Our thanks are especially due to Johan Karlsson, Jan Petersson, Rolf Petters
son and Thomas Weibull. We also want to thank Magnus Bondesson, Christer
Borell, Juliusz Brzezinski, Kenneth Eriksson, Carl-Henrik Fant, Kjell Holmaker,
Lars Homstrom, Eskil Johnson, Martin Lindberg, Jacques de Mare, Bo Nilsson and
Jeffrey Steif 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 distribu
tions in section 18.4 (originally published in Eisenhart, et al: Techniques of Statisti
cal 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).
Lennart Rdde, Berti! Westergren
In the fifth edition of this handbook the sections 15.2 and 15.3 have been removed
and replaced by three sections 15.2 - 15.4. These new sections have been written
by Michael Patriksson of Chalmers University of Technology in Gothenburg,
Sweden. I want to thank Michael Patriksson for his valuable contribution to the
handbook. A new section on fractals has also been added to chapter 3. In other
chapters some changes and corrections have been made.
Since the former edition my dear friend and cowriter Lennart Riide has passed
away and myself have retired from the faculty of the Mathematics Department of
Chalmers University of Technology and the University of Gothenburg.
I shall be grateful for any suggestions about changes, additions, or deletions, as
well as corrections in the Mathematics handbook. It is finally my hope that many
users will find the Mathematics handbook a useful guide to the world of mathe
matics.
Berti! Westergren
8