LN. Bronshtein K.A. Semendyayev. G. Musiol H. Muehlig

Handbook of Mathematics

Springer-Verlag Berlin Heidelberg GmbH

LN. Bronshtein . K.A. Semendyayev . G. Musiol. H. Muehlig

Handbook of Mathematics

With 745 Figures and 142 Tables

i Springer

Ilja N. Bronshtein t Konstantin A. Semendyayev t Gerhard Musiol Heiner Muehlig

Library of Congress Cataloging-in-Publication Data Spravochnik po matematike. English.

Handbook of mathematics I LN. Bronshtein ... let al.].-- 4th ed. p. cm.

Based on the 5th German ed. published by Wissenschaftlicher Verlag Harri Deutsch GmbH, Frankfurt am Main under title: Taschenbuch der Mathematik.

1. Mathematics--Handbooks, manuals, etc. L Bronshtein, LN. (!l'ia Nikolaevich) IL Title.

QA40.S6813 2003 51O--dc22

Based on tl1e Stl1 German edition ofBronstein/Semendjajew/MusioliMhlig "Taschenbuch der Mathematik", 2000. Published by Wissenschaftlicher Verlag Harri Deutsch GmbH, Frankfurt am Main.

ISBN 978-3-540-43491-7 ISBN 978-3-662-05382-9 (eBook) DOI 10.1007/978-3-662-05382-9

This work is subject to copyright. All rights are reserved, whetl1er tl1e whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, 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 permis-sion for use must always be obtained from Springer-Verlag. Violations are !iable for prosecution under German Copyright Law.

springeronline.com Springer-Verlag Berlin Heidelberg 2004 Originally published by Springer-Verlag Berlin Heidelberg New York in 2004.

The use of general descriptive names, registered names, trademarks, etc. in this pub!ication does not imply, even in the absence of a specific statement, that such names are exempt from tl1e relevant protective laws and regulations and therefore free for general use. The authors and the publisher makes no representation, express or imp!ied, with regard to the accuracy of the information contained in tl1is book and cannot accept any legal responsibilty or liability for any errors or omissions that may be made. Typesetting by autl10rs Final processing: PTP-Berlin Protago-TeX-Production GmbH, Berlin Cover-Design: design & production GmbH, Heidelberg Translation: Gabriela Szep, Budapest Printed on acid-free paper 62/3111 - 5 43 2 1 - SPIN 11331360

Preface to the Fourth English Edition The "Handbook of '\Iathematics" by thf' mathematician. 1. N. BRONSHTEIN and the engineer, K. A. SEMENDYAYEV was designed for engineers and students of technieal universities. It appeared for the first time in Russian and was widel~' distribnted both as a referenee book and as a text book for colleges and universities. It was later translated into German and the many editions have made it a permanent fixture in German-speaking countries, where generations of engineers, natural seientists and others in technical training or already working with applications of mathematics have used it. On behalf of the publishing housp Harri Deutsch, arevision and a substantially enlarged edition was prepared in 1992 by Gerhm'd !\Iusiol and Heiner '\!iihlig, with the goal of giving "Bronshtein" the mod-ern practical coverage requested by numerous students, university teachers and practitioners. The original style successfully used by the authors has been maintained. It ean be eharacterized as "short, easily understandable, comfortable to use, but featuring mathematical accuraey (at a level of detail consistent with the needs of engineers)"'. Since 2000, the revised and extended fifth German edition of the revision has )wen on the market. Acknowledging the suceess that "B RONSTEIN" has experieneed in the German-speaking countries, Springer-Verlag HeidelbergjGermany is publishing a fourth English edition, wh ich corresponds to the improved a]](1 extended fifth German edition. In the work at hand, the c1assical areas of Engineering l'vIathematics required for current practice are presented, such as "Arithmetic", "Fullctions", "Geometry", "Linear Algebra", "Algebra and Diserete :VIathematics". (inclueling "Logic", "Set Theory", "Classieal Aigebraic Structures", "Elementary Num-ber Theory". ,. Cryptology", "~Universal Algebra", "Boolean Algebra and Swich Algebra", "Algorithms of graph Theory". "Fuzzy Logic"), 'Differentiation", "Integral Calculus", "Differential Equations", "Calculus of Variations", "Linear Integral Equations", "Functional Analysis", "Vector Analysis and Vector Fields", "Function Theory", "Integral Transformations", "Probability Theory and IVlathemat-ieal Statistics". Fields of mathematics that have gained importance with regards to the inereasing mathematical modeling and penetration of technical and scientific proeesses also receive special atten-tion. Included amongst these chapters are "Stochastic Processes and Stochastic Chains" as weil as "Calculus of Errors", "Dynamical Systems anel Chaos", "Optimization", "Numerical Analysis". "Us-ing the Computer" ancl "Computer Algebra Systems" . The book is enhanced with over a thousand complementary illustrations and many tables. Special functions, series expansions, indefinite, definite and elliptic integrals as weil as integral transformations and statistical distributions are supplied in an extensive appendix of tables. In order to make the reference book more effective, clarity and fast access through a clear structure were the goals, especially through visual dues as weil as by a detailed technical index and colored tabs. An exteneleel bibliograph}' also directs users to further resomees. We would like to cordially thank all readers and professional colleagues who helped us with their valu-able statements, remarks ami suggestions on the German edition of the book during the revision pro-cess. Special thanks go to l\1rs. Professor Dr. Gabriela Szp (Budapest), who made this English debut version possible. Furthermore our thanks go to the co-authors for the critical treatIlHmt of their chap-ters.

Dresden, June 2003

Prof. Dr. GERHARD l\IIUSIOL Prof. Dr. HEINER l\IIHLIG

Coauthors Some chapters and sections originated through a cooperation with the co-authors (see next page).

'See Preface to the first russian edition

VI

Chapter or Section Spherical Trigonometry (3.4.1-3.4.3) Spherical Curves (3.4.3.4) Logic, Set Theory, Classic Algebraic Structures, Applications of Groups, Rings and Fields, Vektor Spaces (5.1, 5.2, 5.3, besides 5.3.4, 5.3.7.4-5.3.7.6) Groups Representations, Applications of Groups (5.3.4, 5.3.5.4-5.3.5.6) Elementary Number Theory, Cryptology, Graphs (5.4,5.5,5.8), Fuzzy-Logic (5.9) Non-Linear Partial Differential Equations, Solitonen (9.2.4) Linear Integral Equations (11) Functional Analyzis (12) Elliptic Functions (14.6) Dynamical Systems and Chaos (17) Optimization (18) Computer Algebra Systems (19.8.4, 20.)

Coauthor Dr. H. NICKEL, Dresden Prof. L. MARSOLEK, Berlin

Prof. Dr. J. BRUNNER, Dresden

Prof. Dr. R. REIF, Dresden

Doz. Dr. U. BAUMANN, Dresden Prof. Dr. A. GRAUEL, Soest

Prof. Dr. P. ZIESCHE, Dresden Dr. 1. STEINERT, Dsseldorf Prof. Dr. M. WEBER, Dresden Dr. N. M. FLEISCHER, Moscow Dr. V. REITMANN, Dresden Dr. 1. STEINERT, Dsseldorf Prof. Dr. G. FLACH, Dresden

Contents List of Tables

1 Arithmetic 1.1 Elementary Rules for Calculations . . . . . . . . . . . . .

1.1.1 Numbers ...................... . 1.1.1.1 Natural, Integer, and Rational Numbers . 1.1.1.2 Irrational and Transeendental Numbers 1.1.1.3 Real Numbers .... 1.1.1.4 Continued Fraetions 1.1.1.5 Commensurability

1.1.2 Methods for Proof .... 1.1.2.1 Direct Proof .. 1.1.2.2 Indirect Proof or Proofby Contradiction 1.1.2.3 l\IathematicalInduetion 1.1.2.4 Constructive Proof

1.1.3 Sums and Products 1.1.3.1 Sums........ 1.1.3.2 Products ..... .

1.1.4 Powers, Roots, and Logarithms . 1.1.4.1 Powers .. . 1.1.4.2 Roots ...... . 1.1.4.3 Logarithms ... . 1.1.4.4 Special Logarithms

1.1.5 Algebraic Expressions ... . l.l.5.1 Definitions .... . 1.1.5.2 Algebraic Expressions in Detail

1.1.6 Integral Rational Expressions ...... . 1.1.6.1 Representation in Polynomial Form 1.1.6.2 Factori~ing a Polynomial 1.1.6.3 Special Formulas ......... . 1.1.6.4 Billomial Theorem ........ .

Contents VII

XXXIX

1 1 1 1 2 2 3 4 4 5 5 5 6 6 6 7 7 7 8 9 9

1.1.6.5 Determination of the Greatest Commoll Divisor of Two Polynomials

10 10 11 11 11 11 12 12 14 14 14 15 15 17 17 18 18 18 19 19 19 19 20 20 20

1.1. 7 Rational Expressions . . . . . . . . . . . . . . . . . . 1.1.7.1 Reducing to the Simplest Form ....... . 1.1.7.2 Determination of the Integral Rational Part. 1.1.7.3 Decomposition into Partial Fractions 1.1.7.4 Transformations of Proportions

1.1.8 Irrational Expressions . . . 1.2 Finite Series. . ........ .

1.2.1 Definition of a Finite Series 1.2.2 Arithmetic Series .. 1.2.3 Geometrie Se ries .. 1.2.4 Special Finite Series . 1.2.5 Mean Values .....

1.2.5.1 Arithmetie Mean or Arithmetic Average 1.2.5.2 Geometrie Mean or Geometrie Average 1.2.5.3 Harmonie :'vIean . 1.2.5.4 Quadratic Mean ............ .

VIII Contents

1.2.3.5 Relations Between the :Yleans of Two Positive Values . 20

I 1.3 Business Mathematics. . . . . . . . . . . . . 21 1.3.1 Calculation ofInterest or Percentage . 21 1.3.2 Calculation of Compound Interest 22 1.3.2.1 Interest ..... . 22 1.3.2.2 Compound Interest 22

1.3.3 Amortization Calculus . . . 23 1.3.3.1 Amortization... 23 1.3.3.2 Equal Principal Repayments 23 1.3.3.3 Equal Annuities . 24

1.3.4 Annuity Calculations . . . . . . . . . 25 1.3.4.1 Annuities . . . . . . . . . 25 1.3.4.2 Future Amount of an Ordinary Annuity . 25 1.3.4.3 Balance after n Annuity Payments 25

1.3.5 Depreciation... 26 1.4 Inequalities......... 28

1.4.1 Pure Inequalities 28 1.4.1.1 Definitions. 28 1.4.1.2 Properties of Inequalities of Type land II . 29

1.4.2 Speciallnequalities . . . . . . . . . . . . . . . . . . 30 1.4.2.1 Triangle Inequality for Real Numbers . . . 30 1.4.2.2 Triangle Inequality for Complex Numbers . 30 1.4.2.3 Inequalities far Absolute Values of Differences of Real and Complex

:"Jumbers. . . . . . . . . . . . . . . . 30 1.4.2.4 Inequality for Arithmetic and Geometrie Means 30 1.4.2.5 Inequality far Arithmetic and Quadratic Means. . 30 1.4.2.6 Inequalities for Different Means of Real :'-Iumbers . 30 1.4.2.7 Bernoulli's Inequality . . . . 30 1.4.2.8 Binomiallnequality..... 31 1.4.2.9 Cauehy-Schwarz Inequality 31 1.4.2.10 Chebyshev Inequality . . . 31 1.4.2.11 Generalized Chebyshev Inequality 32 1.4.2.12 Hlder Inequality . . . . . . . . . 32 1.4.2.13 :V1inkowski Inequality . . . . . . . 32

1.4.3 Solution of Linear and Quadratie Inequalities 33 1.4.3.1 General Remarks . . . 33 1.4.3.2 Linear Inequalities .. . . . . 33 1.4.3.3 Quadratie Inequalities . . . . 33 1.4.3.4 General Case for Inequalities of Seeond Degree 33

1.5 Complex ~umbers. . . . . . . . . . . . . 34 1.5.1 Imaginary and Complex Numbers 34

1.5.1.1 Imaginary Unit . . 34 1.5.1.2 Complex Numbers .. 34

1.5.2 Geometrie Representation ... 34 1.5.2.1 Vector Representation 34 1.5.2.2 Equality of Complex Numbers 34 1.5.2.3 Trigonometrie Farm of Complex Numbers . 35 1.5.2.4 Exponential Form of a Complex Number 35 1.5.2.5 Conjugate Complex ~umbers 36

1.5.3 Caleulation with Complex Numbers 36 1.5.3.1 Addition and Subtraction 36 1.5.3.2 Multiplieation....... 36

1.5.3.3 Division. . ...... . 1.3.3.4 General Rules for the Basic Operations 1.5.3.3 Taking Powers of Complex Numbers .. 1.5.3.6 Taking of the n-th Root of a Complex Number

1.6 Algebraic and Transcendental Equations .......... . 1.6.1 Transforming Algebraic Equations to Normal Form.

1.6.1.1 Definition ............ . 1.6.1.2 Systems of n Algebraic Equations 1.6.1..3 Superftuous Roots ........ .

1.6.2 Equations of Degree at Most FOUf .... . 1.6.2.1 Equations of Degree One (Linear Equations) 1.6.2.2 Equations of Degree Two (Quadratic Equations) 1.6.2.3 Equations of Degree Three (Cubic Equations) 1.6.2.4 Equations of Degree FOUf . 1.6.2.5 Equations of Higher Degree ....... .

1.6.3 Equations of Degree n . 1.6.3.1 General Properties of Algebraic Equations 1.6.3.2 Equations with Real Coefficients ..... .

1.6.4 Redueing Transcendcntal Equations to Algebraic Equations 1.6.4.1 Definition....

2 Functions

1.6.4.2 Exponcntial Equations . 1.6.4.3 Logarithmic Equations . 1.6.4.4 Trigonometrie Equations 1.6.4.5 Equations with Hyperbolic Functions

2.1 Notion of Functions ..... . 2.1.1 Definition of a FUIlction .

2.1.1.1 Function .... 2.1.1.2 Real Functions 2.1.1.3 Functions of Several Variables 2.1.1.4 Complex Functions 2.1.1.5 Further Functions . 2.1.1. 6 Functionals 2.1.1.7 Functions and Mappings

2.1.2 ;'vlethods for Defining aReal Function 2.1.2.1 Defining a Function .... 2.1.2.2 Analytic Representation of a Function .

2.1.3 Certain Types of Functions .. 2.1.3.1 Monotone Functions 2.1.3.2 Bounded Functions 2.1.3.3 Even Functions 2.1.3.4 Odd Functions ... 2.1.3.5 Representation with Even and Odd Functions . 2.1.3.6 Periodic Functions 2.1.3.7 Inverse Functions .. . ...... .

2.1. 4 Limi ts of Functions . . . . .. ....... . 2.1.4.1 Definition of the Limit of a Function . 2.1.4.2 Definition by Limit of Sequences .. 2.1.4.3 Cauchy Conditioll for Convergence . 2.1.4.4 Infinity as a Limit of a Function ... 2.1.4.5 Left-Hand and Right-Hand Limit of a Function .

Cuntents IX

37 37

37 38 38 38 38 38 39 39 39 39 40 42 43 43 43 44 45 45 45 46 46 46

47 47 47 47 47 47 47 47 47 48 48 48 48 49 49 50 50 50 50 50 51 51 51 52 52 52 52

I

X Contents

2.1.4.6 Limit of a Function as x Tends to Infinity 2.1.4.7 Theorems About Limits ofFunctions .. 2.1.4.8 Calculation of Limits . . . . . . . . . . . 2.1.4.9 Order of Magnitude of Functions and Landau Order Symbols

2.1.5 Continuity of a Function ............ . 2.1.5.1 Notion of Continuity and Discontinuity 2.1.5.2 Definition of Continuity ....... . 2.1.5.3 Most Frequent Types of Discontinuities 2.1.5.4 Continuity and Discontinuity of Elementary Functions . 2.1.5.5 Properties of Continuous Functions

2.2 Elementary Functions .... . 2.2.1 Algebraic Functions .... .

2.2.1.1 Polynomials ... . 2.2.1.2 Rational Functions 2.2.1.3 Irrational Functions .

2.2.2 Transcendental Functions ... 2.2.2.1 Exponential Functions 2.2.2.2 Logarithmic Functions 2.2.2.3 Trigonometrie Functions 2.2.2.4 Inverse Trigonometrie Functions 2.2.2.5 Hyperbolic Functions ..... 2.2.2.6 Inverse Hyperbolic Functions .

2.2.3 Composite Functions 2.3 Polynomials . . . . . . .

2.3.1 Linear Function . . . . 2.3.2 Quadratie Polynomial. 2.3.3 Cubic Polynomials .. 2.3.4 Polynomials of n-th Degree . 2.3.5 Parabola of n-th Degree "

2.4 Rational Functions .. . . . . . . . 2.4.1 Special Fractional Linear Function (Inverse Proportionality) . 2.4.2 Linear Fractional Function ... 2.4.3 Curves of Third Degree, Type I. . 2.4.4 Curves of Third Degree, Type II . 2.4.5 Curves ofThird Degree, Type III . 2.4.6 Reciprocal Powers . . . . . . . . .

2.5 Irrational Functions . . . . . . . . . . . . 2.5.1 Square Root of a Linear Binomial 2.5.2 Square Root of a Quadratic Polynomial 2.5.3 Power Function ............ .

2.6 Exponential Functions and Logarithmic Functions 2.6.1 Exponential Functions 2.6.2 Logarithmie Functions 2.6.3 Error Curve ..... . 2.6.4 Exponential Sum .. . 2.6.5 Generalized Error Function . 2.6.6 Product of Power and Exponential Functions

2.7 Trigonometrie Functions (Funetions of Angles) 2.7.1 Basie Notion ................. .

2.7.1.1 Definition and Representation .. . 2.7.1.2 Range and Behavior of the Functions

2.7.2 Important Formulas for Trigonometrie Functions

53 53 54 55 57 57 57 57 58 59 60 60 60 61 61 61 61 61 61 61 62 62 62 62 62 62 63 63 64 64 64 65 65 66 67 68 69 69 69 70 71 71 71 72 72 73 74 74 74 74 77 79

Contents XI

2.7.2.1 Relations Between the Trigonometrie Functions of the Same Angle (Addition Theorems) ........................ .

2.7.2.2 Trigonometrie Functions of the Sum and Differenee of Two Angles 2.7.2.3 Trigonometrie Functions of an Integer Multiple of an Angle 2.7.2.4 Trigonometrie Funetions of Half-Angles ...... . 2.7.2.5 Sum and Difference of Two Trigonometrie Funetions 2.7.2.6 Produets ofTrigonometrie Funetions 2.7.2.7 Powers ofTrigonometric Funetions

2.7.3 Deseription of Oseillations ...... . 2.7.3.1 Formulation of the Problem .. 2.7.3.2 Superposition of Oseillations .. 2.7.3.3 Vector Diagram for Oseillations 2.7.3.4 Damping of Oseillations ....

2.8 Inverse Trigonometrie Functions . . . . . . . . . 2.8.1 Definition of the Inverse Trigonometrie Functions . 2.8.2 Reduetion to the Principal Value .... 2.8.3 Relations Between the Prineipal Values .. 2.8.4 Formulas for Negative Arguments ..... 2.8.5 Sum and Differenee of aresin x and aresin y 2.8.6 Sum and Differenee of areeos x and arceos y 2.8.7 Sum and Differenee of aretanx and arctan y 2.8.8 Special Relations for aresin x, arecos x, aretan x

2.9 Hyperbolie Functions ................. . 2.9.1 Definition of Hyperbolie Functions ...... . 2.9.2 Graphieal Representation of the Hyperbolie Functions

2.9.2.1 Hyperbolie Sine ... 2.9.2.2 Hyperbolie eosine .. 2.9.2.3 Hyperbolic Tangent .. 2.9.2.4 Hyperbolie Cotangent

2.9.3 Important Formulas for the Hyperbolie Functions . 2.9.3.1 Hyperbolie Functions of One Variable .. 2.9.3.2 Expressing a Hyperbolie Function by Another One with the Same

Argument .............................. . 2.9.3.3 Formulas for Negative Arguments ................. . 2.9.3.4 Hyperbolie Functions of the Sum and Differenee of Two Arguments

(Addition Theorems) ............ . 2.9.3.5 Hyperbolie Functions of Double Arguments. 2.9.3.6 De '\1oivre Formula for Hyperbolic Functions 2.9.3.7 Hyperbolie Functions of Half-Argument ... 2.9.3.8 Sum and Differenee of Hyperbolie Functions 2.9.3.9 Relation Between Hyperbolie and Trigonometrie Funetions with Com-

plex Arguments z 2.10 Area Funetions

2.10.1 Definitions ..... . 2.10.1.1 Area Sine . 2.10.1.2 Area Cosine 2.10.1.3 Area Tangent 2.10.1.4 Area Cotangent

2.10.2 Determination oE Area Functions Using Natural Logarithm 2.10.3 Relations Between Different Area Functions . 2.10.4 Sum and Differenee of Area Funetions 2.10.5 Formulas for Negative Arguments ..... .

79 79 79 80 81 81 82 82 82 82 83 83 84 84 84 85 86 86 86 86 87 87 87 88 88 88 88 89 89 89

89 89

89 90 90 90 90

91 91 91 91 91 92 92 92 93 93 93

I

XII Contents

2.11 Curves ofOrder Three (Cubic Curves) 2.11.1 Semicubic Parabola 2.11.2 "Vitch of Agnesi . . . . . . . . 2.11.3 Cartesian Folium (Folium of Descartes) 2.11.4 Cissoid ......... . 2.11.5 Strophoide ... .

2.12 Curves of Order Four (Quartics) 2.12.1 Conchoid of Nicomedes 2.12.2 General Conchoid 2.12.3 Pascal's Limac;on 2.12.4 Cardioid ..... 2.12.5 Cassinian Curve . 2.12.6 Lemniscate ... .

2.13 Cycloids ..... . 2.13.1 Common (Standard) Cycloid 2.13.2 Prolate and Curtate Cycloids or Trochoids 2.13.3 Epicycloid ................. . 2.13.4 Hypocycloid and Astroid ......... . 2.13.5 Prolate and Curtate Epicycloid and Hypocycloid

2.14 Spirals .......... . 2.14.1 Archimedean Spiral 2.14.2 Hyperbolic Spiral . 2.14.3 Logarithmic Spiral 2.14.4 Evolvent of the Circle 2.14.5 Clothoid ....

2.15 Various Other Curves . 2.15.1 Catenary Curve 2.15.2 Tractrix .

2.16 Determination of Empirical Curves . 2.16.1 Procedure ......... .

2.16.1.1 Curve-Shape Comparison 2.16.1.2 Rectification ....... . 2.16.1.3 Determination of Parameters.

2.16.2 Useful Empirical Formulas .. . 2.16.2.1 Power Functions .. . 2.16.2.2 Exponential Functions 2.16.2.3 Quadratic Polynomial 2.16.2.4 Rational Linear Function . 2.16.2.5 Square Root of a Quadratic Polynomial 2.16.2.6 General Error Curve ..... . 2.16.2.7 Curve ofOrder Three, Type II . 2.16.2.8 Curve of Order Three, Type III 2.16.2.9 Curve of Order Three, Type I . 2.16.2.10 Product ofPower and Exponential Functions 2.16.2.11 Expon8ntial Sum .. 2.16.2.12 Numerical Example .

2.17 Scales and Graph Paper. 2.17.1 Sc ales ............ . 2.17.2 Graph Paper ......... .

2.17.2.1 Semilogarithmic Paper 2.17.2.2 Double Logarithmic Paper 2.17.2.3 Graph Paper with a Reciprocal Scale

93 93 94 94 95 95 96 96 96 96 98 98 99

100 100 100 101 102 102 103 103 104 104 104 105 105 105 106 106 106 106 107 107 107 108 108 109 109 110 110 110 111 111 112 112 112 114 114 115 115 115 116

Contents XIII

2.17.2.4 Remark ...... . 2.18 Functions of Several Variables ... .

2.18.1 Definition ana Representation 2.18.1.1 Representation of Functions of Several Variables 2.18.1.2 Geometrie Representation of Functions of Several Variables

2.18.2 Different Domains in the Plane ... . 2.18.2.1 Domain of a Function ........ . 2.18.2.2 Two-Dimensional Domains ..... . 2.18.2.3 Three or ~1ultidimensional Domains . 2.18.2.4 ~!ethods to Determine a Function . 2.18.2.5 Varius Ways to Define a Function . 2.18.2.6 Dependence of Functions

2.18.3 Limits .......... . 2.18.3.1 Definition ...... . 2.18.3.2 Exact Definition ... . 2.18.3.3 Generalization for Several Variables 2.18.3.4 Iterateel Limit ...... .

2.18.4 Continuity ............... . 2.18.5 Properties f ContillUous Functions ..

2.18.5.1 Theorem Oll Zeros ofBolzano. 2.18.5.2 Intermediate Value Theorem. 2.18.5.3 Theorem About the Bouneleelness of a Function . 2.18.5.4 Weierstrass Theorem (About the Existence of Maximum and Mini-

mum) 2.19 Nomography ...

2.19.1 :\"omograms .. 2.19.2 :\"et Charts ... 2.19.3 Alignment Charts

2.19.3.1 Alignment Charts with Three Straight-Line Seales Through a Point 2.19.3.2 Alignment Charts with Two Parallel and One Inclineel Straight-Line

&~ ................................ . 2.19.3.3 Alignrnent Charts with Two Parallel Straight Lines and a Curveel

Srale . .. . .......... . 2.19.4 :\"et Charts for !\Iore Than Three Variables ................. .

3 Geornetry 3.1 Plane Geometry .............. .

3.1.1 Basie Notation ...... . 3.1.1.1 Point, Line, Ray, Segment 3.1.1.2 Angle ........ . 3.1.1.3 Angle Between Two Intersecting Lines . 3.1.1.4 Pairs of Angles with Intersecting Paralleis . 3.1.1.5 Angles Measureel in Degrees and in Radians .

3.1.2 Geometrieal Definition of Circular anel Hyperbolic Functions . 3.1.2.1 Definition of Circular or Trigonometrie Functions . 3.1.2.2 Definitions of the Hyperbolic Functions

3.1.3 Plane Triangles .............. . 3.1.3.1 Statements ab out Plane Triangles 3.1.3.2 Symmetry ..

3.1.4 Plane QuaeIrangles ...... . 3.1.4.1 Parallelogram. 3.1.4.2 Rectangle and Square.

116 117 117 117 117 118 118 118 118 119 120 121 122 122 122 122 122 122 123 123 123 123

123 123 123 123 124 125

125

126 127

128 128 128 128 128 128 129 130 130 130 131 131 131 132 134 134 135

I

XIV Contents

3.1.4.3 Rhombus ..... . 3.1.4.4 Trapezoid...... 3.1.4.5 General Quadrangle 3.1.4.6 Inscribed Quadrangle . 3.1.4.7 Circumscribing Quadrangle

3.1.5 Polygons in the Plane ....... . 3.1.5.1 General Polygon ..... . 3.1.5.2 Regular Convex Polygons .. 3.1.5.3 Some Regular Convex Polygons

3.1.6 The Cirde and Related Shapes ..... . 3.1.6.1 Circle . . . . . . . . . . . . . . 3.1.6.2 Circular Segment and Circular Sector 3.1.6.3 Annulus

3.2 Plane Trigonometry . . . . . . . . . . . . . . . . . . . 3.2.1 Triangles .................... .

3.2.1.1 Calculations in Right-Angled Triangles in the Plane 3.2.1.2 Calculations in General Triangles in the Plane

3.2.2 Geodesie Applications ..... . 3.2.2.1 Geodetic Coordinates .. . 3.2.2.2 Angles in Geodesy ... . 3.2.2.3 Applications in Surveying

3.3 Stereometry............ 3.3.1 Lines and Planes in Space . 3.3.2 Edge, Corner, Solid Angle. 3.3.3 Polyeder or Polyhedron . . 3.3.4 Solids Bounded by Curved Surfaces

3.4 Spherical Trigonometry . . . . . . . . . . . 3.4.1 Basic Concepts of Geometry on the Sphere

3.4.1.1 Curve, Are, and Angle on the Sphere 3.4.1.2 Special Coordinate Systems 3.4.1.3 Spherical Lune or Biangle 3.4.1.4 Spherical Triangle . . . . . . 3.4.1. 5 Polar Triangle . . . . . . . . 3.4.1.6 Euler Triangles and Non-Euler Triangles 3.4.1. 7 Trihedral Angle ....... .

3.4.2 Basic Properties of Spherical Triangles ...... . 3.4.2.1 General Statements . . . . . . . . . . . . 3.4.2.2 Fundamental Formulas and Applications 3.4.2.3 Further Formulas ............ .

3.4.3 Calculation of Spherical Triangles . . . . . . . . . 3.4.3.1 Basic Problems, Accuracy Observations. 3.4.3.2 Right-Angled Spherical Triangles .... 3.4.3.3 Spherical Triangles with Oblique Angles . 3.4.3.4 Spherical Curves . . . .

3.5 Vector Algebra and Analytical Geometry 3.5.1 Vector Algebra ......... .

3.5.1.1 Definition of Vectors .. 3.5.1.2 Calculation Rules for Vectors . 3.5.1.3 Coordinates of a Vector ... . 3.5.1.4 Directional Coefficient ... . 3.5.1.5 Scalar Product and Vector Product 3.5.1.6 Combination ofVector Products ..

135 135 136 136 136 137 137 137 138 138 138 140 140 141 141 141 141 143 143 145 147 150 150 150 151 154 158 158 158 160 161 161 162 162 163 163 163 164 166 167 167 168 169 172 180 180 180 181 182 183 183 184

Contents XV

3.5.1.7 Vector Equations ................... . 3.5.1.8 Covariant and Contravariant Coordinates of a Vector 3.5.1.9 Geometrie Applieations of Vector Algebra .... .

3.5.2 Analytical Geometry of the Plane ............. . 3.5.2.1 Basic Concepts, Co ordinate Systems in the Plane. 3.5.2.2 Coordinate Transformations . 3.5.2.3 Special Notation in the Plane 3.5.2.4 Line 3.5.2.5 Circle .. . 3.5.2.6 Ellipse .. . 3.5.2.7 Hyperbola . 3.5.2.8 Parabola .. 3.5.2.9 Quadratic Curves (Curves of Second Order or Conie Sections) .

3.5.3 Analytical Geometry of Space ............ . 3.5.3.1 Basic Concepts, Spatial Co ordinate Systems 3.5.3.2 Transformation of Orthogonal Coordinates 3.5.3.3 Special Quantities in Space ......... . 3.5.3.4 Line and Plane in Space .......... . 3.5.3.5 Surfaces of Second Order, Equations in Normal Form. 3.5.3.6 Surfaces of Second Order or Quadratic Surfaces, General Theory

3.6 Differential Geometry . . . . . . . . . . . . . . 3.6.1 Plane Curves ............. .

3.6.1.1 Ways to Define a Plane Curve 3.6.1.2 Local Elements of a Curve 3.6.1.3 Special Points of a Curve .. . 3.6.1.4 Asymptotes of Curves ... . 3.6.1.5 General Discussion of a Curve Given by an Equation 3.6.1.6 Evolutes alld Evolvellts .... 3.6.1.7 Envelope of a Family of Curves .

3.6.2 Space Curves ..... . ...... . 3.6.2.1 Ways to Define aSpace Curve 3.6.2.2 Moving Trihedral ... 3.6.2.3 Curvature and Torsion .

3.6.3 Surfaces ............. . 3.6.3.1 ''''ays to Define a Surface 3.6.3.2 Tangent Plane and Surface Normal 3.6.3.3 Line Elements of a Surface ..... 3.6.3.4 Curvature of a Surfaee . . . . . . . 3.6.3.5 Ruled Surfaces and Developable Surfaces 3.6.3.6 Geodesic Lines on a Surface ...... .

4 Linear Algebra 4.1 Matrices ....... .

4.1.1 Notion of Matrix 4.1.2 Square ~Iatrices . 4.1.3 Vectors ..... . 4.1.4 Arithmetical Operations with Matrices 4.1.5 Rules of Calculation for Matrices . 4.1.6 Vector and Matrix Norms .

4.1.6.1 Vector Norms . 4.1.6.2 ~latrix Norms .

4.2 Determinants

187 187 189 189 189 190 191 194 197 198 200 203 205 207 207 210 212 214 220 223 225 225 225 225 231 234 235 236 237 238 238 238 240 243 243 244 245 247 250 250

251 251 251 252 253 254 257 258 258 259 259

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XVI Contents

4.2.1 Definitions.... 259 4.2.2 Subdeterminants 259 4.2.3 Rules of Calculation for Determinants 260 4.2.4 Evaluation of Determinants . 261

4.3 Tensors..................... 262 4.3.1 Transformation of Co ordinate Systems. 262 4.3.2 Tensors in Cartesian Coordinates. 262 4.3.3 Tensors with Special Properties 264

4.3.3.1 Tensors of Rank 2 . . . . 264 4.3.3.2 Invariant Tensors . . . . 265

4.3.4 Tensors in Curvilinear Co ordinate Systems 266 4.3.4.1 Covariant and Contravariant Basis Vectors 266 4.3.4.2 Coyariant and Contravariant Coordinates of Tensors of Rank 1 266 4.3.4.3 Cm'ariant, Contrayariant and Mixed Coordinates of Tensors of Rank

2 . . . . . . . . . . . 267 4.3.4.4 Rules of Calculation ......... 268

4.3.5 Pseudotensors.................. 268 4.3.5.1 Symmetry with Respect to the Origin 269 4.3.5.2 Introduction to the Notion of Pseudotensors 270

4.4 Systems of Linear Equations . . 271 4.4.1 Linear Systems, Pivoting 271

4.4.1.1 Linear Systems 271 4.4.1.2 Pivoting 271 4.4.1.3 Linear Dependenee 272 4.4.1.4 Calculation of the Inverse of a Matrix 272

4.4.2 Solution of Systems of Linear Equations 272 4.4.2.1 Definition and Solvability . 272 4.4.2.2 Application of Piyoting . 274 4.4.2.3 Cramer's Rule . . . . . . . 275 4.4.2.4 Gauss's Algorithm .. . . 276

4.4.3 Overdetermined Linear Equation Systems 277 4.4.3.1 Overdetermined Linear Systems ofEquations and Linear Mean Square

Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .. 277 4.4.3.2 Suggestions for Numerical Solutions of Mean Square Value Problems 278

4.5 Eigenvalue Problems for Matrices . 278 4.5.1 General Eigenvalue Problem . . . . 278 4.5.2 Special Eigenvalue Problem. . . . . 278

4.5.2.1 Characteristic Polynomial 278 4.5.2.2 Real Symmetrie Matrices, Similarity Transformations 280 4.5.2.3 Transformation of Principal Axes of Quadratic Forms 281 4.5.2.4 Suggestions for the Numerical Calculations of Eigenvalues 283

4.5.3 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . 285

5 Algebra and Discrete Mathematics 5.1 Logic .............. .

5.1.1 Propositional Calculus .... . 5.1.2 Formulas in Predicate Calculus .

5.2 Set Theory ............. . 5.2.1 Concept of Set, Special Sets. 5.2.2 Operations with Sets .... 5.2.3 Relations and Mappings .. 5.2.4 Equivalence and Order Relations.

286 286 286 289 290 290 291 294 296

Contenls XVII

5.2.5 Cardinality of Sets .. 298 5.3 Classical Algebraic Structures 298

5.3.1 Operations. 298 5.3.2 Semigroups 299 5.3.3 Groups . 299 5.3.3.1 Definition and Basic Properties 299 5.3.3.2 Subgroups and Direct Products 300 5.3.3.3 1'vJ.appings Between Groups 302

5.3.4 Group Representations 303 5.3.4.1 Definitions. 303 5.3.4.2 Particular Representations 303 5.3.4.3 Direct Sum of Representations 305 5.3.4.4 Direet Product of Reprcsentations . 305 5.3.4.5 Reducible and Irreducible Representations 305 5.3.4.6 Schur's Lemma 1 306 5.3.4.7 Clebsch-Gordan Se ries 306 5.3.4.8 Irreducible Representations of the Symmetrie Group SM 306

5.3.5 Applications of Groups 307 5.3.5.1 Symmetry Operations, Symmetry Elements. 307 5.3.5.2 Symmetry Groups or Point Groups 308 5.3.5.3 Symmetry Operations with Moleeules 308 5.3.5.4 Symmetry Groups in Crystallography 310 5.3.5.5 Symmetry Groups in Quantum Mechanics 312 5.3.5.6 Further Applications of Group Theory in Physics . 312

5.3.6 Rings and Fields . 313 5.3.6.1 Definitions. 313 5.3.6.2 Subrings, Ideals 313 5.3.6.3 Hornomorphism, Isomorphism, Homomorphism Theorem 314

5.3.7 Vector Spaces 314 5.3.7.1 Definition 314 5.3.7.2 Linear Dependence 315 53.7.3 Linear Mappings 315 5.3.7.4 Subspaces, Dimension Formula . 315 5.3.7.5 Euclidean Vector Spaces, Euclidean Norm. 316 5.3.7.6 Linear Operators in Vector Spaces 316

5.4 Elementary Number Theory 318 5.4.1 Divisibilit~ 318

5.4.1.1 Divisibility and Elementary Divisibility Rules . 318 5.4.1.2 Prime N umbers 318 5.4.1.3 Criteria for Divisibility 320 5.4.1.4 Greatest Common Divisor and Least Common Multiple 321 5.4.1.5 Fibonacci Numbers 323

5.4.2 Linear Diophantine Equations 323 5.4.3 Congruences and Residne Classes 325 5.4.4 Theorems of Fermat, Euler, and Wilson 329 5.4.5 Codes 329

5.5 Cryptology . 332 5.5.1 Problem of Cryptology 332 5.5.2 Cryptosystems . 332 5.5.3 :Ylathematic:al F oundation 332 5.5.4 Security of Cryptosysterns 333

5.5.4.1 IVlethods of Conventional Cryptography 333

I

XVIII Contents

5.5.4.2 Linear Substitution Ciphers 5.5.4.3 Vigenere Cipher ..... . 5.5.4.4 Matrix Substitution ... .

5.5.5 Methods of Classical Cryptanalysis 5.5.5.1 Statistical Analysis .. 5.5.5.2 Kasiski-Friedman Test

5.5.6 One-Time Pad ......... . 5.5.7 Public Key Methods ...... .

5.5.7.1 Diffie-Hellman Key Exchange 5.5.7.2 One-Way Function ..... . 5.5.7.3 RSA Method ........ .

5.5.8 DES Algorithm (Data Encryption Standard) 5.5.9 IDEA Algorithm (International Data Encryption Algorithm)

5.6 Universal Algebra ............... . 5.6.1 Definition . . . . . . . . . . . . . . . . 5.6.2 Congruence Relations, Factor Aigebras 5.6.3 Homomorphism ..... . 5.6.4 Homomorphism Theorem . . . 5.6.5 Varieties ........... . 5.6.6 Term Aigebras, Free Aigebras

5.7 800lean Algebras and Switch Algebra 5.7.1 Definition ....... . 5.7.2 Duality Principle ...... . 5.7.3 Finite Boolean Algebras .. . 5.7.4 Boolean Aigebras as Orderings 5.7.5 Boolean Functions, Boolean Expressions . 5.7.6 Normal Forms ... . 5.7.7 Switch Algebra ........ .

5.8 Algorithms of Graph Theory . . . . . . 5.8.1 Basic Notions and Notation ... 5.8.2 Traverse of Undirected Graphs .

5.8.2.1 Edge Sequences or Paths 5.8.2.2 Euler Trails ..... 5.8.2.3 Hamiltonian Cycles . .

5.8.3 Trees and Spanning Trees . 5.8.3.1 Trees...... 5.8.3.2 Spanning Trees

5.8.4 Matchings ....... . 5.8.5 Planar Graphs ..... . 5.8.6 Paths in Directed Graphs . 5.8.7 Transport Networks ... .

5.9 Fuzzy Logic ........... . 5.9.1 Basic Notions of Fuzzy Logic

5.9.1.1 Interpretation of Fuzzy Sets 5.9.1.2 Membership Functions on the Real Line ... 5.9.1.3 Fuzzy Sets ............... .

5.9.2 Aggregation of Fuzzy Sets .............. . 5.9.2.1 Concepts for Aggregation of Fuzzy Sets .. . 5.9.2.2 Practical Aggregator Operations of Fuzzy Sets 5.9.2.3 Compensatory Operators. 5.9.2.4 Extension Principle .. 5.9.2.5 Fuzzy Complement ....

334 334 334 335 335 335 336 336 336 337 337 337 338 338 338 338 339 339 339 339 340 340 341 341 341 342 343 344 346 346 349 349 350 351 352 352 353 354 355 355 356 358 358 358 359 361 363 363 364 366 366 366

Contents XIX

5.9.3 Fuzzy-Valued Relations ........ . 5.9.3.1 Fuzzy Relations ....... . 5.9.3.2 Fuzzy Product Relation R 0 S

5.9.4 Fuzzy Inference (Approximate Reasoning) . 5.9.5 Defuzzification Methods ..... 5.9.6 Knowledge-Based Fuzzy Systems.

5.9.6.1 :vlethod of Mamdani 5.9.6.2 Method of Sugeno ... . 5.9.6.3 Cognitive Systems .. . 5.9.6.4 Knowledge-Based Interpolation Systems

6 Differentiation 6.1 Differentiation of Functions of One Variable ......... .

6.1.1 Differential Quotient ................. . 6.1.2 Rules of Differentiation for Functions of One Variable.

6.1.2.1 Derivatives ofthe Elementary Functions . 6.1.2.2 Basic Rules of Differentiation ..... .

6.1.3 Derivatives of Higher Order. . . . . . . . . . . . . 6.1.3.1 Definition of Derivatives of Higher Order 6.1.3.2 Derivatives of Higher Order of some Elementary Functions . 6.1.3.3 Leibniz's Formula ...................... . 6.1.3.4 Higher Derivatives of Functions Given in Parametric Form. 6.1.3.5 Derivatives of Higher Order of the Inverse Function .

6.1.4 Fundamental Theorems of Differential Calculus . 6.1.4.1 Monotonicity ... 6.1.4.2 Fermat's Theorem .......... . 6.1.4.3 Rolle's Theorem ............ . 6.1.4.4 Mean Value Theorem of Differential Calculus 6.1.4.5 Tavlor's Theorem of Functions of One Variable 6.1.4.6 Ge~eralized Mean Value Theorem of Differential Calculus (Cauchy's

Theorem) ..................... . 6.1.5 Determination of the Extreme Values and Infiection Points ......... .

6.1.5.1 Maxima and Minima ........................ . 6.1.5.2 Necessary Conditions for the Existence of a Relative Extreme Value 6.1.5.3 Relative Extreme Values of a Differentiable, Explicit Function . 6.1.5.4 Determination of Absolute Extrema ........ . 6.1.5.5 Determination of the Extrema of Implicit Functions

6.2 Differentiation of Fllnctions of Several Variables. 6.2.1 Partial Derivatives ...................... .

6.2.1.1 Partial Derivative of a Fllnction .......... . 6.2.1.2 GeometricalMeaning for Functions of Two Variables. 6.2.1.3 Differentials of x and f(x) .... . 6.2.1.4 Basic Properties of the Differential .... . 6.2.1.5 Partial Differential ............ .

6.2.2 Total Differential and Differentials of Higher Order. 6.2.2.1 \'otion of Total Differential of a Function of Several Variables (Com-

plete Differential) .................. . 6.2.2.2 Derivatives and Differentials of Higher Order ... . 6.2.2.3 Taylor's Theorem for Functions of Several Variables

6.2.3 Rules of Differentiation for Functions of Several Variables 6.2.3.1 Differentiation of Composite Fnnctions 6.2.3.2 Differentiation of Implicit Fllnctions ...

367 367 369 370 371 372 372 373 373 375

377 377 377 378 378 378 383 383 383 383 385 385 386 386 386 386 387 387

388 388 388 388 389 390 390 390 390 390 391 391 392 392 392

392 393 394 395 395 396

I

XX Gantent"

6.:2.4 Substitution ofVariables in Differential Expressions ami Coordinate Transfor-mations 397 6.2.4.1 Fundion of One Variable. . . . . . . . 397 6.2.4.2 Function of Two Variables . 398

6.2.5 Extreme Values of Functions of Several Variables 399 6.2.5.1 Definition................ 399 6.2 .. 5.2 Geometrie Representation . . . . . . . 399 6.2.5.3 Determination of Extreme \'alues of Functions of Two Variables. 400 6.2.5.4 Determination of the Extreme Values of a Funetion of n Variables. 400 6.2.5.3 Solution of Approximation Problems ..... 401 6.2.5.6 Extreme \'alue Problem with Siele Conditions. . . . . . . . . . .. 401

7 Infinite Series 7.1 Sequenees ofNllmbcrs.

7.1.1 Properties of Seqllenees of 0Iumbers 7.1.1.1 Definition of Sequence of Numbers . 7.1.1.2 :'\[onotone Sequences of N umbers 7.1.1.3 Bouneled Sequences ..

7.1.2 Limits of Sequences of Numbers 7.2 l\umber Scries ..

7.2.1 General Convergence Theorems 7.2.1.1 Com'ergence and Divergence ofInfinite Series . 7.2.1.2 General Theorems ab out the Convergence ofSeries .

7.2.2 Convergence Criteria for Scries with Positive Terms. 7.2.2.1 Comparison Criterion .. 7.2.2.2 D'Alembert's Ratio Test 7.2.2.3 Root Test of Cauchy 7.2.2.4 Integral Test of Cauchy .

7.2.3 Absolute anel Conelitional Convergence 7.2.3.1 Definition 7.2.3.2 Properties of Absolutely Convergent Series 7.2.3.3 Alternating Scries ...... .

7.2.4 Some Special Series . . .. . ....... . 7.2.4.1 The Values of Some Important. Number Series 7.2.4.2 Bernoulli anel Euler Numbers

7.2.5 Estimation of thc Remainder ..... . 7.2.5.1 Estimation with Majorant .. 7.2.5.2 Alternating Convergent Series 7.2.5.3 Special Series

7.3 Function Series 7.3.1 Definitions ...... . 7.3.2 Uniform Convergence .

7.3.2.1 Definition, Weierstrass Theorem 7.3.2.2 Properties of lJniformly Convergent Series

7.3.3 Power series ............... . 7.3.3.1 Definition, Convergence ......... . 7.3.3.2 Calculations with Power Series ...... . 7.3.3.3 Taylor Se ries Expansion, Maclaurin Series .

7.3.4 Approximation Formulas 7.3.5 Asymptotic Power Series .....

7.3.5.1 Asymptotic Behavior .. 7.3.3.2 Asymptotic Power Series

402 402 402 402 402 402 403 404 404 404 404 405 405 405 406 406 407 407 407 408 408 408 410 411 411 412 412 412 412 412 412 413 411 4'.4 4H 415 416 4;:7 417 418

Contents XXI

7.4 Fourier Series .... 7.4.1 Trigonometrie Sum and Fourier Series

7.4.1.1 Basic Not.ions ....... . 7.4.1.2 :\108t Important Properties of the Fourier Series

7.4.2 Determination of Coefficients for Symmetrie Functions 7.4.2.1 Different Kinds of Symmetries ......... . 7.4.2.2 Forms ofthe Expansion into a Fourier Series ..

7.4.3 Determination of the Fourier Coefficients with Numerical Methods 7.4.4 Fourier Series and Fourier Integrals ....... . 7.4.5 Remarks on the Table of Some Fourier Expansions ........ .

8 Integral Calculus 8.1 Indefini te Integrals

8.1.1 Primitive Function or Antiderivative . 8.1.1.1 Indefinite Integrals ..... 8.1.1.2 Integrals of Elementary Functions

8.1.2 Rules of Integration ......... . 8.1.3 Integration of Rational Functions .... .

8.1.3.1 Integrals of Integer Rational Functions (Polynomials) 8.1.3.2 Integrals of Fractional Rational Functions .. . 8.1.3.3 Four Cases of Partial Fraction Decomposition .... .

8.1.4 Integration of Irrational Functions ............... . 8.1.4.1 Suhst.itution to Reduce to Integration of Rational Functions 8.1.4.2 Integration of Binomial Integrands . 8.1.4.3 Elliptic Integrals ...... .

8.1.5 Int.egration of Trigonometrie Functions 8.1.5.1 Substitution ......... . 8.1.5.2 Simplified Methods .

8.1.6 Integration of FUl"ther Transcendental Functions 8.1.6.1 Integrals with Exponential Functions 8.1.6.2 Integrals with Hyperbolic Functions . 8.1.6.3 Application ofIntegration by Part.s . 8.1.6.4 Integrals of Transcendent.al Functions

8.2 Definite Integrals ..... . 8.2.1 Basic Notions, Rules and Theorems ..... .

8.2.1.1 Definition and Existence of the Definite Integral 8.2.1.2 Properties of Definite Integrals.. . ..... . 8.2.1.3 Further Theorems about the Limits ofIntegration 8.2.1.4 Evaluation of the Definite Integral ........ .

8.2.2 Application of Definite Integrals . . .. . ........ . 8.2.2.1 General Principles for Application of the Definite Integral 8.2.2.2 Applications in Geometry ..... . 8.2.2.3 Applications in :'vlechanics and Physics ..

8.2.3 Improper Integrals. Stieltjes and Lebesgue Integrals 8.2.3.1 Generalizatioll of the Notion of the Integral 8.2.3.2 Integrals with Infinite Integration Limits 8.2.3.3 Integrals with l:nhounded Integrand.

8.2.4 Parametric Integrals .............. . 8.2.4.1 Definition of Parametrie Integrals . 8.2.4.2 Differentiation Under the Symbol ofIntegration 8.2.4.3 Integration Under the Symbol of Integration ..

8.2.G Integration by Series Expansion, Special Non-Elementary Functions .

418 418 418 419 420 420 421 422 422 423

425 425 425 426 426 427 430 430 430 430 433 433 434 435 436 436 436 437 437 438 438 438 438 438 438 439 441 443 445 445 446 449 451 451 452 454 457 457 457 457 458

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XXII Contents

8.3 Line Integrals . . . . . . . . . . . . . . 8.3.1 Line Integrals of the First Type .

8.3.1.1 Definitions....... 8.3.1.2 Existence Theorem .. 8.3.1.3 Evaluation ofthe Line Integral ofthe First Type 8.3.1.4 Application of the Line Integral of the First Type.

8.3.2 Line Integrals of the Second Type 8.3.2.1 Definitions...................... 8.3.2.2 Existence Theorem ................ . 8.3.2.3 Calculation of the Line Integral of the Second Type.

8.3.3 Line Integrals of General Type . . . . . . . . . . . . . . 8.3.3.1 Definition.................... 8.3.3.2 Properties of the Line Integral of General Type 8.3.3.3 Integral Along a Closed Curve ........ .

8.3.4 Independence of the Line Integral of the Path of Integration 8.3.4.1 Two-Dimensional Case ..... . 8.3.4.2 Existence of a Primitive Function ... . 8.3.4.3 Three-Dimensional Case ........ . 8.3.4.4 Determination of the Primitive Function 8.3.4.5 Zero-Valued Integral Along a Closed Curve

8.4 Multiple Integrals ............... . 8.4.1 Double Integrals .............. .

8.4.1.1 Notion of the Double Integral .. 8.4.1.2 Evaluation ofthe Double Integral 8.4.1.3 Applications of the Double Integral

8.4.2 Tripie Integrals ............. . 8.4.2.1 Notion of the Tripie Integral .. . 8.4.2.2 Evaluation ofthe Tripie Integral. 8.4.2.3 Applications of the Tripie Integral

8.5 Surface Integrals. . . . . . . . . . . . . . . . . . . 8.5.1 Surface Integral of the First Type .....

8.5.1.1 Notion ofthe Surface Integral ofthe First Type. 8.5.1.2 Evaluation of the Surface Integral of the First Type 8.5.1.3 Applications of the Surface Integral of the First Type

8.5.2 Surface Integral of the Second Type . . . . . . . . . . . . . 8.5.2.1 Notion of the Surface Integral of the Second Type 8.5.2.2 Evaluation of Surface Integrals of the Second Type 8.5.2.3 An Application ofthe Surface Integral ...... .

460 461 461 461 461 462 462 462 464 464 465 465 465 466 466 466 467 467 467 468 469 469 469 470 472 474 474 474 477 477 477 478 479 480 481 481 482 484

9 Differential Equations 485 9.1 Ordinary Differential Equations ............ 485

9.1.1 First-Order Differential Equations . . . . . . . 486 9.1.1.1 Existence Theorems, Direction Field . 486 9.1.1.2 Important Solution Methods . . . . . 487 9.1.1.3 Implicit Differential Equations . . . . 490 9.1.1.4 Singular Integrals and Singular Points. 491 9.1.1.5 Approximation Methods for Solution ofFirst-Order Differential Equa-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 494 9.1. 2 Differential Equations of Higher Order and Systems of Differential Equations 495

9.1.2.1 Basic Results . . . . . . . . . . . . . . . 495 9.1.2.2 Lowering the Order. . . . . . . . . . . . 497 9.1.2.3 Linear n-th Order Differential Equations 498

Contents XXIII

9.1.2.4 Solution of Linear Differential Equations with Constant Coefficients 500 9.1.2.5 Systems of Linear Differential Equations with Constant Coeffieients 503 9.1.2.6 Linear Second-Order Differential Equations . 505

9.1.3 Boundary Value Problems ....................... 512 9.1.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . 512 9.1.3.2 Fundamental Properties of Eigenfunctions and Eigenvalues 513 9.1.3.3 Expansion in Eigenfunctions 514 9.1.3.4 Singular Cases. .. ...... 514

9.2 Partial Differential Equations. . . . . . . . . . . 515 9.2.1 First-Order Partial Differential Equations 515

9.2.1.1 Linear First-Order Partial Differential Equations 515 9.2.1.2 Non-Linear First-Order Partial Differential Equations 517

9.2.2 Linear Seeond-Order Partial Differential Equations . . . . . . . 520 9.2.2.1 Classifieation and Properties of Second-Order Differential Equations

with Two Independent Variables. . . . . . . . . . . . . . . . . .. 520 9.2.2.2 Classifieation and Properties ofLinear Second-Order Differential Equa-

tions with more than two Independent Variables . . . . . . . . .. 521 9.2.2.3 Integration }.Iethods for Linear Second-Order Partial Differential Equa-

tions . . . . . . . . . . . . . .. .................. 522 9.2.3 Some further Partial Differential Equations From :\fatural Seien ces and Engi-

neering . . . . . . . . . . .. .............. 532 9.2.3.1 Formulation of the Problem and the Boundary Conditions . . . .. 532 9.2.3.2 VVave Equation . . . . . . .. ................... 534 9.2.3.3 Heat Conduetion and Diffusion Equation for Homogencous Media. 535 9.2.3.4 Potential Eqllation . . . . . . . .. ................ 536 9.2.3.5 Sehrdinger's Equation . . . . . . . . . . . . . . . . . . . . . . .. 536

9.2.4 N on-Linear Partial Differential Equations: Solitons, Periodic Patterns and Chaos 544 9.2.4.1 Formulation ofthe Physical-Mathematical Problem 544 9.2.4.2 Korteweg de Vries Equation (KdV) . . . 546 9.2.4.3 Non-Linear Schrdinger Equation (NLS) ...... 547 9.2.4.4 SineGordon Equation (SG) . . . . . . . . . . . . . 547 9.2.4.5 Further :\fon-linear Evolution Equations with Soliton Solutions 549

10 Calculus ofVariations 10.1 Defining the Problem 10.2 Historical Problems .

10.2.1 Isoperimetrie Problem 10.2.2 Brachistochrone Problem

10.3 Variational Problems of One Variable 10.3.1 Simple Variational Problems and Extremal Curves 10.3.2 Euler Differential Equation ofthe Variational Calculus 10.3.3 Variational Problems with Side Conditions ..... . 10.3.4 Variation al Problems with Higher-Order Derivatives . 10.3.5 Variational Problem with Several Unknown Functions 10.3.6 Variational Problems llsing Parametric Representation .

10.4 Variation al Problems with Functions of Several Variables. 10.4.1 Simple Variational Problem ..... 10.4.2 More General Variational Problems

10.5 Numerical Solution of Variational Problems 10.6 Supplemcntary Problems .....

10.6.1 First and Second Variation 10.6.2 Application in Physics "

550 550 551 551 551 551 551 552 553 554 555 555 556 556 558 558 559 559 560

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XXIV Contents

11 Linear Integral Equations 561 11.1 Introduction and Classification . . . . . . . . . . . 561 11.2 Fredholm Integral Equations of the Second Kind . 562

11.2.1 Integral Equations with Degenerate Kernel 562 11.2.2 Successive Approximation Method, Neumann Series 565 11.2.3 Fredholm Solution Method, Fredholm Theorems 567

11.2.3.1 Fredholm Solution Method . . . . . . . . . 567 11.2.3.2 Fredholm Theorems. . . . . . . . . . . . . 569

11.2.4 Numerical iI.!ethods for Fredholm Integral Equations of the Second Kind 570 11.2.4.1 Approximation of the Integral 570 11.2.4.2 Kernel Approximation . . . . 572 11.2.4.3 Collocation '.1ethod. . . . . . 574

11.3 Fredholm Integral Equations of the First Kind. 575 11.3.1 Integral Equations with Degenerate Kerneis . 575 11.3.2 Analytic Basis. . . . . . . . . . . . . . . . . 576 11.3.3 Reduction of an Integral Equation into a Linear System of Equations 578 11.3.4 Solution of the Homogeneous Integral Equation of the First Kind . . 579 11.3.5 Construction ofTwo Special Orthonormal Systems for a Given Kernel 580 11.3.6 Iteration Method .. . . 582

11.4 Volterra Integral Equations . . . . 583 11.4.1 Theoretical Foundations . 583 11.4.2 Solution by Differentiation 584 11.4.3 Solution of the Volterra Integral Equation of the Second Kind by ~eumann

Series. . . . . . . . . . . . . . . . . . . . . . . . . . . 585 11.4.4 Comiolution Type Volterra Integral Equations ............. 585 11.4.5 ~umerical Methods for Volterra Integral Equations of the Second Kind 586

11.5 Singular Integral Equations . . . . . . . . . . . . 588 11.5.1 Abel Integral Equation . . . . . . . . . . . . . 588 11.5.2 Singular Integral Equation with Cauchy Kernel 589

11.5.2.1 Formulation of the Problem . . . . . 589 11.5.2.2 Existence of a Solution . . . . . . . . 590 11.5.2.3 Properties of Cauchy Type Integrals. 590 11.5.2.4 The Hilbert Boundary Value Problem 591 11.5.2.5 Solution of the Hilbert Boundary Value Problem (in short: Hilbert

Problem) ........ ............ 591 11.5.2.6 Solution of the Characteristic Integral Equation .......... 592

12 Functional Analysis 12.1 Vector Spaces ............. .

12.1.1 Notion of a Vector Spaee ... . 12.1.2 Linear and Affine Linear Subsets 12.1.3 Linearly Independent Elements. 12.1.4 Convex Subsets and the Convex Hull .

12.1.4.1 Convex Sets ...... . 12.1.4.2 Cones ......... .

12.1.5 Linear Operators and Functionals 12.1.5.1 Mappings ....... . 12.1.5.2 Homomorphism and Endomorphism . 12.1.5.3 Isomorphie Vector Spaces ..

12.1.6 Complexification of Real Veetor Spaces 12.1.7 Ordered Vector Spaces ...... .

12.1.7.1 Cone and Partial Ordering ..

594 594 594 595 596 597 597 597 598 598 598 599 599 599 599

Content8 XXV

12.1. 7.2 Order Bounded Sets 12.1.7.3 Positive Operators 12.1. 7.4 Vector Lattices

12.2 Metric Spaces 12.2.1 Notion of aMetrie: Spae:e ..

12.2.1.1 Balls, :-Jeighborhoods and Open Sets. 12.2.1.2 Convergence of Sequences in :vletrie: Spaees 12.2.1.3 Closed Sets and Closure ......... . 12.2.1.4 Dcnse Subsets and Separable Metrie Spaees .

12.2.2 Complete :vletrie: Spaees .. 12.2.2.1 Cauchy Sequene:es .............. . 12.2.2.2 Complete Metric Spaees ... . 12.2.2.3 Some Fundamental Theorems in Complete Metrie: Spaees 12.2.2.4 Some Applie:ations of the Contraction Mapping Principle 12.2.2.5 Completion of a \Ietrie Spae:e

12.2.3 Continuous Operators .. . 12.3 !\iormed Spaces ....... . ..... .

12.3.1 Notion of a :'jormed Space .... . 12.3.1.1 Axioms of a :-Jormed Spaee 12.3.1.2 Some Properties of Normed Spaces

12.3.2 Banach Spaces. . . . . . . . . . . . . 12.3.2.1 Series in ~ormed Spae:es .. 12.3.2.2 Examples of Banaeh Spaees 12.3.2.3 Sobolev Spaces

12.3.3 Ordered \'ormed Spae:es 12.3.4 Normed Algebras ....

12.4 Hilbert Spaces .. 12.4.1 Notion of a Hilbert Space

12.4.1.1 Scalar Produet 12.4.1.2 Unitary Spaees and Some of their Properties 12.4.1.3 Hilbert Space ....... .

12.4.2 Orthogonality. . ......... . 12.4.2.1 Properties of Orthogonality 12.4.2.2 Orthogonal Systems .

12.4.3 Fourier Series in Hilbert Spae:es ... . 12.4.3.1 Best Approximation ... . 12.4.3.2 Parseval Equation, Riesz-Fischer Theorem

12.4.4 Existence of a Basis, Isomorphie: Hilbert Spae:es ... 12.5 Continuous Linear Operators and Functionals .....

12.5.1 Boundedness, "iorm and Continuity of Linear Operators 12.5.1.1 Boundedness and the Norm of Linear Operators 12.5.1.2 The Spae:e of Linear Continuous Operators 12.5.1.3 Convergence of Operator Sequences ... .

12.5.2 Linear Continuous Operators in Banach Spaces .. . 12.5.3 Elements of the Speetral Theory of Linear Operators

12.5.3.1 Resolvent Set and the Resolvent of an Operator. 12.5.3.2 Spectrum of an Operator

12.5.4 Continuous Linear Functionals . . . .. . ....... . 12.5.4.1 Definition . . . . . . . ..... . 12.5.4.2 Continuous Linear Functionals in Hilbert Spaces, Riesz Representa-

timl Theorem . . . . . . . . 12.5.4.3 Continuous Linear Functionals in LP ............... .

600 600 600 602 602 603 604 604 605 605 605 606 606 606 608 608 609 609 609 610 610 610 610 611 611 612 613 613 613 613 613 614 614 614 615 615 616 616 617 617 617 617 618 618 620 620 620 621 621

622 622

I

XXVI Contents

12.5.5 Extension of a Linear Functional ..... . 12.5.6 Separation of Convex Sets ........ . 12.5.7 Second Adjoint Space and Reflexive Spaces

12.6 Adjoint Operators in Normed Spaces ....... . 12.6.1 Adjoint of a Bounded Operator ...... . 12.6.2 Adjoint Operator of an Unbounded Operator 12.6.3 Self-Adjoint Operators ........ .

12.6.3.1 Positive Definite Operators .. 12.6.3.2 Projectors in a Hilbert Space .

12.7 Compact Sets and Compact Operators .... 12.7.1 Compact Subsets of a Normed Space .. 12.7.2 Compact Operators .......... .

12.7.2.1 Definition of Compact Operator 12.7.2.2 Properties of Linear Compact Operators 12.7.2.3 Weak Convergence of Elements

12.7.3 Fredholm Alternative ........ . 12.7.4 Compact Operators in Hilbert Space . 12.7.5 Compact Self-Adjoint Operators ..

12.8 Non-Linear Operators ............ . 12.8.1 Examples of Non-Linear Operators .. 12.8.2 Differentiability of Non-Linear Operators 12.8.3 Newton's Method ........ . 12.8.4 Schauder's Fixed-Point Theorem. 12.8.5 Leray-Schauder Theory . . . . . . 12.8.6 Positive Non-Linear Operators .. 12.8.7 Monotone Operators in Banach Spaces

12.9 Measure and Lebesgue Integral ... 12.9.1 Sigma Algebra and Measures . 12.9.2 Measurable Functions .....

12.9.2.1 Measurable Function 12.9.2.2 Properties of the Class of Measurable Functions

12.9.3 Integration ................ . 12.9.3.1 Definition of the Integral ... . 12.9.3.2 Some Properties of the Integral 12.9.3.3 Convergence Theorems.

12.9.4 LP Spaces ............... . 12.9.5 Distributions ............. .

12.9.5.1 Formula of Partial Integration 12.9.5.2 Generalized Derivative .. . 12.9.5.3 Distributions ....... . 12.9.5.4 Derivative of a Distribution

13 Vector Analysis and Vector Fields 13.1 Basic Notions of the Theory of Vector Fields .

13.1.1 Vector Functions of a Scalar Variable 13.1.1.1 Definitions ......... . 13.1.1.2 Derivative of a Vector Function 13.1.1.3 Rules of Differentiation for Vectors 13.1.1.4 Taylor Expansion for Vector Functions

13.1.2 Scalar Fields ................... . 13.1.2.1 Scalar Field or Scalar Point Function . 13.1.2.2 Important Special Cases of Scalar Fields

622 623 624 624 624 625 625 626 626 626 626 626 626 627 627 627 628 628 629 629 630 630 631 631 631 632 633 633 634 634 634 635 635 635 636 637 637 637 638 638 639

640 640 640 640 640 640 641 641 641 641

Contents XXVII

13.1.2.3 Co ordinate Definition of a Field ..... 642 13.1.2.4 Level Surfaces and Level Lines of a Field 642

13.1.3 Vector Fields. . . . . . . . . . . . . . . . . . . 643 13.1.3.1 Vector Field or Vector Point Function . . 643 13.1.3.2 Important Cases of Vector Fields .... 643 13.1.3.3 Coordinate Representation of Vector Fields . 644 13.1.3.4 Transformation of Coordinate Systems 645 13.1.3.5 Vector Lines . . . . . . . 646

13.2 Differential Operators of Space . . . . . . . . . . . . . . 647 13.2.1 Directional and Space Derivatives ........ 647

13.2.1.1 Directional Derivative of a Scalar Field 647 13.2.1.2 Directional Derivative of a Vector Field 648 13.2.1.3 Volume Derivative ... , 648

13.2.2 Gradient of a Scalar Field .......... 648 13.2.2.1 Definition of the Gradient .... 648 13.2.2.2 Gradient and Volume Derivative . 649 13.2.2.3 Gradient and Directional Derivative . 649 13.2.2.4 Further Properties of the Gradient .. 649 13.2.2.5 Gradient ofthe Scalar Field in Different Coordinates. 649 13.2.2.6 Rules of Calculations 650

13.2.3 Vector Gradient .......... 650 13.2.4 Divergence ofVector Fields .... 651

13.2.4.1 Definition of Divergence 651 13.2.4.2 Divergence in Different Coordinates 651 13.2.4.3 Rules for Evaluation of the Divergence 651 13.2.4.4 Divergence of a Central Field . 652

13.2.5 Rotation of Vector Fields . . . . . . . . . . 652 13.2.5.1 Definitions of the Rotation . 652 13.2.5.2 Rotation in Different Coordinates 653 13.2.5.3 Rules for Evaluating the Rotation 653 13.2.5.4 Rotation of a Potential Field 654

13.2.6 Nabla Operator, Laplace Operator ..... 654 13.2.6.1 Nabla Operator .......... 654 13.2.6.2 Rules for Calculations with the :"Jabla Operator . 654 13.2.6.3 Vector Gradient . . . . . . . . . 655 13.2.6.4 Nabla Operator Applied Twice . 655 13.2.6.5 Laplace Operator . . . . . . . . 655

13.2.7 Review of Spatial Differential Operations 656 13.2.7.1 Fundamental Relations and Results (see Table 13.2) 656 13.2.7.2 Rules of Calculation for Spatial Differential Operators 656 13.2.7.3 Expressions ofVector Analysis in Cartesian, Cylindrical, and Spher-

ical Coordinates. . . . . . . . . . 657 13.3 Integration in Vector Fields . . . . . . . . . . . . . 658

13.3.1 Line Integral and Potential in Vector Fields 658 13.3.1.1 Line Integral in Vector Fields ... 658 13.3.1.2 Interpretation of the Line Integral in Mechanics . 659 13.3.1.3 Properties of the Line Integral . . . . . . . . . . 659 13.3.1.4 Line Integral in Cartesian Coordinates ..... 659 13.3.1.5 Integral Along a Closed Curve in a Vector Field . 660 13.3.1.6 Conservative or Potential Field 660

13.3.2 Surface Integrals. . . . . . . . . 661 13.3.2.1 Vector of a Plane Sheet ..... 661

XXVIII Cuntents

13.3.2.2 Evaluation of the Surface Integral . . . . . . .. ......... 661

I 13.3.2.3 Surface Integrals and Flow of Fields . . . . . .. ......... 662 13.3.2.4 Surface Integrals in Cartesian Coordinates as Surface Integral of Sec-ond Type . . . . . . . . . . . . . . . . . . . . . 662

13.3.3 Integral Theorems. . . . . . . . . . . . . . . . . . . . . . 663 13.3.3.1 Integral Theorem and Integral Formula of Gauss 663 13.3.3.2 Integral Theorem of Stokes . 664 13.3.3.3 Integral Theorems of Green 664

13.4 Evaluation of Fields . . . . . . . . . . . . . . 665 13.4.1 Pure Source Fields .......... 665 13.4.2 Pure Rotation Field or Zero-Divergence Field 666 13.4.3 Vector Fields with Point-Like Sources . . . . 666

13.4.3.1 Coulomb Field of a Point-Like Charge. 666 13.4.3.2 Gravitational Field of a Point Mass 667

13.4.4 Superposition of Fields . . . . . . . . . . 667 13.4.4.1 Discrete Source Distribution . . 667 13.4.4.2 Continuous Source Distribution 667 13.4.4.3 Conclusion ......... 667

13.5 Differential Equations of Vector Field Theory 667 13.5.1 Laplace Differential Equation . 667 13.5.2 Poisson Differential Equation . 668

14 Function Theory 669 14.1 Functions of Complex Variables .......... 669

14.1.1 Continuity, Differentiability. . . . . . . . . 669 14.1.1.1 Definition of a Complex Function 669 14.1.1.2 Limit of a Complex Function . . . 669 14.1.1.3 Continuous Complex Functions . 669 14.1.1.4 Differentiability of a Complex Function 669

14.1.2 Analytic Functions . . . . . . . . . . . . 670 14.1.2.1 Definition of Analytic Functions . 670 14.1.2.2 Examples of Analytic Functions . 670 14.1.2.3 Properties of Analytic Functions . 670 14.1.2.4 Singular Points . . . . . . . . . . 671

14.1.3 Conformal 'vfapping . . . . . . . . . . . . . 672 14.1.3.1 Notion and Properties of Conformal Mappings 672 14.1.3.2 Simplest Conformal Mappings . . 673 14.1.3.3 The Schwarz Reflection Principle 679 14.1.3.4 Complex Potential .. . . . . . . 679 14.1.3.5 Superposition Principle. . . . . . 681 14.1.3.6 Arbitrary Mappings of the Complex Plane 682

14.2 Integration in the Complex Plane .......... 683 14.2.1 Definite and Indefinite Integral. . . . . . . . . . . . 683

14.2.1.1 Definition of the Integral in the Complex Plane. 683 14.2.1.2 Properties and Evaluation of Complex Integrals 684

14.2.2 Cauchy Integral Theorem. . . . . . . . . . . . . . . . . . 686 14.2.2.1 Cauchy Integral Theorem for Simply Connected Domains 686 14.2.2.2 Cauchy Integral Theorem for Multiply Connected Domains 686

14.2.3 Cauchy Integral Formulas. . . . . . . . . . . . . . . . . 687 14.2.3.1 Analytic Function on the Interior of a Domain 687 14.2.3.2 Analytic Function on the Exterior of a Domain 687

14.3 Power Series Expansion of Analytic Functions . . . . . . . . . . 687

Content" XXIX

14.3.1 Convergenc:r of Se ries with Complex Terms . . . .. . ..... . 14.3.1.1 Convergence of a "Jumber Sequence with Complex Terms 14.3.1.2 Convergence of an Infinite Series with Complex Terms 14.3.1.3 Power Serie:; with Complex Terms

14.3.2 Taylor Series ............ . 14.3.3 Principle of Analytic Continuation .... . 14.3.4 Laurent Expansion ............ . 14.3.5 Isolated Singular Points and the Residue Theorem

14.3.5.1 Isolated Singular Points 14.3.5.2 l\lcromorphic Functions 14.3.5.3 Elliptic Functions . 14.3.5.4 Rcsidue 14.3.5.5 Residue Theorem .

14.4 Evaluation of Real Integrals by Complex Integrals 14.4.1 Application of Cauchy Integral Formulas 14.4.2 Application of thc Residue Theorem 14.4.3 Application of the Jordan Lemma ... .

14.4.3.1 Jordan Lemma ....... . 14.4.3.2 Examples of the Jordan Lemma

14.5 Algebraic and Elementary Transcendental Functions 14.5.1 Algebraic Functions ..... 14.5.2 Elementary Transc:endental Functions . 14.5.3 Description of Curves in Complex Form

14.6 Elliptic Functions ......... . 14.6.1 Relation to Elliptic Integrals 14.6.2 Jacobian Functions .. 14.6.3 Theta Function .... 14.6.4 Weierstrass Functions .

15 Integral Transformations 15.1 Notion ofIntegral Transformation .......... .

15.2

15.1.1 General Definition of Integral Transformations 15.1.2 Special Integral Transformations .. . 15.1.3 Inverse Transformations .......... . 15.1.4 Linearit.v ofIntegral Transformations .... . 15.1.;) Integral Transformations for Functions of Several Variables 15.1.6 Applications ofIntegral Transformations Laplace Transformation . . .. ..... . 15.2.1 Properties uf the Laplace Transformation

15.2.1.1 Laplace Transformation, Original and Image Space . 15.2.1.2 Rules for the Evaluation of the Laplac:e Transformation 15.2.1.3 Transforms of Special Functions .. . 15.2.1.4 Dirae d Function and Distributions ...... .

13.2.2 Inverse Transformation intu the Original Space 15.2.2.1 Inverse Transformation with the Help ofTables. 15.2.2.2 Partial Fraction Decomposition 15.2.2.3 Series Expansion . . . .. . ......... . 15.2.2.4 Inverse Integral ..

13.2.3 Solution of Differential Equations using Laplace Transformation. 15.2.3.1 Ordinary Linear Differential Equations with Constant Coefficients 15.2.3.2 Ordinary Linear Differential Equations with Coefficients Depending

on the Variable ........................... .

687 687 688 688 689 689 690 690 690 691 691 691 692 692 692 693 693 693 694 696 696 696 699 700 700 701 703 703

705 705 705 705 705 705 707 707 708 708 708 709 712 715 716 716 716 717 718 719 719

720

I

XXX Contents

15.2.3.3 Partial Differential Equations . 15.3 Fourier Transformation .. . ......... .

15.3.1 Properties of the Fourier Transformation 15.3.1.1 Fourier Integral . . . . . . . . . 15.3.1.2 Fourier Transformation and Inverse Transformation 15.3.1.3 Rules of Calculation with the Fourier Transformation 15.3.1.4 Transforms of Special Functions ........... .

15.3.2 Solution of Differential Equations using the Fourier Transformation 15.3.2.1 Ordinary Linear Differential Equations 15.3.2.2 Partial Differential Equations

15.4 Z-Transformation ............. . 15.4.1 Properties of the Z-Transformation .. .

15.4.1.1 Discrete Functions .... . . 15.4.1.2 Definition of the Z-Transformation. 15.4.1.3 Rules of Calculations ....... . 15.4.1.4 Relation to the Laplace Transformation 15.4.1.5 Inverse ofthe Z-Transformation ... .

15.4.2 Applications of the Z-Transformation ..... . 15.4.2.1 General Solution of Linear Difference Equations 15.4.2.2 Second-Order Difference Equations (Initial Value Problem) 15.4.2.3 Second-Order Difference Equations (Boundary Value Problem)

15.5 Wavelet Transformation. 15.5.1 Signals ......... . 15.5.2 Wavelets ........ . 15.5.3 Wavelet Transformation 15.5.4 Discrete Wavelet Transformation.

15.5.4.1 Fast Wavelet Transformation. 15.5.4.2 Discrete Haar Wavelet Transformation

15.5.5 Gabor Transformation 15.6 Walsh Functions ....

15.6.1 Step Functions. . . . . 15.6.2 Walsh Systems. . . . .

16 Probability Theory and Mathematical Statistics 16.1 Combinatorics ....

16.1.1 Permutations 16.1.2 Combinations 16.1.3 Arrangements 16.1.4 Collection of the Formulas of Combinatorics (see Table 16.1) .

16.2 Probability Theory ........... . 16.2.1 Event, Frequency and Probability .. .

16.2.1.1 Events ............ . 16.2.1.2 Frequencies and Probabilities 16.2.1.3 Conditional Probability, Bayes Theorem

16.2.2 Random Variables, Distribution Functions 16.2.2.1 Random Variable ............ . 16.2.2.2 Distribution Function .......... . 16.2.2.3 Expected Value and Variance, Chebyshev Inequality 16.2.2.4 Multidimensional Random Variable

16.2.3 Discrete Distributions ......... . 16.2.3.1 Binomial Distribution ... . 16.2.3.2 Hypergeometrie Distribution.

721 722 722 722 723 725 728 729 729 730 731 732 732 732 733 734 735 736 736 737 738 738 738 739 739 741 741 741 741 742 742 742

743 743 743 743 744 745 745 745 745 746 748 749 749 749 751 752 752 753 754

Contents XXXI

16.2.3.3 Poisson Distribution 16.2.4 Continuous Distributions ...

16.2.4.1 "Iormal Distribution 16.2.4.2 Standard Normal Distribution, Gaussian Error Function . 16.2.4.3 Logarithmic Normal Distribution 16.2.4.4 Exponential Distribution .. . 16.2.4.5 Weibull Distribution .... . 16.2.4.6 X2 (Chi-Square) Distribution. 16.2.4.7 Fisher F Distribution .... . 16.2.4.8 Student t Distribution ... .

16.2.5 Law of Large Numbers, Limit Theorems . 16.2.6 Stochastic Processes and Stochastic Chains

16.2.6.1 Basic \lotions, Markov Chains 16.2.6.2 Poisson Process ...... .

16.3 Mathematical Statistics ........... . 16.3.1 Statistic Function or Sam pie Function

16.3.1.1 Population, Sampie, Random Vector . 16.3.1.2 Statistic Function or Sampie Function .

16.3.2 Descriptiye Statistics .............. . 16.3.2.1 Statistical Summarization and Analysis of Given Data 16.3.2.2 Statistical Parameters ............. .

16.3.3 Important Tests ...................... . 16.3.3.1 Goodness ofFit Test for a Normal Distribution. 16.3.3.2 Distribution of the Sampie Mean .. . 16.3.3.3 Confidence Limits for the Mean .. . 16.3.3.4 Conficlence Interval for the Variance . 16.3.3.5 Structure of Hypothesis Testing .. .

16.3.4 Correlation and Regression .......... . 16.3.4.1 Linear Correlation of two Measurable Characters .

755 756 756 757 757 758 759 760 761 761 762 763 763 766 767 767 767 768 770 770 771 772 772 774 775 776 777 777 777

16.3.4.2 Linear Regression for two Measurable Characters . 778 16.3.4.3 Multidimensional Regression. 779

16.3.5 Monte Carlo Methods . . . . 781 16.3.5.1 Simulation. . . . . . . 781 16.3.5.2 Random Numbers. . . 781 16.3.5.3 Example of a Monte Carlo Simulation . 782 16.3.5.4 Application ofthe Monte Carlo Method in Numerical Mathematics 783 16.3.5.5 Further Applications of the Monte Carlo Method . 785

16.4 Calculus of Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785 16.4.1 Measurement Error ancl its Distribution. . . . . . . . . . . . 785

16.4.1.1 Qualitative Characterization of Measurement Errors 785 16.4.1.2 Density Function of the Measurement Error. . . . . 786 16.4.1.3 Quantitative Characterization of the Measurement Error 788 16.4.1.4 Determining the Result of a Measurement with Bounds on the Error 790 16.4.1.5 Error Estimation for Direct Measurements with the Same Accuracy 791 16.4.1.6 Error Estimation for Direct Measurements with Different Accuracy 791

16.4.2 Error Propagation ancl Error Analysis . 792 16.4.2.1 Gauss Error Propagation Law 792 16.4.2.2 Error Analysis. . . . . . . . . 794

I

XXXII Contents

17 Dynamical Systems and Chaos 17.1 Ordinary Differential Equations and Ylappings

17.1.1 Dynamical Systems .. 17 .1.1.1 Basic Notions . . . . . . . . . 17 .1.1. 2 Invariant Sets . . . . . . . . .

17.1.2 Qualitative Theory of Ordinary Differential Equations 17.1.2.1 Existence of Flows, Phase Space Structure 17.1.2.2 Linear Differential Equations . 17.1.2.3 Stability Theory .. 17.1.2.4 Invariant Manifolds .. . 17.1.2.5 Poincare Mapping ... . 17.1.2.6 Topological Equivalence of Differential Equations .

17.1.3 Discrete Dynamical Systems .............. . 17.1.3.1 Steady States, Periodic Orbits and Limit Sets. 17.1.3.2 Invariant Manifolds ............ . 17.1.3.3 Topological Conjugacy of Discrete Systems

17.1.4 Structural Stability (Robustness) ......... . 17.1.4.1 Structurally Stable Differential Equations . 17.1.4.2 Structurally Stable Discrete Systems 17.1.4.3 Generic Properties ... .

17.2 Quantitative Description of Attractors .. . 17.2.1 Probability Measures on Attractors

17.2.1.1 Invariant Measure ..... 17.2.1.2 Elements of Ergodie Theory

17.2.2 Entropies ..... . 17.2.2.1 Topological Entropy 17.2.2.2 Metric Entropy

17.2.3 Lyapunov Exponents ... . 17.2.4 Dimensions ........ .

17.2.4.1 Metric Dimensions 17.2.4.2 Dimensions Defined by Invariant Measures 17.2.4.3 Local Hausdorff Dimension According to Douady and Oesterle 17.2.4.4 Examples of Attractors .. .

17.2.5 Strange Attractors and Chaos ... . 17.2.6 Chaos in One-Dimensional Mappings

17.3 Bifurcation Theory and Routes to Chaos ... 17.3.1 Bifurcations in Morse-Smale Systems

17.3.1.1 Local Bifurcations in Neighborhoods of Steady States 17.3.1.2 Local Bifurcations in a Neighborhood of a Periodic Orbit 17.3.1.3 Global Bifurcation ..... .

17.3.2 Transitions to Chaos ......... . 17.3.2.1 Cascade of Period Doublings . 17.3.2.2 Intermittency ........ . 17.3.2.3 Global Homoclinic Bifurcations 17.3.2.4 Destruction of a Trus .....

18 Optimization 18.1 Linear Programming ...................... .

18.1.1 Formulation of the Problem and Geometrical Representation 18.1.1.1 The Form of a Linear Programming Problem 18.1.1.2 Examples and Graphical Solutions .....

18.1.2 Basic )lotions of Linear Programming, Normal Form .

795 795 795 795 797 798 798 799 801 804 806 808 809 809 810 811 811 811 812 812 814 814 814 815 817 817 817 818 820 820 822 824 825 826 827 827 827 828 833 836 837 837 837 8::\8 839

844 844 844 844 845 847

Cuntent.s XXXIII

18.1.2.1 Extreme Points allCl Basis 8-H 18.1.2.2 )/ormal Form of thc Linear Programming Problem 848

18.1.3 Simplex l\!ethod . .. 849

18.1.3.1 Simplex Tableau 849 18.1.3.2 Transition to t.he New Simplex Tableau .. 850 18.1.3.3 Det.ermination of an Initial Simplex Tableau 852 18.1.3,4 Revised Simplex '\Iethod ... 853 18.1.3.5 Duality in Linear Programming 854

18.1.4 Special Linear Programming Problems. 855 18.1.4.1 Transportation Problem 855 18.1.4.2 Assignment Prohlem 858 18.1.4.3 Distribution Problem 858 18.1.4.4 Travelling Salesman . 859 18.1.4.5 Scheduling Problem. 859

18.2 l\'on-linear Optimization 859 18.2.1 Formulation of the Problem, Theoretical Basis 859

18.2.1.1 Formulation of the Problem 859 18.2.1.2 Optimality Conclitions .. 860 18.2.1.3 Duality in Optimization 861

18.2.2 Special Non-linear Optimization Problems. 861 18.2.2.1 Convex Optimization . 861 18.2.2.2 Quaclratic Optimization 862

18.2.3 Solution '.lctlJOcls for Quac!ratic Optimization Problems 863 18.2.3.1 Wolfe's l\lethocl .. 863 18.2.3.2 Hilclreth c!'Esopo Methoc! 865

18.2,4 :'\ umerical Seareh Proceclures . 865 18.2.4.1 One-Dimensional Search 865 18.2.4.2 l\[inimum Search in n-Dimensional Euclidf'an Vector Space 866

18.2.5 Met.hocls for U nco!lstrainecl Problems 866 18.2.5.1 .'vlethoc! of Strcpest Descent .. 867 18.2.5.2 Application oE tlH' Newton Method 867 18.2.5.3 Conjugat.e Grac!ient l\!ethods 867 18.2.5,4 .'vlethod oE DaYielon, Fleteher and Powell (DFP) 868

18.2.6 Gradient. :'Ilethods for Problems with Inequality Type Constraint.s) 868 18.2.6.1 l\lethod oE Feasihle Direct.ions 869 18.2.6.2 Gradient. Projection l\Iethod 870

18.2.7 Pcnalt~, FunC'tion and Barrier :'I'lethods . 872 18.2.7.1 Penalty Function I\Icthod 872 18.2.7.2 Barrier :'Ilethod 873

18.2.8 Cutting Plane "'lethods . 874 18.3 Discret{' Dynamic Programming 875

18.3.1 Discrete Dvnamic Decision Models . 875 18.3.1.1 TI-Stage Decision Processes 870 18.3.1.2 DYllamic Programmillg Problem . 875

18.3.2 Examples oE Discrrte Decisioll Models 876 183.2.1 Purchasillg Problem 876 18.3.2.2 Kllapsack Prohlem 876

18.3.3 Bellman FUllctional Equatiolls .. 876 18.3.3.1 Propcrties of the Cost Functioll 876 18.3.3.2 Formulation of the Functional EquatiolJs 877

18.3,4 Bellman Optirnality Prilleipie 878 18.3.,) Bellman FUllctiollal Equationl\!ethod ... 878

I

XXXIV Gonienis

18.3.5.1 Determination of Minimal Costs .......... . 18.3.5.2 Determination of the Optimal Policy ....... .

18.3.6 Examples of Applications ofthe Functional Equation Method 18.3.6.1 Optimal Purchasing Policy . 18.3.6.2 Knapsack Problem ................. .

19 Numerical Analysis 19.1 Numerical Solution of Non-Linear Equations in a Single Unknown

19.1.1 Iteration Method .......... . 19.1.1.1 Ordinary Iteration Method . 19.1.1.2 Newton's Method ... . 19.1.1.3 Regula Falsi ...... .

19.1.2 Solution of Polynomial Equations 19.1.2.1 Horner's Scheme ... 19.1.2.2 Positions of the Roots .. 19.1.2.3 Numerieal Methods ...

19.2 Numerieal Solution of Equation Systems. 19.2.1 Systems of Linear Equations ...

19.2.1.1 Triangular Decomposition of a Matrix. 19.2.1.2 Cholesky's Method for aSymmetrie Coeffieient Matrix. 19.2.1.3 Orthogonalization Method 19.2.1.4 Iteration Methods ..... .

19.2.2 Non-Linear Equation Systems ... . 19.2.2.1 Ordinary Iteration Method . 19.2.2.2 Newton's Method ..... . 19.2.2.3 Derivative-Free Gauss-Newton Method

19.3 Numerical Integration ........ . 19.3.1 General Quadrature Formulas 19.3.2 Interpolation Quadratures ..

19.3.2.1 Rectangular Formula 19.3.2.2 Trapezoidal Formula 19.3.2.3 Simpson's Formula . 19.3.2.4 Hermite's Trapezoidal Formula

19.3.3 Quadrature Formulas of Gauss ..... . 19.3.3.1 Gauss Quadrature Formulas .. 19.3.3.2 Lobatto's Quadrature Formulas

19.3.4 Method of Romberg ........... . 19.3.4.1 Algorithm of the Romberg Method 19.3.4.2 Extrapolation Principle ...... .

19.4 Approximate Integration of Ordinary Differential Equations 19.4.1 Initial Value Problems ..... .

19.4.1.1 Euler Polygonal Method 19.4.1.2 Runge-Kutta Methods . 19.4.1.3 Multi-Step Methods .. 19.4.1.4 Predictor-Corrector Method 19.4.1.5 Convergence, Consistency, Stability

19.4.2 Boundary Value Problems ......... . 19.4.2.1 Difference Method ........ . 19.4.2.2 Approximation by Using Given Functions . 19.4.2.3 Shooting Method ............ .

19.5 Approximate Integration of Partial Differential Equations 19.5.1 Difference Method ................. .

878 878 879 879 880

881 881 881 881 882 883 884 884 885 886 887 887 887 890 890 892 893 893 894 894 895 895 896 896 896 897 897 897 897 898 898 898 899 901 901 901 901 902 903 904 905 905 906 907 908 908

Contents XXXV

19.5.2 Approximation by Given Functions ........... . 19.5.3 Finite Element Method (FEM) .............. .

19.6 Approximation, Computation of Adjustment, Harmonie Analysis 19.6.1 Polynomial Interpolation ......... .

19.6.l.1 Newton's Interpolation Formula . 19.6.l.2 Lagrange's Interpolation Formula 19.6.l.3 Aitken-Neville Interpolation ...

19.6.2 Approximation in Mean ....... . 19.6.2.1 Continuous Problems, Normal Equations 19.6.2.2 Diserete Problems, Normal Equations, Householder's Method 19.6.2.3 .'vfultidimensional Problems .... 19.6.2.4 Non-Linear Least Squares Problems .......... .

19.6.3 Chebyshev Approximation .................... . 19.6.3.1 Problem Definition and the Alternating Point Theorem 19.6.3.2 Properties of the Chebyshev Polynomials ...... . 19.6.3.3 Remes Algorithm ................... . 19.6.3.4 Diserete Chebyshev Approximation and Optimization

19.6.4 Harmonie Analysis ................ . 19.6.4.1 Formulas for Trigonometrie Interpolation 19.6.4.2 Fast Fourier Transformation (FFT)

19.7 Representation of Curves and Surfaees with Splines . 19.7.1 Cubic Splines ..... .

19.7.1.1 Interpolation Splines 19.7.l.2 Smoothing Splines .

19.7.2 Bicubic Splines ....... . 19.7.2.1 Use of Bieubic Splines 19.7.2.2 Bicubic Interpolation Splines . 19.7.2.3 Bicubic Smoothing Splines ..

19.7.3 Bernst.einBezier Representation of Curves and Surfaces 19.7.3.1 Principle of the B-B Curve Representation 19.7.3.2 B- B Surface Representation

19.8 Using the Computer.. . ...... . 19.8.1 Internal Symbol Representation .. .

19.8.1.1 Number Systems ..... . 19.8.l.2 Internal )/umber Representation .

19.8.2 '\Iumerical Problems in Calculations with Computers. 19.8.2.1 Introduction, Error Types ......... . 19.8.2.2 Normalized Decimal Numbers and Round-Off . 19.8.2.3 Accuracy in Numerical Calculations .

19.8.3 Libraries of Numerical :vlethods 19.8.3.1 NAG Library . 19.8.3.2 IMSL Library .... . 19.8.3.3 Aachen Library ... .

19.8.4 Application of Computer Algebra Systems. 19.8.4.1 .'v!athematiea 19.8.4.2 .'vIapie .............. .

20 Computer Algebra Systems 20.1 Introduction ................... .

20.1.1 Brief Characterization of Computer Algebra Systems. 20.1.2 Examples of Basic Application Fields

20.1.2.1 Manipulation of Formulas ......... .

909 910 914 914 914 915 915 916 916 918 919 919 920 920 921 922 923 924 924 925 928 928 928 929 930 930 930 932 932 932 933 933 933 933 935 936 936 936 938 941 941 942 943 943 943 946

950 950 950 950 950

I

XXXVI Contents

20.1.2.2 Numerical Calculations ........... . 20.1.2.3 Graphical Representations ......... . 20.1.2.4 Programming in Computer Algebra Systems

20.1.3 Structure of Computer Algebra Systems . 20.1.3.1 Basic Structure Elements.

20.2 lvIathematica ............... . 20.2.1 Basic Structure Elements ..... . 20.2.2 Types of Numbers in Mathematica .

20.2.2.1 Basic Types of Numbers in Mathematica 20.2.2.2 Special Numbers ............ . 20.2.2.3 Representation and Conversion of Numbers .

20.2.3 Important Operators 20.2.4 Lists .................. .

20.2.4.1 Notions ........... . 20.2.4.2 Nested Lists, Arrays or Tables 20.2.4.3 Operations with Lists . 20.2.4.4 Special Lists ........ .

20.2.5 Vectars and Matrices as Lists .... . 20.2.5.1 Creating Appropriate Lists . 20.2.5.2 Operations with Matrices and Vectors .

20.2.6 Functions .......... . 20.2.6.1 Standard Functions . 20.2.6.2 Special Functions 20.2.6.3 Pure Functions

20.2.7 Patterns ....... . 20.2.8 FunctionalOperations . 20.2.9 Programming ..... . 20.2.10 Supplement about Syntax, Information, Messages.

20.2.10.1 Contexts, Attributes 20.2.10.2 Information 20.2.10.3 :Vlessages .. .

20.3 Maple ............. . 20.3.1 Basic Structure Elements

20.3.1.1 Types and Objects 20.3.1.2 Input and Output.

20.3.2 Types of Numbers in Maple . 20.3.2.1 Basic Types of Numbers in Maple 20.3.2.2 Special Numbers ........ . 20.3.2.3 Representation and Conversion of Numbers .

20.3.3 Important Operators in Maple 20.3.4 Aigebraic Expressions ........ . 20.3.5 Sequences and Lists ......... . 20.3.6 Tables, Arrays, Vectors and Matrices .

20.3.6.1 Tables and Arrays ..... 20.3.6.2 One-Dimensional Arrays 20.3.6.3 Two-Dimensional Arrays . 20.3.6.4 Special Commands far Vectars and Matrices

20.3.7 Functions and Operators 20.3.7.1 Functions ...... . 20.3.7.2 Operators ...... . 20.3.7.3 Differential Operators 20.3.7.4 The Functional Operator map

951 952 952 952 952 953 953 954 954 955 955 956 956 956 957 957 958 958 958 959 960 960 960 961 961 962 963 964 964 964 965 965 965 965 966 967 967 967 967 968 969 969 970 970 971 971 972 972 972 973 973 974

20.3.8 Programming in }'laple ............. . 20.3.9 Supplement about Syntax, Information and Help

20.3.9.1 Using the Maple Library 20.3.9.2 Environment Variable .. 20.3.9.3 Information and Help ...

20.4 Applications of Computer Algebra Systems 20.4.1 Manipulation of Algebraic Expressions .

20.4.1.1 Mathematica ... . 20.4.1.2 Maple ............ .

20.4.2 Solution of Equations and Systems of Equations 20.4.2.1 \lathematica .. . 20.4.2.2 Maple ...... .

20.4.3 Elements of Linear Algebra . 20.4.3.1 \lathematica ... 20.4.3.2 \laple ....

20.4.4 Differential and Integral Calculus 20.4.4.1 \lathematica ..... 20.4.4.2 Maple ...

20.5 Graphics in Computer Algebra Systems 20.5.1 Graphics with Mathematica

20.5.1.1 Basic Elements of Graphics . 20.5.1.2 Graphics Primitives ..... 20.5.1.3 Syntax of Graphical Representation 20.5.1.4 GraphicalOptions . . ..... . 20.5.1.5 Two-Dimensional Curves ..... . 20.5.1.6 Parametrie Representation of Curves 20.5.1. 7 Representation of Surfaces and Space Curves

20.5.2 Graphics with J\laple

21 Tables

20.5.2.1 Two-Dimensional Graphics . 20.5.2.2 Three-Dtmensional Graphics .

21.1 Frequently Used Constants 21.2 Natural Constants . 21.3 Important Se ries Expansions 21.4 Fourier Series ...... . 21.5 Indefinite Integrals .... .

21.5.l Integral Rational Funct.ions . 21.5.1.1 Integrals with X = ax + b 21.5.1.2 Integrals with X = ax2 + bx + c 21.5.1.3 Integrals with X = a2 x2 . 21.5.1.4 Integrals with X = a:l x3 . 21.5.1.5 Integrals with X = a4 + x4 . 21.5.1.6 Integrals with X = a4 - x4 . 21.5.1.7 Some Cases of Part.ial Fraction Decomposition

21.5.2 Integrals ofIrrat.ional Functions ..... 21.5.2.1 Integrals with IX and a2 b2 x . 21.5.2.2 Other Integrals with IX .... 21.5.2.3 Integrals with lax + b . 21.5.2.4 Integrals \'iith lax + band l Ix + 9 . 21.5.2.5 Integrals with l(j2 - x2 . ......

Contents XXXVII

974 975

975 975 975 975 976 976 978 981 981 983 984 984 986 989 989 992 994 995 995 995 995 996 998 999

1000 1002 1002 1004

1007 1007 1007 1009 1014 1017 1017 1017 1019 1020 1022 1023 1023 1023 1024 1024 1024 1025 1026 1027

I

XXXVIII Contents

21.6

21.7

21.8 21.9 21.10 21.11 21.12

21.13 21.14 21.15

21.16 21.17 21.18 21.19

21.5.2.6 Integrals with V x 2 + a2 ... . 21.5.2.7 Integrals with ~ ... . 21.5.2.8 Integrals with vax2 + bx + c . 21.5.2.9 Integrals with other Irrational Expressions 21.5.2.10 Reeursion Formulas for an Integral with Binomial Differential

21.5.3 Integrals of Trigonometrie Fnnctions .. . 21.5.3.1 Integrals with Sine Fnnction ...... . 21.5.3.2 Integrals with Cosine Function ..... . 21.5.3.3 Integrals with Sine and Cosine Function . 21.5.3.4 Integrals with Tangent Fnnction .. 21.5.3.5 Integrals with Cotangent Fnnction ..

21.5.4 Integrals of other Transcendental Fnnctions .. 21.5.4.1 Integrals with Hyperbolic Functions . 21.5.4.2 Integrals with Exponential Functions 21.5.4.3 Integrals with Logarithmie Functions 21.5.4.4 Integrals with Inverse Trigonometrie Functions 21.5.4.5 Integrals with Inverse Hyperbolie Functions .

Definite Integrals .. . . . . . . . . . . . . . . . . . 21.6.1 Definite Integrals of Trigonometrie Fnnetions 21.6.2 Definite Integrals of Exponential Functions 21.6.3 Definite Integrals of Logarithmic Fnnctions . 21.6.4 Definite Integrals of Algebraic Functions ... Elliptic Integrals. . . . . . . . . . . . . . . . . . . . 21.7.1 EllipticIntegral of the First Kind F ('P, k), k = sin Cl! 21. 7.2 EllipticIntegral of the Second Kind E ( 'P, k), k = sin Cl! 21. 7.3 Complete Elliptic Integral, k = sin Cl! Gamma Function . . . . . . . . . . . . . Bessel Funetions (Cy lindrical Funetions) . Legendre Polynomials of the First Kind Laplace Transformation . . . . . . . . Fourier Transformation . . . . . . . . 21.12.1 Fourier Cosine Transformation 21.12.2 Fourier Sine Transformation . 21.12.3 Fourier Transformation . . . . 21.12.4 Exponential Fourier Transformation Z Transformation . . . . . . . Poisson Distribution. . . . . . . . . . . . . Standard Normal Distribution ...... . 21.15.1 Standard Normal Distribution for 0.00:::; x :::; 1.99 21.15.2 Standard Normal Distribution for 2.00 < x < 3.90 X2 Distribution .... - -Fisher F Distribution . Student t Distribution. Random Numbers .

22 Bibliography

Index

Mathematic Symbols

1029 1030 1032 1034 1034 1035 1035 1037 1039 1043 1043 1044 1044 1045 1047 1048 1049 1050 1050 1051 1052 1053 1055 1055 1055 1056 1057 1058 1060 1061 1066 1066 1072 1077 1079 1080 1083 1085 1085 1086 1087 1088 1090 1091

1092

1103

A

List 01 Tables XXXIX

List ofTables 1.1 Definition of powers . . . . . . . . . . . . . . . . . . . . . . 1.2 Pascal's triangle . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Auxiliary values for the solution of equations of degree three

2.1 Domain and range of trigonometricfunctions ....... . 2.2 Signs of trigonometrie funetions ..... . ....... . 2.3 Values of trigonometrie functions for 0,30,45,60 and 90. 2.4 Reduction formulas and quadrant relations of trigonometrie funetions . 2.5 Relations between the trigonometrie functions of the same argument in the interval

0< Cl < 7r /2 . . . . . . . . . . . . . ................... . 2.6 Domains and ranges of the inverses of trigonometrie funetions . . . . . . . . . . 2.7 Relations between two hyperbolie functions with the same arguments for x > 0 . 2.8 Domains and ranges of the area functions . . . . . . . . . . . . . . . . . . . 2.9 For the approximate determination of an empirically given function relation

3.1 Names of angles in degree and radian measure ...... . 3.2 Properties of some regular polygons ........... . 3.3 Defining quantities of a right angled-triangle in the plane. 3.4 Defining quantities of a general triangle, basic problems . 3.5 Grade and Gon Division .... . ........... . 3.6 Directional angle in a segment with correet sign for arctan 3.7 Regular polyeders with edge length a. . . . . . . . 3.8 Defining quantities of a spherical right-angled triangle .. 3.9 First and second basic problems for spherical oblique triangles 3.10 Third basic problem for spherical oblique triangles ..... . 3.11 Fourth basic problem for spherieal oblique triangles ..... . 3.12 Fifth and sixth basic problemes for a spherieal oblique tri angle . 3.13 Sealar product ofbasis vectors . 3.14 Veetor prod uet of basis vectors . 3.15 Sealar product of reciprocal bas