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Mathematical handbook for scientists and engineers: definitions, theorems, and formulas for reference and review

Granino Arthur Korn, Theresa M. Korn
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A reliable source of definitions, theorems, and formulas, this authoritative handbook provides convenient access to information from every area of mathematics. Coverage includes Fourier transforms, Z transforms, linear and nonlinear programming, calculus of variations, random-process theory, special functions, combinatorial analysis, numerical methods, game theory, and much more.
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Dover Publications
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\  MATHEMATICAL^  HANDBOOK FOR x  SCIENTISTS AND ENGINEERS Definitions,Theorems, and Formulas for Reference and Review  Granino A. Korn and Theresa M. Korn  DOVER BOOKS ON MATHEMATICS Handbook of Mathematical Functions, Milton Abramowitz and Irene A. Stegun. (61272-4) $32.95  Theory of Approximation, N. I. Achieser. (67129-1) $8.95 Tensor Analysis on Manifolds, Richard L. Bishop and Samuel I. Goldberg. (64039-6) $9.95  Tables of Indefinite Integrals, G. Petit Bois. (60225-7) $8.95 Vector and Tensor Analysis with Applications, A. I. Borisenko and I. E. Tarapov. (63833-2) $10.95 The History of the Calculus and Its Conceptual Development, Carl B. Boyer. (605094) $9.95 The Qualitative Theory of Ordinary Differential Equations: An  Introduction, Fred Brauer and John A. Nohel. (65846-5) $12.95 Principlesof Statistics, M. G. Bulmer. (63760-3) $9.95 The Theory of Spinors, Elie Cartan. (64070-1) $8.95 Advanced Number Theory, Harvey Cohn. (64023-X) $10.95  StatisticsManual, Edwin L. Crow, FrancisDavis, and Margaret Maxfield. (60599-X) $8.95  Fourier Series and Orthogonal Functions, Harry F. Davis. (65973-9) $13.95  eoMPUTABiLiTY and Unsolvabiltty, Martin Davis. (61471-9) $12.95 Asymptotic Methods inAnalysis, N. G. de Bruijn. (64221-6) $9.95 Problems in Group Theory, John D. Dixon. (61574-X) $10.95 The Mathematics of Games of Strategy, Melvin Dresher. (64216-X) $7.95 Asymptotic Expansions, A. ErdSlyi. (60318-0) $6.95  Complex Variables: Harmonic and Analytic Functions, Francis J. Flanigan. (61388-7) $10.95 On Formally Undecidable Propositions of Principia Mathematica and  Related Systems, Kurt Godel. (66980-7) $6.95  A History of Greek Mathematics, Sir Thomas Heath. (24073^8, 24074-6) Two-volume set $29.90  Probability: Elements of the Mathematical Theory, C. R. Heathcote. (41149-4) $8.95  Introduction to Numerical Analysis, Francis B. Hildebrand. (65363-3) $16.95  Methods of Appued Mathematics, Francis B. Hildebrand. (67002-3) $12.95 Topology, John G. Hocking and GailS. Young. (65676-4) $13.95 Mathematics and Logic,;  Mark Kac and Stanislaw M. Ulam. (67085-6) $7.95  Mathematical Methods and Theory in Games, Programming, and Economics, Samuel Karlin. (67020-1) $24.95  Mathematical Foundations of Information Theory, A. I. Khinchin. (60434-9) $5.95  Arithmetic Refresher, A. Albert Klaf. (21241-6) (continued on back nap)  MATHEMATICAL HANDBOOK FOR SCIENTISTS AND ENGINEERS Definitions, Theorems, and Formulas for Reference and Review  Granino A. Korn and  Theresa M. Korn  DOVER PUBLICATIONS, INC. Mineola, New York  Copyright Copyright © 1961,1968 byGranino A.Korn and Theresa M. Korn All rights reserved under Pan American and International Copyright Conventions.  Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario.  Bibliographical Note This Dover edition, first published in 2000, is an unabridged republication of the work originally published in 1968 by McGrawHill, Inc., New York. A number of typographical errors have been corrected.  Library of Congress Cataloging-in-Publication Data Korn, Granino Arthur, 1922-  Mathematical handbook for scientists and engineers : defini tions, theorems, and formulas for reference and review / Granino A. Korn and Theresa M. Korn. p. cm.  Originally published: 2nd, enl.and rev. ed. NewYork: McGrawHill, cl968.  Includes bibliographical references andindex. ISBN 0-486-41147-8 (pbk.) 1. Mathematics—Handbooks, manuals, etc. I. Korn, Theresa M. II. Title.  QA40 .K598 2000 510'.2'1—dc21 00-030318  Manufactured in the United States of America  Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501  This book is dedicated to the memory of Arthur Korn (1870-1945), who excelled as a mathematician as well as a  theoretical physicist and as a communications engineer.  PREFACE TO THE  SECOND EDITION  This new edition of the Mathematical Handbook has been substantially enlarged, and much of our original material has been carefully, and we hope usefully, revised and expanded. Completelynew sections deal with z transforms, the matrix notation for systems of differential equations (state equations), representation of rotations, mathematical program ming, optimal-control theory, random processes, and decision theory. The chapter on numerical computation was almost entirely rewritten, and the revised appendixes include a discussion of Polya's counting theorem, several new tables of formulas and numerical data, and a new, much larger integral table.  Numerous illustrations have been added  throughout the remaining text.  The handbook is again designed, first, as a comprehensive reference collection of mathematical definitions, theorems, and formulas for scientists, engineers, and students. Subjects of both undergraduate and graduate level are included. The omission of all proofs and the concise tabular presentation of related formulas have made it possible to incorpo rate a relatively large amount of reference material in one volume.  The handbook is, however, not intended for reference purposes alone; it attempts to present a connected survey of mathematical methods useful beyond specialized applications. Each chapter is arranged so as to permit review of an entire mathematical subject. Such a presentation  is made more manageable and readable through the omission of proofs; numerous references provide access to textbook material for more detailed  studies. Special care has been taken to point out, by means of suitable introductions, notes, and cross references, the interrelations of various topics and their importance in scientific and engineering applications. The writers have attempted to meet the individual reader's require ments by arranging the subject matter at three levels: 1. The most important formulas and definitions have been collected  in tables and boxed groups permitting rapid reference and review.  PREFACE TO THE SECOND EDITION  viii  2. The main text presents, in large print, a concise, connected review of each subject.  3. More detailed discussions and advanced topics are presented in  small print. This arrangement makes it possible to include such material without cluttering the exposition of the main review.  We believe that this arrangement has proved useful in the first edition.  The arrangement of the introductory chapters was left unchanged, although additions and changes were made throughout their text. Chap ters 1 to 5 review traditional college material on algebra, analytic geom  etry, elementary and advanced calculus, and vector analysis; Chapter 4 also introducesLebesgue and Stieltjes integrals, and Fourier analysis. Chapters 6, 7, and 8 deal with curvilinear coordinate systems, functions of a complex variable, Laplace transforms, and other functional transforms; new mate rial on finite Fourier transforms and on z transforms was added.  Chapters 9 and 10 deal with ordinary and partial differential equations and include Fourier- and Laplace-transform methods, the method of charac  teristics, and potential theory; eigenvalue problems as such are treated in Chapters 14 and 15. Chapter 11 is essentially new; in addition to ordi nary maxima and minima and the classical calculus of variations, this chapter now contains material on linear and nonlinear programming and on optimal-control theory, outlining both the maximum-principle and dynamic-programming approaches.  Chapter 12, considerably expanded in this edition, introduces the elements of modern abstract language and outlines the construction of mathematical models such as groups, fields, vector spaces, Boolean algebras,  and metric spaces. The treatment of function spaces continues through Chapter 14to permit a modest functional-analysis approach to boundaryvalue problems and eigenvalue problems in Chapter 15, with enough essential definitions to enable the reader to use modern advanced texts and periodical literature.  Chapter 13treats matrices; we have added severalnew sections review ing matrix techniques for systems of ordinary differential equations (state equations of dynamical systems), including an outline of Lyapunov sta bility theory. Chapter 14 deals with the important topics of linear vector spaces, linear transformations (linear operators), introduces eigen value problems, and describes the use of matrices for the representation of mathematical models. The material on representation of rotations was  greatly enlarged because of its importance for space-vehicle design as well as for atomic and molecular physics. Chapter 15 reviews a variety of topics related to boundary-value problems and eigenvalue problems, including Sturm-Liouville problems, boundary-value problems in two and three dimensions, and linear integral equations, considering functions as vectors in a normed vector space.  ix  PREFACE TO THE SECOND EDITION  Chapters 16 and 17 respectively outline tensor analysis and differential geometry, including the description of plane and space curves, surfaces, and curved spaces.  In view of the ever-growing importance of statistical methods, the completely revised Chapter 18 presents a rather detailed treatment of probability theory and includes material on random processes, correlation functions, and spectra. Chapter 19 outlines the principal methods of mathematical statistics and includes extensive tables of formulas involving special sampling distributions. A subchapter on Bayes tests and esti mation was added.  The new Chapter 20 introduces finite-difference methods and difference equations and reviews a number of basic methods of numerical compu tation. Chapter 21 is essentially a collection of formulas outlining the properties of higher transcendental functions; various formulas and many illustrations have been added.  The appendixes present mensuration formulas, plane and spherical trigonometry, combinatorial analysis, Fourier- and Laplace-transform tables, a new, larger integral table, and a new set of tables of sums and series. The treatment of combinatorial analysis was enlarged to outline the use of generating functions and a statement of Polya's counting theorem. Several new tables of formulas and functions were added. As before, there is a glossary of symbols, and a comprehensive and detailed index permits the use of the handbook as a mathematical dictionary. The writers hope and believe that this handbook will give the reader an opportunity to scan the field of mathematical methods, and thus to widen his background or to correlate his specialized knowledge with more general developments. We are very grateful to the many readers who have helped to improve the handbook by suggesting corrections and additions; once again, we earnestly solicit comments and suggestions for improvements, to be addressed to us in care of the publishers. Granino A. Korn  Theresa M. Korn  CONTENTS  Preface  vii  Chapter 1.  Real and Complex Numbers.  Elementary Algebra.  1.1. Introduction. The Real-number System 1.2. Powers, Roots, Logarithms, and Factorials. 1.3. Complex Numbers  Sum and Product Notation  1  2 4 7  1.4. Miscellaneous Formulas  10  1.5. Determinants  12  1.6. Algebraic Equations: General Theorems 1.7. Factoring of Polynomials and Quotients of Polynomials.  15  Partial Frac  tions  19  1.8. Linear, Quadratic, Cubic, and Quartic Equations 1.9. Systems of Simultaneous Equations 1.10. Related Topics, References, and Bibliography  Chapter 2.  2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7.  22 24 27  Plane Analytic Geometry  Introduction and Basic Concepts The Straight Line Relations Involving Points and Straight Lines Second-order Curves (Conic Sections) Properties of Circles, Ellipses, Hyperbolas, and Parabolas Higher Plane Curves Related Topics, References, and Bibliography.  29  ...  Chapter 3. Solid Analytic Geometry 3.1. Introduction and Basic Concepts  30 37 39 41 48 53 56  57 58  3.2. The Plane  67  3.3. 3.4. 3.5. 3.6.  69 70 ?4 82  The Straight Line Relations Involving Points, Planes, and Straight Lines Quadric Surfaces Related Topics, References, and Bibliography  Chapter 4. Functions and Limits. Differential and Integral Calculus  83  4.1. Introduction 4.2. Functions  85 85  4.3. Point Sets, Intervals, and Regions 4.4. Limits, Continuous Functions, and Related Topics  87 90  4.5. Differential Calculus  95 si  CONTENTS  xii  4.6. Integrals and Integration 4.7. Mean-value Theorems.  102  Values of Indeterminate Forms.  Weierstrass's  Approximation Theorems 4.8. Infinite Series, Infinite Products, and Continued Fractions. . . . 4.9. Tests for the Convergence and Uniform Convergence of Infinite Series and Improper Integrals  4.10. Representation of Functions by Infinite Series and Integrals. Series and Taylor's Expansion  118 121 127  Power 131 134 144  4.11. Fourier Series and Fourier Integrals  4.12. Related Topics, References, and Bibliography  Chapter 5. Vector Analysis  145  5.1. Introduction  146  5.2. Vector Algebra  147  5.3. Vector Calculus: Functions of a Scalar Parameter 5.4. Scalar and Vector Fields  151 153  5.5. 5.6. 5.7. 5.8.  157 162 164 166  Differential Operators Integral Theorems Specification of a Vector Field in Terms of Its Curl and Divergence. Related Topics, References, and Bibliography  .  Chapter 6. Curvilinear Coordinate Systems  168  6.1. Introduction  168  6.2. Curvilinear Coordinate Systems  169  6.3. Representation of Vectors in Terms of Components 6.4. Orthogonal Coordinate Systems. Vector Relations in Terms of Orthog onal Components 6.5. Formulas Relating to Special Orthogonal Coordinate Systems . . . 6.6. Related Topics, References, and Bibliography  171  Chapter 7. Functions of a Complex Variable 7.1. Introduction  7.2. Functions of a Complex Variable.  174 177 177  187 188  Regions of the Complex-number  Plane  7.3. Analytic (Regular, Holomorphic) Functions 7.4. Treatment of Multiple-valued Functions 7.5. Integral Theorems and Series Expansions 7.6. Zeros and Isolated Singularities 7.7. Residues and Contour Integration 7.8. Analytic Continuation 7.9. Conforms! Mapping 7.10. Functions Mapping Specified Regions onto the Unit Circle . . . . 7.11. Related Topics, References, and Bibliography  188  192 193 196 198 202 205 206 219 219  Chapter 8. The Laplace Transformation and Other Functional Transformations  221  8.1. Introduction  222  8.2. The Laplace Transformation 8.3. Correspondence between Operations on Object and Result Functions . 8.4. Tables of Laplace-transform Pairs and Computation of Inverse Laplace  222 225  Transforms  228  xiii  8.5. 8.6. 8.7. 8.8.  CONTENTS  "Formal" Laplace Transformation of Impulse-function Terms . . . Some Other Integral Transformations Finite Integral Transforms, Generating Functions, and z Transforms . Related Topics, References, and Bibliography  Chapter 9.  Ordinary Differential Equations  233 233 237 240  243  9.1. Introduction  244  9.2. 9.3. 9.4. 9.5. 9.6. 9.7.  247 252 265 277 284 286  First-order Equations Linear Differential Equations Linear Differential Equations with Constant Coefficients Nonlinear Second-order Equations Pfaffian Differential Equations Related Topics, References, and Bibliography  Chapter 10.  Partial Differential Equations  287  10.1. Introduction and Survey 10.2. Partial Differential Equations of the First Order 10.3. Hyperbolic, Parabolic, and Elliptic Partial Differential Equations. Characteristics  10.4. Linear Partial Differential Equations of Physics. 10.5. Integral-transform Methods 10.6. Related Topics, References, and Bibliography  Chapter 11.  288 290 302  Particular Solutions.  Maxima and Minima and Optimization Problems .  330  11.1. Introduction 11.2. Maxima and Minima of Functions of One Real Variable 11.3. Maxima and Minima of Functions of Two or More Real Variables.  321 332 333  11.4. Linear Programming, Games, and Related Topics 11.5. Calculus of Variations. Maxima and Minima of Definite Integrals. 11.6. Extremals as Solutions of Differential Equations: Classical Theory.  311 324 328  335 . .  11.7. Solution of Variation Problems by Direct Methods 11.8. Control Problems and the Maximum Principle 11.9. Stepwise-control Problems and Dynamic Programming  344 346  357 358  11.10. Related Topics, References, and Bibliography  369 371  Chapter 12. Definition of Mathematical Models: Modern (Abstract) Algebra and Abstract Spaces  373  12.1. Introduction  374  12.2. Algebra of Models with a Single Defining Operation: Groups. . . . 12.3. Algebra of Models with Two Defining Operations: Rings, Fields, and Integral Domains 12.4. Models Involving More Than One Class of Mathematical Objects: Linear Vector Spaces and Linear Algebras 12.5. Models Permitting the Definition of Limiting Processes: Topological Spaces  378  382  384 386  12.6. Order 12.7. Combination of Models: Direct Products, Product Spaces, and Direct Sums.  391  12.8. Boolean Algebras 12.9. Related Topics, References, and Bibliography  393 400  392  CONTENTS  Chapter 13.  Matrices.  Quadratic and Hermitian Forms  xiv  ....  402  13.1. Introduction  403  13.2. Matrix Algebra and Matrix Calculus 13.3. Matrices with Special Symmetry Properties 13.4. Equivalent Matrices. Eigenvalues, Diagonalization, Topics  403 410  and Related  13.5. Quadratic and Hermitian Forms 13.6. Matrix Notation for Systems of Differential Equations (State Equa tions). Perturbations and Lyapunov Stability Theory 13.7. Related Topics, References, and Bibliography  412 416 420 430  Chapter 14. Linear Vector Spaces and Linear Transformations (Linear Operators). Representation of Mathematical Models in Terms of Matrices  14.1. 14.2. 14.3. 14.4.  14.5. 14.6. 14.7. 14.8. 14.9. 14.10. 14.11.  Introduction. Reference Systems and Coordinate Transformations . Linear Vector Spaces Linear Transformations (Linear Operators) Linear Transformations of a Normed or Unitary Vector Space into Itself. Hermitian and Unitary Transformations (Operators) . . . Matrix Representation of Vectors and Linear Transformations (Opera tors) Change of Reference System Representation of Inner Products. Orthonormal Bases Eigenvectors and Eigenvalues of Linear Operators Group Representations and Related Topics Mathematical Description of Rotations Related Topics, References, and Bibliography  Chapter 15. Linear Integral Equations, Boundary-value Problems, and Eigenvalue Problems Introduction. Functional Analysis Functions as Vectors. Expansions in Terms of Orthogonal Functions. Linear Integral Transformations and Linear Integral Equations . Linear Boundary-value Problems and Eigenvalue Problems Involving Differential Equations 15.5. Green's Functions. Relation of Boundary-value Problems and Eigen value Problems to Integral Equations 15.6. Potential Theory 15.7. Related Topics, References, and Bibliography  15.1. 15.2. 15.3. 15.4.  Chapter 16. Representation of Mathematical Models: Tensor Algebra and Analysis  431  433 435 439  441 447 449 452 457 467 471 482  484 486 487 492 502 515 520 531  533  16.1. Introduction  534  16.2. Absolute and Relative Tensors  537  16.3. 16.4. 16.5. 16.6. 16.7. 16.8.  540 543 543 545 546 549  Tensor Algebra: Definition of Basic Operations Tensor Algebra: Invariance of Tensor Equations Symmetric and Skew-symmetric Tensors Local Systems of Base Vectors Tensors Defined on Riemann Spaces. Associated Tensors . . . . Scalar Products and Related Topics  xv  CONTENTS  16.9. Tensors of Rank Two (Dyadics) Defined on Riemann Spaces.... 16.10. The Absolute Differential Calculus.  Covariant Differentiation.  .  .  16.11. Related Topics, References, and Bibliography Chapter 17.  Differential Geometry  551 552  560 561  17.1. Curves in the Euclidean Plane  562  17.2. 17.3. 17.4. 17.5.  565 569 579 584  Curves in Three-dimensional Euclidean Space Surfaces in Three-dimensional Euclidean Space Curved Spaces Related Topics, References, and Bibliography  Chapter 18.  Probability Theory and Random Processes  18.1. Introduction  18.2. 18.3. 18.4. 18.5. 18.6. 18.7. 18.8. 18.9. 18.10.  Definition and Representation of Probability Models One-dimensional Probability Distributions Multidimensional Probability Distributions Functions of Random Variables. Change of Variables Convergence in Probability and Limit Theorems Special Techniques for Solving Probability Problems Special Probability Distributions Mathematical Description of Random Processes Stationary Random Processes. Correlation Functions and Spectral Densities  585 587  588 593 602 614 620 623 625 637 641  18.11. Special Classes of Random Processes. Examples 18.12. Operations on Random Processes 18.13. Related Topics, References, and Bibliography  650 659 662  Chapter 19.  664  Mathematical Statistics  19.1. Introduction to Statistical Methods  665  19.2. Statistical Description. Definition and Computation of Randomsample Statistics 19.3. General-purpose Probability Distributions  668 674  19.4. Classical Parameter Estimation  676  19.5. Sampling Distributions  680  19.6. Classical Statistical Tests  686  19.7. Some Statistics, Sampling Distributions, and Tests for Multivariate Distributions  19.8. Random-process Statistics and Measurements 19.9. Testing and Estimation with Random Parameters 19.10. Related Topics, References, and Bibliography  Chapter 20. Numerical Calculations and Finite Differences . . . .  697  703 708 713  715  20.1. Introduction  718  20.2. Numerical Solution of Equations 20.3. Linear Simultaneous Equations, Matrix Inversion, and Matrix Eigen  719  value Problems  729  20.4. Finite Differences and Difference Equations  737  20.5. Approximation of Functions by Interpolation 20.6. Approximation by Orthogonal Polynomials, Truncated Fourier Series,  746  and Other Methods  20.7. Numerical Differentiation and Integration  755  770  CONTENTS  xvi  20.8. Numerical Solution of Ordinary Differential Equations 20.9. Numerical Solution of Boundary-value Problems, Partial Differential Equations, and Integral Equations 20.10. Monte-Carlo Techniques 20.11. Related Topics, References, and Bibliography  785 797 800  Chapter 21.  804  Special Functions  777  21.1. Introduction  806  21.2. The Elementary Transcendental Functions 21.3. Some Functions Defined by Transcendental Integrals  806 818  21.4. The Gamma Function and Related Functions  21.5. Binomial Coefficients and Factorial Polynomials.  822  Bernoulli Poly  nomials and Bernoulli Numbers  21.6. Elliptic Functions, Elliptic Integrals, and Related Functions. . 21.7. Orthogonal Polynomials 21.8. Cylinder Functions, Associated Legendre Functions, and Spherical Harmonics  824  827 848 857  21.9. Step Functions and Symbolic Impulse Functions 21.10. References and Bibliography  874 880  Appendix A.  Formulas Describing Plane Figures and S o l i d s . . . .  881  Appendix B.  Plane and Spherical Trigonometry  885  Appendix C.  Permutations, Combinations, and Related Topics.  894  Appendix D.  Tables of Fourier Expansions and Laplace-transform  Pairs  Appendix E.  900  Integrals, Sums, Infinite Series and Products, and  Continued Fractions  Appendix F.  Numerical Tables  925  989  Squares Logarithms Trigonometric Functions Exponential and Hyperbolic Functions Natural Logarithms Sine Integral Cosine Integral  990 993 1010 1018 1025 1027 1028  Exponential and Related Integrals Complete Elliptic Integrals Factorials and Their Reciprocals  1029 1033 1034  Binomial Coefficients Gamma and Factorial Functions Bessel Functions  1034 1035 1037  Legendre Polynomials  1060  Error Function Normal-distribution Areas Normal-curve Ordinates Distribution of t  1061 1062 1063 1064  Distribution of x2  1065  xvii  CONTENTS  Distribution of F  1066  Random Numbers . . . Normal Random Numbers  1070 1075  sin x/x Chebyshev Polynomials  1080 1089  Glossary of Symbols and Notations  1090  Index  1097  MATHEMATICAL HANDBOOK FOR SCIENTISTS AND ENGINEERS  CHAPT  ER 1  REAL AND COMPLEX NUMBERS ELEMENTARY ALGEBRA  1.1. Introduction.  The Real-  1.4. Miscellaneous Formulas  number System 1.4-1. The Binomial Theorem and 1.1-1. Introduction 1.1-2. Real Numbers  1.1-3. Equality Relation 1.1-4. Identity Relation 1.1-5. Inequalities 1.1-6. Absolute Values  Related Formulas  1.4-2. Proportions 1.4-3. Polynomials.  Symmetric  Functions 1.5. Determinants  1.2. Powers, Roots, Logarithms, and Factorials.  Sum and Product  Notation  1.5-1. Definition  1.5-2. Minors and Cofactors.  Expan  sion in Terms of Cofactors 1.2-1. Powers and Roots  1.5-3. Examples: Second- and Third-  1.2-2. Formulas for Rationalizing the Denominators of Fractions  order Determinants  1.2-3. Logarithms  1.5-4. Complementary Minors. Laplace Development  1.2-4. Factorials 1.2-5. Sum and Product Notation  1.5-5. Miscellaneous Theorems  1.2-6. Arithmetic Progression 1.2-7. Geometric Progression  1.5-6. Multiplication of Determinants 1.5-7. Changing the Order of Determi nants  1.3. Complex Numbers 1.6.  1.3-1. Introduction  1.3-2. Representation of Complex  Algebraic Equations: General Theorems  Numbers as Points or Position  1.6-1. Introduction  Vectors.  1.6-2. Solution of an Equation. 1.6-3. Algebraic Equations  Polar Decomposition  1.3-3. Representation of Addition, Multiplication, and Division, Powers and Roots  Roots  1.6-4. Relations between Roots and Coefficients  ELEMENTARY ALGEBRA  1.1-1  1.6-5. Discriminant of an Algebraic Equation 1.6-6. Real Algebraic Equations and Their Roots  (a) (b) (c) (d) (e) (/) (g)  Complex Roots Routh-Hurwitz Criterion Descartes's Rule An Upper Bound Rolle's Theorem Budan's Theorem Sturm's Method  1.8-2. Solution of Quadratic Equations 1.8-3. Cubic Equations: Cardan's Solu tion  1.8-4. Cubic Equations: Trigonometric Solution  1.8-5. Quartic Equations: DescartesEuler Solution  1.8-6. Quartic Equations: Ferrari's Solution  1.9. Systems of Simultaneous Equa tions  1.7. Factoring of Polynomials and Quotients of Polynomials. Par  1.9-1. Simultaneous Equations 1.9-2. Simultaneous Linear Equations: Cramer's Rule  tial Fractions  1.7-1. Factoring of a Polynomial 1.7-2. Quotients of Polynomials. Remainder. Long Division 1.7-3. Common Divisors and Common  Roots of Two Polynomials 1.7-4. Expansion in Partial Fractions 1.8. Linear, Quadratic, Cubic, and Quartic Equations  1.8-1. Solution of Linear Equations  1.9-3. Linear Independence 1.9-4. Simultaneous Linear Equations: General Theory 1.9-5. Simultaneous Linear Equations: n Homogeneous Equations in n Unknowns  1.10. Related Topics, References, and Bibliography  1.10-1. Related Topics 1.10-2. References and Bibliography  1.1. INTRODUCTION THE REAL-NUMBER SYSTEM  1.1-1. This chapter deals with the algebra* of real and complex numbers, i.e., with the study of those relations between real and complex numbers which involve a finite number of additions and multiplications. This is  considered to include the solution of equations based on such relations, even though actual exact numerical solutions may require infinite num bers of additions and/or multiplications.  The definitions and relations  presented in this chapter serve as basic tools in many more general mathematical models (see also Sec. 12.1-1).  1.1-2. Real Numbers. The axiomatic foundations ensuring the selfconsistency of the real-number system are treated in Refs. 1.1 and 1.5, and lead to the acceptance of the following rules governing the addition and multiplication of real numbers. * See also footnote to Sec. 12.1-2.  REAL NUMBERS  1.1-2  a + b and ab are real numbers (algebraic numbers, rational numbers, integers, positive integers) if this is true for both a and b (closure) a+ b= b+ a ab = ba (commutative laws) a+(b + c) = (a + b)+c = a + b + c  a(bc) = (ab)c = abc (associative laws) 0 •1 = a (multiplicative identity) a(b + c) = ab + ac (distributive law)  (1.1-1)  a + c = b + c implies a = b  ca = cb, c^O implies a = b (cancellation laws)  The real number 0 (zero, additive identity) has the properties a + 0 = a  a  0 = 0  (1.1-2)  for every real a.  The (unique) additive inverse —a and the (unique) multiplicative inverse (reciprocal) a-1 = 1/a of a real number a are respectively defined by a + (—a) = a — a = 0  aa~  = 1  (o^O)  (1.1-3)  Division by 0 is not admissible.  In addition to the "algebraic" properties (1), the class of the positive integers 1,2, ... hasthe properties of being simply ordered (Sec. 12.6-2; n is"greater than" m or n > mii and only if n = m + x wherea; is a positive integer) and well-ordered  (every nonempty set of positive integers has a smallest element). A set of positive integers containing (1) 1 and the "successor" n + 1 ofeach ofits elements n, or (2) all integers less than nforany n,contains all positive integers (Principle ofFinite Induction). The properties of positive integers may be alternatively denned by Peano's Five Axioms, viz., (1) 1 is a positive integer, (2) each positive integer n has a unique suc cessor S(n)t (3) S(n) * 1, (4) S(n) = S(m) implies n = m, (5) the principle of finite induction holds. Addition and multiplication satisfying the rules (1) are denned by the "recursive" definitions n + 1 = S(n)} n + S(m) = S(n + m); n •1 = n, n • S(m) — n - m -\- n.  Operations onthe elements m - n of the class of all integers (positive, negative, orzero) are interpreted asoperations oncorresponding pairs (m, n)of positive integers m, n such that (m - n) + n = m, where 0, defined by n + 0 = n, corresponds to (n, n), for all n. An integer is negative if and only if it is neither positive nor zero. The study of the properties of integers is called arithmetic.  Operations on rational numbers m/n (n j* 0) are interpreted as operations on corresponding pairs (m, n) of integers m, n such that (m/n)n = m. m/n is positive if and only if mn is positive.  Real algebraic (including rational and irrational) numbers, corresponding to (real) roots of algebraic equations with integral coefficients (Sec. 1.6-3) and real  1.1-3  ELEMENTARY ALGEBRA  4  transcendental numbers, for which no such correspondence exists, may be intro duced in terms of limiting processes involving rational numbers (Dedekind cuts, Ref. 1.5).  The class of all rational numbers comprises the roots of all linear equations (Sec.  1.8-1) with rational coefficients, and includes the integers. The class of all real algebraic numbers comprises the real roots of all algebraic equations (Sec. 1.6-3) with algebraic coefficients, including the rational numbers. The class of all real numbers contains the real roots of all equations involving a finite or infinite number of additions and multiplications of real numbers and includes real algebraic and tran scendental numbers (see also Sec. 4.3-1).  A real number a is greater than the real number b (a > 6, b < a) if and only if a = b 4- x, where x is a positive real number (see also Sees. 1.1-5 and 12.6-2).  1.1-3. Equality Relation (see also Sec. 12.1-3). An equation a = b implies b = a (symmetry of the equality relation), and a + c = b + c, ac = be [in general, f(a) = f(b) if f(a) stands for an operation having a unique result], a = b and b = c together imply a = c (transitivity of the equality relation), ab j* 0 implies a j& 0, b ^A 0. 1.1-4. Identity Relation.  In general, an equation involving operations on a  quantity x or on several quantities X\, xz, . . . will hold only for special values of x or special sets of values xi, x2, . . • (see also Sec. 1.6-2). // it is desired to stress the fact that an equation holds for all values of x or of xi, xi, . . . within certain ranges of interest, the identity symbol = may be used instead of the equality symbol = [EXAMPLE: (x —l)(x -f 1) = x2 —1], and/or the ranges of the variables in ques tion may be indicated on the right of the equation, a = b (better a — b) is also used with the meaning "a is defined as equal to 6."  1.1-5. Inequalities (see also Sees. 12.6-2 and 12.6-3). a > b implies b < a, a + c> b + cy ac > be (c > 0), -a < -b, 1/a < \/b (a > 0, b > 0). A real number a is positive (a > 0), negative (a < 0), or zero (a = 0). Sums and products of positive numbers are positive, a < A, b < B implies a + b < A + B.  a >b,b > c implies a > c. The absolute  1.1-6. Absolute Values (see also Sees. 1.3-2 and 14.2-5).  value \a\ of a real number a is defined as equal to a if a > 0 and equal to -a if a < 0.  \a\ > 0  Note  \a\ = 0 implies a = 0  I  ||a| - |b|| < \a + b\ < \a\ + \b\  \\a\ - |ft|| < \a - b\ < \a\ + \b\ } (1.1-4)  \ab\ = \a\ \b\  |a| < A,  a  b  =g  \b\ <B implies |a| + |6|<A+£  Q>* 0) and  (1.1-5) \ab\<AB  (1.1-6)  1.2. POWERS, ROOTS, LOGARITHMS, AND FACTORIALS. SUM AND PRODUCT NOTATION  1.2-1. Powers and Roots. The nth power of any real number (base) a is defined as the product of n factors equal to a, where the exponent  5  LOGARITHMS  1.2-3  n is a positive integer. The resulting relations apaq = ap+q  (ap)q = apq  (1.2-1)  are postulated to apply for all real values of p and q and thus serve to define powers involving exponents other than positive integers:  (a^O)  a0 = 1 ap  (1.2-2)  A pth root a1/p = \/a of the radicand a is a solution of the equation xp = a. \/a == \fa is the square root of a, and \/a is its cube root. Powers and roots with irrational exponents can be introduced through  limiting processes (seealso Sec. E-6). In general, y/a is not unique, and some or all of the roots of a given real radicand a may not be real numbers (Sec. 1.3-3). If a is real and positive, many authors specifically denote the real positive solution values of x2 = a, xz = a, xA = a, . . . as  y/a~ == Va, \/a, \/a} . . . . The real solutions of x2 = a, xA = a, . . . are then written as ± \Za, ± y/a, .... To emphasize a choice of the positive square root, one may write yfa. For q ^ 0 ap _  V  ap~q  a* =  aq  (aby p  =  =^  <^Ta = Va  =-«/*)*  VaVb  apbp  (1.2-3)  ap  "' bp  &b  0*0)  1.2-2. Formulas for Rationalizing the Denominators of Fractions.  V& V& ± Vc  - &-c r^; (VS T V5)  &V  v^  b  (1.2-4)  \A+V^ (1.2-5)  -^5 ±<#S  *>±c  ( ^ T v^ + -^?)  (1.2-6)  1.2-3. Logarithms. The logarithm £ = logc a to the base c > 0 (cj^ 1) of the number (numerus) a > 0 may be defined by cx =  a  or  ciogo° = a  (1.2-7)  ELEMENTARY ALGEBRA  1.2-4  Refer to Table 7.2-1 and Sec. 21.2-10 for a more general discussion of  logarithms.  logc a may be a transcendental number (Sec. 1.1-2). Note  logc 1=0 logc C = 1 logc cp = p (logarithmic property) logc (ab) = logc a + logc b  logcf|j =logc a-- logc b logc (ap) = p logc a  logc' a = logc a logc' c =  (1.2-8) (P*0)  l0gc (^) = - logc a V  logc a  logc C'  logc' c =  X"&C "  (c' 5* 1)  1  logc C'  (1.2-9)  (change op base)  Of particular interest are the "common" logarithms to the base 10 and the natural (Napierian) logarithms to the base  e= lim (1 +iV =2.71828182 • • •  (1.2-10)  e is a transcendental number. loge a may be written In a, log a or log nat a. logio a is sometimes written log a. Note  logc a=S^°-5 =l0ge 10 logio a= (2.30259 • • •) logio a log io e  logio a=jjg^ =logio 6logc a=(0.43429 •••) log* a 1.2-4. Factorials.  (1.2-11)  The factorial n! of any integer n > 0 is defined by n  0! = 1  n\ = J] fc = 1 *2 *3 * * • (n - l)n  (n > 0) (1.2-12)  * = i  Refer to Sec. 21.4-2 for approximation formulas.  1.2-5. Sum and Product Notation. negative, or zero) n and m > n  For any two integers (positive,  m  Y a} = a„ + an+i + • • • + am_i + am  (m —n + 1 terms) (1.2-13)  f[ ay se anan+i • • • am-iam  (m —n + 1 factors)  (1.2-14)  7  COMPLEX NUMBERS  1.3.1  Note m  m'  2/ 2atk=2 a aik n n «•*=n n««a.*-!*) V .=nfo +1)  y j2 =n(n +l)(2n +1)  V =n2(n +1)» (1.2-16)  Refer to Chap. 4 for infinite series; and see Sec. E-3.  1.2-6. Arithmetic Progression. If a0 is the first term and d is the common difference between successive terms a,-, then aj = ao+jd(j = 0, 1, 2, . . .) n  «»= 2,a* =^Hr^ (2a° +™*) =rj^1 (a°+ *»> c1-2"17) y=o  1.2-7. Geometric Progression. If a0 is the first term and r is the  common ratio of successive terms, then (see Sec. 4.10-2 for infinite geo metric series)  a} = a** (J = 0, 1, 2, . . .) — r»+i  Sn =2/ai =2/aor'"= a° t^i i=o  ao — ttnr  y=o  1.3. COMPLEX NUMBERS  1.3-1. Introduction (see also Sec. 7.1-1). Complex numbers (some  times called imaginary numbers) are not numbers in the elementary sense used in connection with counting or measuring; they constitute a new class of mathematical objects defined by the properties described below (see also Sec. 12.1-1).  Each complex number c may be made to correspond to a unique pair (a, b) of real numbers a, 6, and conversely. The sum and product of two complex numbers cx <-> (ah bx) and c2 <-> (a2, b2) are defined as  ci + c2 <-• (ai + a2, bi + b2) and Cic2 <-> (aia2 - bj)2, aib2 + a2bx), respec  tively. The real numbers a are "embedded" in this class of complex numbers as the pairs (a, 0). The unit imaginary number i defined as i <-> (0, 1) satisfies the relations  *'2 = (-*')2 = -1  *= V=l  -i = - V^T  (1.3-I)  ELEMENTARY ALGEBRA  1.3-2  Each complex number c <-> (a, b) may be written as the sum c = a + ib of a real number a <-» (a, 0) and a pure imaginary number z'6 <-• (0, 6), The real numbers a = Re (c) and 6 = Im (c) are respectively called the  real part of c and the imaginary part of c. Two complex numbers c = a + ib and c* = a — ib having equal real parts and equal and opposite imaginary parts are called complex conjugates. Two complex numbers Ci = a\ + ibi and c* = a2 + i&2 are equal if and only if their respective real and imaginary parts are equal, i.e., c\ = c2 if and only if ai = a2,6i = &2. c = a + ib = 0 implies a = 6 = 0. Addition and multiplication of complex numbers satisfies all rules of Sees. 1.1-2 and 1.2-1, with =  -1 7-4n+3 =  f4«+4 = i  (n = 0, 1, 2, . . .)  (1.3-2)  ci ± c2 = (ai ± a2) + *'(&i ± 62) CiC2 = (aia2 — bj)2) + i(aj)2 + a2bi)  ci _ a\ + ib\ _ (aia2 + bj)2) + i(a2bi —aib2) (c2 ^ 0) c2  a2 + ib2  a22 + b22  (1.3-3)  (ci + c2)* = c\ + c% (cic2)* = c\ct (ci/c2)* = c*/c? (c, * 0) (c±c*)  a = Re (c) =  b = Im (c) =  (c*)* 2i  The class of all complex numbers contains the roots of all equations based on addi tions and multiplications involving complex numbers and includes the real numbers.  1.3-2. Representation of Complex Numbers as Points or Position Vectors. Polar Decomposition (see also Sec. 7.2-2). Complex  Fig. 1.3-1. Representation of complex numbers as points or position vectors. The x axis and the y axis are called the real axis and the imaginary axis, respectively.  OPERATIONS WITH COMPLEX NUMBERS  1.3-3  numbers z = x + iy are conveniently represented as points (z) = (re, y) or corresponding position vectors (Sees. 2.1-2 and 3.1-5) in the Argand or Gauss plane (Fig. 1.3-1). The (rectangular cartesian, Sec. 2.1-3) x axis and y axis are referred to as the real axis and imaginary axis, respectively. The abscissa and ordinate of each point (z) respectively represent the real part x and the imaginary part y of z. The correspond ing polar coordinates (Sec. 2.1-8)  r = \z\ = y/x2 + y2 = \/zz* = |**| <p = arg (z) = arctan (y/x) = - arg (z*)  (1.3-4)  are, respectively, the absolute value (norm, modulus) and the argu ment (amplitude) of the complex number z. x = r cos <p  y = r sin <p  Note  z = x + iy  = r(cos <p + i sin <p)  (1.3-5)  TTie absolute values of complex numbers satisfy the relations (1.1-4) to (1.1-6); if z is a real number, the definition of \z\ reduces to that of Sec. 1.1-6.  For any two sets of (real or) complex numbers ah a2, . . . , an; Pi, 02, . . . , Pn (see also Sees. 14.2-6 and 14.2-6a), n  n  n  /] oi*Pi\ < Y M2 y IftI2 (Cauchy-Schwarz inequality) (1.3-6) *»1  1.3-3. Representation of Addition, Multiplication, and Division. Powers and Roots. Addition of complex numbers corresponds to addition of the corresponding position vectors (see also Sees. 3.1-5 and 5.2-2). Given zx = ri(cos ?i + i sin <pi), z2 = r2(cos <p2 + i sin <p2), Z\Z2 Zx z2 zp  = rir2[cos (<pi + <p2) + i sin (<px + ^2)] r*  = —[cos (<pi —<p2) + i sin (<pi —<p2)]  (z2 9^ 0)  (1.3-7)  = rp (cos <p + i sin <p)*> = rp[cos {p<p) + i sin (p<p)] (De Moivre^ theorem)  (See Sec. 21.2-9 for the case of complex exponents.) All formulas of Sees. 1.2-1 to 1.2-7 hold for complex numbers (see also Sees. 1.4-1 to 1.4-3).  10  ELEMENTARY ALGEBRA  1.4-1  Note nr-  V 1 = cos  „  V  2klT  .  . .  h %sin  2&7T  (2fc + Pit  — + isin  —1 = cos-  I (n = 1, 2, . . . ; (1.3-8)  (2fc + ivra0>1'2>'"n"1) (n vafoes) n  In particular,  VI = ±1  (1.3-9)  V11! = ±*  VI =I cos 120° +*sin 120° =M(-l +*V5)  \ cos 120° - t sin 120° =3^(~1 - *V*)  (1.3-10)  {cos 60° + t sin 60° = \i(\ + %V3) -1  cos 60° - i sin 60° = >£(1 - * v3)  1.4. MISCELLANEOUS FORMULAS  1.4-1. The Binomial Theorem and Related Formulas. If a, 6, c are real or complex numbers,  (a ± 6)2 = a2 ± 2ab + b2  la ± by = a3 ± Sa2b + 3ab2 ± 63 (a ± by = a4 ± 4a36 + 6a2&2 ± 4a&3 + 64 (a + b)n with  i  2 (")•"-  (1.4-1)  (n = 1, 2, . . .)  n\  j/ ~j\(n-j)\  (j = 0, 1, 2, . . . < n = 0, 1, 2, . . .)  (Binomial Theorem for integral exponents n; see also Sec. 21.2-12). The  binomial coefficients (n Jare discussed in detail in Sec. 21.5-1. (a + b+ c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc (a2 - b2) = (a + b)(a - b) (a2 + b2) = (a + ib)(a - ib) an _ 5» = (a - &) (a*-1 + an~2b +  (1.4-2) (1.4-3)  + abn~2 + fc""1)  (1.4-4a)  + abn~2 - bn~l)  (1.4-46)  If n is an even positive integer,  an _ hn = (a + b)(a«-1 - an"2& +  11  POLYNOMIALS.  SYMMETRIC FUNCTIONS  1.4-3  If n is an odd positive integer,  an + bn = (a + 6) (a*-1 - an~2b d- • • • - abn~2 + b*-1) (1.4-5) Note also  a4 + a2b2 + 64 = (a2 + ab + b2)(a2 - afc + 62) 1.4-2. Proportions.  (1.4-6)  a:b = c:d or a/6 = c/d implies  ma + nb _ mc + nd  pa + qb  (corresponding addition  pc + qd  and subtraction)  (1.4-7)  In particular,  a± b  c± d  a —b __ c — d  b  d  a + b ~~ c + d  (1.4-8)  1.4-3. Polynomials. Symmetric Functions, (a) A polynomial in (integral rational function of) the quantities xlf x2, . . . , xn is a  sum involving a finite number of terms of the form axiklx2k2 • • • xnkn, where each kj is a nonnegative integer. The largest value of ki + k2 + • ' ' + kn occurring in any term is the degree of the polynomial. A polynomial is homogeneous if and only if all its terms are of the same degree (see also Sec. 4.5-5).  (b) A polynomial in xlt x2, . . . , xn (and more generally, any function of xu X2, . . . , xn) is (completely) symmetric if and only if its value is unchanged by permutations of the x1} x2, . . . , xn for any set of values xlf x2, . . . , xn. The elementary symmetric functions Slt S2, . . . , Sn of xu x2, . . . , xn are the polynomials  & = Xi -j- X2 + • • • + Xn  S2 ss XXX2 + XxXz + • • •  SS ss X1X2X3 + 2:1X2X4 +  Sn S3 XlX2 • • • Xn (1.4-9)  n\  where 5* is the sum of all (n _ k^k^ products combining kfactors Xj without repeti tion ofsubscripts (see also Table C-2). Every polynomial symmetric inxlt x2} . . . , xn can be rewritten as a unique polynomial in Slf S2, . . . , Sn; the coefficients in the new polynomial are algebraic sums of integral multiples of the given coefficients.  Every polynomial symmetric in xlf x2, . . . , xn can also be expressed asa polynomial  in a finite numberof the symmetricfunctions n  ^ Xi ^l  n  82 s= 2J Xi*  n  ' ' ' S* =Y Xik  *^i  (1.4-10)  ffi  The symmetric functions (9) and (10) are related by Newton's formulas  (-l)"kSk + (-1)*-**-*! + (-l)*-%_2s2 + ...» 0  (jfc = 1, 2, • • •) (1.4-11)  where one defines Sk = 0for k > nand k < 0, and So =1(see also Ref. 1.2 for explicit tabulations of Newton's formulas; and see also Sec. 1.6-4). Note that the relations (11) do not involve n explicitly.  12  ELEMENTARY ALGEBRA  1.5-1  1.5. DETERMINANTS  1.5-1. Definition.  The determinant  D — det [aik] =  an  #12  din  a2\  a22  a2n  an\  an2  (1.5-1)  •  of the square array (matrix, Sec. 13.2-1) of n2 (real or complex) numbers (elements) aik is the sum of the n\ terms (—l)raiJfcla2fc8 • • • ankn each corresponding to one of the n\ different ordered sets fci, k2, . . . , kn obtained by r interchanges of elements from the set 1, 2, . . . , n. The number n is the order of the determinant (1). The actual computation of a determinant in terms of its elements is simplified by the use of Sees. 1.5-2 and 1.5-5a.  Note that n  n  |£|2 < 11 / I0**'2 (Hadamard's INEQUALITY) (1.5-•2) = 1 *=1  1.5-2. Minors and Cofactors.  Expansion in Terms of Cofactors.  The (complementary) minor Dik of the element aik in the nth-order determinant (1) is the (n — l)8t-order determinant obtained from (1) on erasing the ith row and the kth column. The cofactor A** of the element ane is the coefficient of aik in the expansion of Z), or  A,, = (-l)-A* = ^  (1.5-3)  A determinant D may be represented in terms of the elements and cofac tors of any one row or column as follows: n  D = det[o,*] == y  n  aijkij = Y ajkAjk  i-1  0" = 1, 2, . . . , n)  (1-5-4)  A=i  (simple Laplace development)  Note also that n  n  y atjAih = T OjtAu = 0 i^\  i = l  k- 1  (i * h)  (1.5-5)  13  MISCELLANEOUS THEOREMS  1.5-5  1.5-3. Examples: Second- and Third-order Determinants. On  I O21 On  012  0i«  (1.5-6)  flllfl&22 — 0>2lQ>l2  0221  Ois  O21  O22  O28  O31  CI32  O33  — a\\a22azz — 011*123032 + 012023031 — #13022031 H~ 018021032 — 012021033  = Oll(022033 —O32O28) —02i(Oi2088 —0820i3) + 03l(Oi2028 ~ O22O13) = Oii(0220»8 —08202s) —Oi2(02l088 —08i028) + Oi8(02l082 —081022) etc.  1.5-4. Complementary Minors.  Laplace Development.  (1.5-7)  The mth-order deter  minant M obtained from the nth-order determinant D by deleting all rows except for the m rows labeled iu i2) . . . , im (m < ri), and all columns except for the m columns labeled klt k2f . . . , km is an ra-rowed minor of D.  The m-rowed minor M  and the (n - w)-rowed minor M' of D obtained by deleting the rows and columns conserved in M are complementary minors; in the special case m = n, M' = 1. The algebraic complement M" of M is defined as ( —l)»'i+<« *' *+»'m+*i+*» •••+*«ilf/.  Given any m rows (or columns) ofD, D is equal to the sum ofthe products MM" ofall m-rowed minors M using these rows (or columns) and their algebraic complements M" (Laplace development by rows or columns).  An nth-order determinant Dhas (*J m-rowed principal minors whose diagonal elements are diagonal elements of D.  1.5-5. Miscellaneous Theorems, (a) The value D of a determinant (1) is not changed by any of the following operations:  1. The rows are written ascolumns, and the columns asrows [interchange of i and k in Eq. (1)].  2. An even number of interchanges of any two rows or two columns.  3. Addition of the elements of any row (or column), all multiplied, if desired, by the same parameter a, to the respective corresponding ele ments of another row (or column, respectively). EXAMPLES: All  Ol2  a2i  a22  CLnl  an%  a2n  an  a2\  Ctnl  c&i2  a22  an2  flin  a2n  an + aai2 a2i + aa22  • • •  ai2 a22  aT  • • • ain • • • a2n  (1.5-3) ani + aan2  an2  (b) An odd number of interchanges of any two rows or two columns is equivalent to multiplication of the determinant by —1.  (c) Multiplication of all the elements of any ~one row or column by a factor a is equivalent to multiplication ofthe determinant by a.  1.5-6  ELEMENTARY ALGEBRA  14  (d) If the elements of the jth row (or column) of an nth-order determinant D mm  m  are represented as sums ^ cri, } cr2, ...,}. c>-*> &2S eQ.ual to the sum r=l  r=l  r=l  m  y Dr of mnth-order determinants Dr. The elements of each DT are identi cal with those of D, except for the elements of the jth row (or column, respec tively), which are cri, cf2, . . . , crn. EXAMPLE:  On + &11  O12 + &12  Osi  O22  Onl  a„2  • • • Oi» + bm 02n  On  Oi2  flln  6ll  612  6m  021  O22  02n  O21  O22  02n  0„i  0„2  Onl  On2  Onn  (1.6-9)  (e) A determinant is equal to zero if 1. All elements of any row or column are zero.  2. Corresponding elements of any two rows or columns are equal, or proportional with the same proportionality factor.  1.5-6. Multiplication of Determinants (see also Sec. 13.2-2). The product of two nth-order determinants det [aik] and det [bik] is 71  «  det [aik] det [bik] =det [ Ya^J =det [ 2, ^'J y-i  y-i  n  n  - det [^ «fiu] =det [£ a*6,*] (1-5-10) 'y-i  y=i  1.5-7. Changing the Order of Determinants.  A given determinant may be  expressed in terms of a determinant of higher order as follows: On  O12  O21  O22  Onl  On2  *  •  dm  On  O12  *  '  a2n  021  022  '  •  o„„  Onl  On2  0  0  *  OmCKi  •  '  02n<*2  •  •  annan  (1.6-11)  =  *  • • •  0  1  where the ct% are arbitrary. This process can be repeated as desired. The order of a given determinant may sometimes be reduced through the use of the relation  15  ALGEBRAIC EQUATIONS  On  012  • • •  Oi»  an  ai2  *  •  *  aim  Oil  022  •  • •  o2„  «21  Ct22  '  •  *  ot2m  o„i  0»2  •  onn  ot„i  «n2  '  Otnm  0  0  .  •  0  0  0  •  •  0  611 621  6l2 622  '  1.6-3  ' b\m * &2m  On  O12  Oin  021  022  02n  Onl  0»2  fell 621  6l2 622  bmi  bm2  blm b2m  (1.6-12)  1.6. ALGEBRAIC EQUATIONS: GENERAL THEOREMS  1.6-1. Introduction. The solution of algebraic equations is of particular import ance in connection with the characteristic equations of linear systems in physics (see also Sees. 9.4-1, 9.4-4, and 14.8-5). The general location of the roots needed (e.g., for stability determinations) may be investigated bythemethods ofSec. 1.6-6 and/or Sec. 7.6-9. Numerical solutions are discussed in Sees. 20.2-1 to 20.2-3.  1.6-2. Solution of an Equation. Roots. To solve an equation (see also Sec. 1.1-3)  /(*) = 0  (1.6-1)  for the unknown x means to find values of x [roots of Eq. (1), zeros of f(x)] which satisfy the given equation, x = Xi is a root (zero) of order (multiplicity) m (multiple root if m > 1; see also Sec. 7.6-1) if and only if, for x = xh f(x)/(x - xi)m~l = 0 and f(x)/(x - xt)m ?* 0. A complete solution of Eq. (1) specifies all roots together with their orders. Solutions may be verified by substitution.  1.6-3. Algebraic Equations. An equation (1) of the form  f(x) = aox» + aix*-1 + • • • + an-ix + an = 0 (a„ ^ 0) (1.6-2) where the coefficients a» are real or complex numbers, is called an algebraic equation of degree n in the unknown x. f(x) is a poly nomial of degree n in x (rational integral function; see also Sees.  4.2-2d and 7.6-5). an is the absolute term of the polynomial (2). An algebraic equation ofdegree n has exactly n roots if a root oforder m is counted as m roots (Fundamental Theorem of Algebra). Numbers expressible as roots of algebraic equations with real integral coefficients are algebraic numbers (in general complex, with rational and/or irrational real and  imaginary parts); if the coefficients are algebraic, the roots are still algebraic (see also Sec. 1.1-2). General formulas for the roots of algebraic equations in terms of the coefficients and involving only a finite number of additions, subtractions, multi-  ELEMENTARY ALGEBRA  1.6-4  16  plications, divisions, and root extractions exist only for equations of degree one (linear equations, Sec. 1.8-1), two (quadratic equations, Sec. 1.8-2), three (cubic equations, Sees. 1.8-3 and 1.8-4), and four (quartic equations, Sees. 1.8-5 and 1.8-6).  1.6-4. Relations between Roots and Coefficients. The symmetric functions Sk  and sk (Sec. 1.4-3) of the roots xh x2) . . . , xn of an algebraicequation (2), are related to the coefficients a0, ait . . . , a„ as follows: (k = 0, 1, 2, • • • , n)  (1.6-3)  Go  kak + ajb_isi + ak-2s2 + • • • + Oo«* =0  (k = 1, 2, • • •)  (1.6-4)  where one defines ak = 0 for k > n and k < 0. The equations (1.6-4) are another version of Newton'sformulas (1.4-11). Note also  Ok  (-D*  a0  A;!  o  s,  1  S2  Si  2  0 • • •  Sz  82  Sl  3  8k  8k-i  a\ 2o2  ao ffli  0 Q>o  0 • • •  3fi&3  a2  Q>\  flo 0  kak  ak-i  «-K.)'  0 • Si  (k = 1, 2, • • • , n)  (1.6-5)  ay  1.6-5. Discriminant of an Algebraic Equation.  The discriminant A of an  algebraic equation (2) is the product of a02n~2 and the squares of all differences (xi —Xk)(i > k) between the roots Xi of the equation (a multiple root of order m is considered as m equal roots with different subscripts),  A = a02n~2 PI (Xi - x*)2 == a°2'  = a02n"2  1 1  Xi x2  Xi2 x2*  X\n'  1  Xn  Xn  Xnn  50  Si  s2  Sn_l  51  S2  S3  Sn  Sn-1  Sn  Sn+1  ' ' *  xf  n(n-l)  = (-D  i2(/,f)  2  (1.6-6)  ao  S2n-2  where R(f, /') is the resultant (Sec. 1.7-3) offix) andits derivative (Sec. 4.5-1)/'(x). Ais a symmetric function of the roots xh x2, . . . , xn and vanishes if and only if f{x) has at least one multiple root [which is necessarily a common root off(x) and f'(x); see also Sec. 1.6-69]. The second determinant in Eq. (6) is called Vandermondef8 determinant.  1.6-6. Real Algebraic Equations and Their Roots. An algebraic equation (2) is called real if and only if all coefficients a* are real; the corresponding real polynomial f(x) is real for all real values of x. The following theorems are useful for determining the general location of roots (e.g., prior to numerical solution, Sec. 20.2-1; see also Sees. 9.4-4  17  REAL ALGEBRAIC EQUATIONS AND THEIR ROOTS  and 14.8-5).  1.6-6  In theorems (b) through (/), a root of order m is counted  as m roots.  (a) Complex Roots. Complex roots of real algebraic equations occur in pairs"of complex conjugates (Sec. 1.3-1). A real algebraic equation ofodd degree must have at least one real root.  (b) Routh-Hurwitz Criterion. The number of roots with positive real parts of a real algebraic equation (2) is equal to the number of sign changes (disregard vanishing terms) in either one of the sequences m rp  T2 T%  1 0, i l, TjT> 7jT> * ' *  (1.6-7)  To, Th TiT2, T2TZ,  or  where  T0 = a0 > 0  Tx = ax  T2 = a3  «1  T» =  a3 ah  ao  0  a2  ai  a4  a3  Tt =  a2  ai  ao  0  0  a3  a2  ai  ao  a5  at  a3  a2  a7  a6  a5  a\  (1.6-8)  Given ao > 0, all roots have negative real parts if and only if T0, Th T2, . . . , Tn are all positive. This is true if and only if all a{ and either  all even-numbered Tk or all odd-numbered Tk are positive (Lienard-Chipart Test).  alternative formulation.  All the roots of a real nth-degree equation  (2) have negative real parts if and only if this is true for the (n —l)Bt-degree equation  afcn-1 + a[xn~2 + a'2xn-* + a^n~4 + + a3xn~z + aan~A +  '  s aixn~l + a2xn~2 ao  azxn  -2  _  ai  ao  abxn ai  = 0  This theorem may be applied repeatedly and yields a simple recursion  scheme useful, for example, for stability investigations. The number of roots with negative real parts is precisely equal to the number of negative multipliers -atP/ai™ (j = 0,1,2, ... ,n - 1;a0(0) = a0 > 0, ai<°> = ai) encountered in successive applicationsof the theorem.  The method becomes  more complicated if one of the a^ vanishes (see Ref. 1.6, which also indicates an extension to complex equations).  (c) Location of Real Roots: Descartes's Rule of Signs. The number of positive real roots of a real algebraic equation (2) either is equal to the number Na ofsign changes in the sequence a0, ah . . . , an ofcoeffi cients, where vanishing terms are disregarded, or it is less than Na by a positive even integer. Application of this theorem to f(-x) yields a similar theorem for negative real roots.  1.6-6  ELEMENTARY ALGEBRA  18  (d) Location of Real Roots: An Upper Bound (Sec. 4.3-3a) for the Real Roots. // the first k coefficients a0, ah . . . , ak-i in a real  algebraic equation (2) are nonnegative (ak is the first negative coefficient) then all real roots of Eq. (2) are smaller than 1 + y/q/a0, where q is the absolute value of the negative coefficient greatest in absolute value. Applica tion of this theorem to f(—x) may similarly yield a lower bound of the real roots.  (e) Location of Real Roots: Rolle's Theorem (see also Sec. 4.7-la). The derivative (Sec. 4.5-1) f(x) of a real polynomial f(x) has an odd number of real zeros between two consecutive real zeros off(x). f(x) = o hasno real root or one real rootbetween two consecutive real roots o, b of f{x) = 0 if f(a) t* 0 and f(b) j± 0 have equal or opposite signs, respectively. At most, one real root of f(x) = 0 is greater than the greatest root or smaller than the smallest root off'(x) = 0.  (f) Location of Real Roots: Sudan's Theorem. For any real algebraic equation (2), let N(x) be the number of signchanges in the sequence of derivatives (Sec. 4.5-1) f(x), f'(x), f"(x), . . . , fn)(x), if vanishing terms are disregarded. Then the number of real roots of Eq. (2) located between two real numbers a and b > a not themselves roots of Eq. (2) is  either N(a) - N(b), or it is less than N(a) - N(b) by a positive even integer. The number of real roots of Eq. (2) locatedbetween a and b is odd or even if f(a) and f(b) have opposite or equal signs, respectively.  (g) Location of Real Roots: Sturm's Method. Given a real algebraic equation (2) without multiple roots (Sec. 1.6-2), let N(x) be the number of sign changes (disregard vanishing terms) in the sequence of functions  /o = f(x) = go(x)fi(x) - f2(x)  fi = f'(x) = gi(x)f2(x) - U(x) h(x) = g*(x)U(x) - U(x)  • • • (1.6-9)  where for i > 1 eachfi(x) is (-1) times the remainder (Sec. 1.7-2) obtained on dividing fi-2(x) by fi-i(x); fn(x) j* 0 is a constant. Then the number of real roots of Eq. (2) located between two real numbers a and b > a not them selves roots of Eq. (2) is equal to N(a) —N(b). Sturm's method applies even if, for convenience in computation, a function /<(x) in the above process is replaced by Fi(x) = fi(x)/k(x), where k(x) is a positive con stant or a polynomial in x positive for a < x < b, and the remaining functions are based on Fi(x) instead of on ft(x). Similar operations may be performed again on any of the F3(x), etc.  If f(x) has multiple roots, f(x) and f'(x) have a common divisor (Sec. 1.7-3); in this case, fn{x) is not a constant, and N{a) - N(b) is the number of real roots between a and 6, where each multiple root is counted only once.  QUOTIENTS OF POLYNOMIALS.  19  REMAINDER  1.7-2  1.7. FACTORING OF POLYNOMIALS AND  QUOTIENTS OF POLYNOMIALS.  PARTIAL FRACTIONS  1.7-1. Factoring of a Polynomial (see also Sec. 7.6-6). If a poly nomial F(x) can be represented as a product of polynomials fi(x), f2(x), • • • >Mx), these polynomials are called factors (divisors) of F(x). If x = Xi is a zero of order m of any factor fi(x), it is also a zero of order M > m of F(x). Every (real or complex) polynomial f(x) of degree n in x can be expressed as a product of a constant and n linear factors (x —ak) in one and only one way, namely, f(x) = aoxn + axxn~l + • • • + an-ix + an  = a0 Yl (x --  xk)  (1.7-1)  &= i  where the xk arethe zeros off(x); a zero xk of order mk (Sec. 1.6-2) contributes mk factors (x —xk) (Factor Theorem). Pairs of factors [x — (ak + uak)], [x —(ak —im)] corresponding to pairs of complex conjugate roots (see also Sec. 1.6-6a) xk —ak + im, xk = ak —io)k may be combined into real quadratic factors [(x — ak)2 + m2].  1.7-2. Quotients of Polynomials. Remainder. Long Division. The quotient F(x)/f(x) of a polynomial F(x) of degree N and a poly nomial f(x) of degree n < N may be expressed in the form F(x) __ Aox" + A&"-•i+ . . • +AN aozn + a.ix"~ i+ . . • + an fix) = (box1*-" + biXN~"--i +  ri(x) . . . + b„. -„) +  (1.7-2)  fix)  where the remainder rx(x) is a polynomial of degree smaller than n.  The coefficients bk and the remainder rx(x) are uniquely determined, e.g., by the process of long division (division algorithm) indicated in Fig. 1.7-1.  In Fig. 1.7-1, each product b0f(x), bif(x), ... is subtracted in turn, with the coefficients b0, bh . . . chosen so as to eliminate the respective coefficients of xN, xN~l, . . . , in successive differences until the remainder  is reached. The remainder rx(x) vanishes if and only if f(x) is a divisor (Sec. 1.7-1) of F(x).  The remainder obtained on dividing any polynomial f(x) by (x —c) is equal to f(c) (Remainder Theorem).  Aoao *"-» + ' ' ')/  ax + 6  (2)  H  ax  + &  (a2 - l)x2 + ax - (a2 - l)a2 + b -[(a2 - l)x2 - (a2 - Da2]  - (x4 - a2x2)  Fig. 1.7-1. Long division.  (x4 - x2 + ax - a4 + a2 + 6) -5- (x2 - a2) =x2 + (a2 - 1) + g2 _ fl2  »  H Cd  ^  g  - 1  —  -(-2x+4)  j* ^  -2x + 3  - (2x3 - 4x2)  j^>  |  (2x3 - 4x2 - 2x + 3) + (x - 2) = 2x2 - 2  (1)  w IT w  EXAMPLES X — >5  At *"-» + •••) * (aox« +ai*--» +ajx-"»+•••) =j**»- +(|j - Ao |±)•»—» + •••+n<»)  (A:-Ao^)x-+(a,-Ao^)x-+ ..  - f\ Aox" + Aoao **-» +  (Ao** + Ai **-i +  21  EXPANSION IN PARTIAL FRACTIONS  1.7-4  1.7-3. Common Divisors and Common Roots of Two Polynomials. If a poly nomial g(x) is a common divisor (factor) of F(x) and f(x), its zeros are common zeros oiF(x) and /(x). In the quotient (2), any common divisor may be factored out and canceled as with numerical fractions.  F(x) and f(x) have at least one common root (and thus a common divisor of degree greater than zero) if and only if the determinant of order N + n A0A1A2 •  Ajv-iAtf  A0A1A2  R(F, /) ao aj a% •  0 ao ai aj  '  0  0  •  •  '  •  •  •  •  •  •  •  •  •  0 0  •  an- io»0  an-ian  *  •  An-iAnQ  An-iAs •  0  ao ai as *  •  (1.7-3  0  On-l an  [resultant of F(x) and f(x)] is equal to zero; otherwise, F(x) and /(x) are relatively prime.  The greatest common divisor (common factor of greatest degree) of F(x) and f(x) is uniquely defined except for a constant factor and may be obtained as follows:  Divide rx(x) into /(x); divide the resulting remainder r2(x) into n(x), and continue until some remainder, rk(x), say, vanishes. Then any constant multiple of rk-\(x) is the desired greatest common divisor.  1.7-4. Expansion in Partial Fractions. Any quotient g(x)/f(x) of a polynomial g(x) of degree m and a polynomial f(x) of degree n > m, without common roots (Sec. 1.7-3) can be expressed as a sum of n partial fractions corresponding to the roots xk (of respective orders mk) oif(x) = 0 as follows:  f(x)  Li Li (x - xky k  y=i  = y r ft**  1  ^2  Lil(z-zk) ~*~ (x-xky +  (1.7-4) +  Vkmk  (x - Xk,:)-J  The coefficients bkj are obtained by one ofthe following methods, or by a com bination of these methods:  1. If mk = 1 (xk is a simple root), then bki = g(xk)/f(xk). 2. Multiply both sides of Eq. (4) by f(x) and equate coefficients of equal powers of x on both sides.  3. Multiply both sides of Eq. (4) byf(x) and differentiate successively. Let <pk(x) = f(x)/(x - xk)mK Then obtain 6*^, bkmk-i, . . . suc cessively from  1.8-1  ELEMENTARY ALGEBRA  22  g(Xk) = bkmk<Pk(Xk) g (Xk) = bkmk(pk(Xk) + bkmk-l(Pk(Xk)  g"(Xk) = bkmk(pk(Xk) + 2bkmk-Kp,k(Xk) + 2bkmk-*Pk(Xk) g^-l)(xk) = bkmk^mk~l)(xk) + mkbkmk-i<Pkimk-2)(xk) + mkbkmk-2(mk - l)iPkimk~z)(xk) + • • • + mk\bknpk(xk) The partial fractions corresponding to any pair of complex conjugate roots ak + io>k, ak —iu>k of order mk are usually combined into x + dki c*i [(» T7Z  ,  x + di  „ \2 +I co,2] .. 21 "T" c*2 [(* - a,)2 + co,2]2 - a,)2 ^ "*2  + ' • • +ckmkI,„_\\2T' 2w *m* [(* - a,)2 + co,2]-*  d.7-5)  The coefficients ckj and dkj may be determined directly by method 2 above. If g(x) and/(x) are real polynomials (Sec. 1.6-6), all coefficients bkj, Ckj, dkj in the resulting partial-fraction expansion are real. Every rational function of x (Sec. 4.2-2c) can be represented as a sum of a  polynomial and a finite set of partial fractions (see also Sec. 7.6-8). Par tial-fraction expansions are important in connection with integration (Sec. 4.6-6c) and integral transforms (Sec. 8.4-5). 1.8. LINEAR, QUADRATIC, CUBIC, AND QUARTIC EQUATIONS  1.8-1. Solution of Linear Equations. The solution of the general equation of the first degree (linear equation) ax = b  or  is  ax - b = 0  (c^O)  x= h  (1.8-1)  (1.8-2)  a  1.8-2. Solution of Quadratic Equations. ax2 + bx + c = 0  The quadratic equation (a ?*Q)  (1.8-3)  has the roots  Xl'2 =  -b ± y/b2 - 4ac Ta  (1.8-4)  The roots Xi and x2 are real and different, real and equal, or complex  conjugates if the discriminant (Sec. 1.6-5) D = b2 - 4ac is, respectively, positive, zero, or negative. Note xi + x2 = —b/a, x\X2 = c/a.  23  CUBIC EQUATIONS: TRIGONOMETRIC SOLUTION  1.8-4  1.8-3. Cubic Equations: Cardan's Solution.  The cubic equation  x3 + ax2 + bx + c = 0  (1.8-5)  is transformed to the "reduced" form  y* + py + q= 0  p= ~j + b a  through the substitution x = y —a/3.  q= 2(%) - ^ + c (1.8-6) "(J)'-* The roots yh y2, y* of the  "reduced" cubic equation (6) are  a ^ t> 2/i = A+ B  2/2,3 =  A + B t .A - B  ,g— ± *—g— ^3  with A=^j- | +Ve ^=^"| " VQ -G)'+(f)"  > (1-8-7)  Q  where the real values of the cube roots are used. The cubic equation has one real root and two conjugate complex roots, three real roots of which at least two are equal, or three different real roots, if Q is positive, zero, or negative, respectively. In the latter case ("irreducible" case), the method of Sec. 1.8-4a may be used. Note that the discriminants (Sec. 1.6-5) of Eq. (5) and Eq. (6) are both equal to - 108Q. 1.8-4. Cubic Equations: Trigonometric Solution,  (a) H Q < 0  ("irreducible" case)  2/i = 2 V-p/3 cos (a/3)  with  2/2,3 = -2 V-p/3 cos (a/3 ± 60°) , + ) cos a = g 2V-(p/3)3  (1.8-8)  (b) If Q > 0, p > 0  2/i = -2 Vp/3 cot 2a  2/2,3 = \/p73 (cot 2a ± i \/3 cosec 2a)  with  \  1 (1.8-9a)  tan a =v^tan (0/2) (|a| <45°)  tan 0=2VTp/3J*/q (|/3| <90°) j  (c) If Q > 0, p < 0  yi = -2 V-P/3 cosec 2a  with  2/2,s = V-p/3 (cosec 2a ± *\/3 cot 2a) \  tan a=Vtan (0/2) (|a| <45°)  I (1.8-9&)  sin /3 =2V(-p/3)3/tf (|0| <90°) j  The real value of the cube root is used.  1.8-5  ELEMENTARY ALGEBRA  1.8-5. Quartic Equations: Descartes-Euler Solution. equation (biquadratic equation)  24  The quartic  x* + ax* + bx2 + ex + d = 0  (1.8-10)  is transformed to the "reduced" form  y* + py2 + qy + r = 0  through the substitution x = y — a/4.  (1.8-11)  The roots 2/1, 2/2, 2/3, 2/4 of the  "reduced" quartic equation (11) are the four sums  ± Vzl ± Vz2 ± V*3  (1.8-12)  with the signs of the square roots chosen so that  Vz~i Vz~2 V*~3 = -q/S  (1.8-13)  where Z\, z2, zs are the roots of the cubic equation  1.8-6. Quartic Equations: Ferrari's Solution. Given any root yi of the resolvent cubic equation corresponding to Eq. (10) yz _ fy/2 + (ac - U)y - a2d + 46d - c2 = 0 (1.8-15) the four roots of the quartic equation (10) are given as roots of the two quadratic equations  (1.8-16)  where the radicand on the right is a perfect square. Note that the discriminants (Sec. 1.6-5) of Eq. (10) and Eq. (15) are equal. 1.9. SYSTEMS OF SIMULTANEOUS EQUATIONS  1.9-1. Simultaneous Equations.  To solve a suitable set (system) of  simultaneous equations  U(xh x2, • • •) = 0  (i = 1, 2, • • •)  (1.9-1)  for the unknowns xh x2, . . . means to determine a set of values of Xi, x2, . . . which satisfy the equations (1) simultaneously. The solu tion is complete if all such sets are found. One can frequently eliminate successive unknowns Xj from a system (1), e.g., by solving one equation for Xj and substituting the resulting expression in the remaining equa tions. The number of equations and unknowns is thus reduced until a  single equation remains to be solved for a single unknown.  The pro-  SIMULTANEOUS LINEAR EQUATIONS  1.9-3  cedure is then repeated to yield a second unknown, etc.  Solutions may  25  be verified by substitution. To eliminate xu say, from two equations fi(xu x2) = 0,f2(xi, x2) = 0 where fi(x1} x2) and f2(xi, x2) are polynomials in x\ and x2 (Sec. 1.4-3), consider both functions as polynomials in x\ and form their resultant R (Sec. 1.7-3). Then x2 must satisfy the equation R — 0 (Sylvester's Dialytic Method of Elimination).  1.9-2. Simultaneous Linear Equations: Cramer's Rule.  Con  sider a set (system) of n linear equations in n unknowns xh x2, . . . , xn auXi + ai2x2 +  • • • + ainXn — bi  n  a21xi + a22x2 + • • • + a2nxn = b2  or  } aikxk = bi &= i  + annxn = bn  aniXi + an2x2 +  (1.9-2)  ,n)  (*-l>2,  such that at least one of the absolute terms bi is different from zero.  If  the system determinant  D = det [aik] =  On  012  a\n  a2\  a22  Q>2n  «ni  an2  • • •  (1.9-3)  ann  differs from zero, the system (2) has the unique solution  xk =§  (k =1, 2,  , n)  (Cramer's rule)  (1.9-4)  where Dk is the determinant obtained on replacing the respective ele ments a^, a2k, . . . , ank in the &th column of D by 6i, b2, . . . , bn, or 71  Dk = £ Aikbi  (k =1, 2, . . . ,n)  (1.9-5)  where Aik is the cofactor (Sec. 1.5-2) of aik in the determinant D (see also Sees. 13.2-3 and 14.5-3).  1.9-3. Linear Independence (see also Sees. 9.3-2, 14.2-3 and 15.2-la). (a) m equations fi(xh x2, . . . , xn) = 0 (i = 1, 2, . . . , m), or m func tions fi(xh x2, . . . , xn) are linearly independent if and only if m  2 Wf(si, x2, . . . ,xn) ss 0implies Xi =X2 = • • • =Xm =0 (1.9-6) Otherwise the m equations or functions are linearly dependent; i.e., at least one of them can be expressed as a linear combination of the others.  1.9-4  26  ELEMENTARY ALGEBRA  As a trivial special case, this is true whenever one or more of the equa tions fi (xi, x2, . . . , xn) = 0 is satisfied identically. n  n homogeneous linear functions } aikXk (i = 1, 2, . . . , n) are linearly independent if and only if det [a,J ^ 0 (see also Sec. 1.9-5). n  More generally, m homogeneous linear functions  /  aikXk (i = 1, 2, . . . , m) are  linearly independent if and only if the m X n matrix [a**] is of rank m (Sec. 13.2-7).  (b) m sets of n numbers £i(1), #2(1), . . . , xn(1); #i(2>, #2<2), . . . , 3n(2); . . . ; £i(m), z2(m), . . . , £n(m) (e.g., solutions of simultaneous equations, or components of m n-dimensional vectors) are linearly independent if and only if 7 X#/° = 0 (j = 1, 2, . . . , n) implies Xi = X2 = • • • = Xm = 0 i-i  (1.9-7)  This is true if and only if them X n matrix [a;/0] is of rank m (Sec. 13.2-7). 1.9-4. Simultaneous Linear Equations: General Theory (see also Sec. 14.8-10). The system of m linear equations in n unknowns X\, x2,  S aikXk = bi  (i = 1, 2, . . . , m)  (1.9-8)  *=»i  possesses a solution if and only if the matrices Oil  a\2  a2i  a22  . .  .  a2n  an  «12  a2\  a22  Jlml  am2  • .  .  Oln  .  a2n  fcl 62  Q>mn  K  y  dm\  am2  .  • •  Omn^  .  . •  (1.9-9)  (system matrix and augmented matrix) are of equalrank (Sec. 13.2-7). Otherwise the equations are inconsistent.  The unique solution of Sec. 1.9-2 applies if r = m = n. If both matrices (9) are of rank r < m, the equations (8) are linearly dependent (Sec. 1.9-3a); m — r equations can be expressed as linear combinations of the remaining r equations and are satisfied by their solution. The r independent equations determine r unknowns as linear functions of the remaining n — r unknowns, which are left arbitrary.  27  REFERENCES AND BIBLIOGRAPHY  1,10-2  1.9-5. Simultaneous Linear Equations: n Homogeneous Equa tions in n Unknowns. In particular, a system of n homogeneous linear equations in n unknowns, n  y OikXk = 0  (i = 1, 2, . . . , n)  (1.9-10)  A= l  has a solution different from the trivial solution X\ = x2 — • • • = xn = 0 if and only if D = det [a.-*] = 0 (see also Sec. 1.9-3a). In this case, there exist exactly n — r linearly independent solutions  *i(1), x2™, . . . , xn™; *i(2), z2(2), . . . , zn<2>; . . . ; *!<»-'>, z2<»-'>, . . . , xn{n~r), where r is the rank of the system matrix (Sec. 1.9-4). The most general solution is, then, n  — r  Xi = £ eta®  (i = 1, 2, . . . ,n)  (1.9-11)  j = l  where the Cj are arbitrary constants (see also Sec. 14.8-10). In the important special case where r = n — 1,  xi = cAki  x2 = cAk2  ....  xn = cAfcn  (1.9-12)  is a solution for any arbitrary constant c, so that all ratios Xi/xk are uniquely determined; the solutions (12) obtained for different values of k are identi cal (see also Sec. 14.8-6). 1.10. RELATED TOPICS, REFERENCES, AND BIBLIOGRAPHY  1.10-1. Related Topics. The following topics related to the study of elementary algebra are treated in other chapters of this handbook: Quadratic and bilinear forms Abstract algebra Matrix algebra Functions of a complex variable Numerical solution of equations, numerical approximations  Chap. Chap. Chap. Chap. Chap.  13 12 13 7 20  1.10-2. References and Bibliography. 1.1. Aitken, A. C: Determinants and Matrices,8th ed., Interscience, New York, 1956. 1.2. Birkhoff, G., and S. MacLane: A Survey of Modern Algebra, 3d ed., Macmillan, New York, 1965.  1.3. Dickson, L. E.: New First Course in the Theory of Equations, Wiley, New York, 1939.  1.4. Kemeny, J. G., et al.: Introduction to Finite Mathematics, Prentice-Hall, Englewood Cliffs, N.J., 1957.  1.5. Landau, E.: The Foundations of Analysis, Chelsea, New York, 1948. 1.6. Middlemiss, R. R.: College Algebra, McGraw-Hill, New York, 1952. 1.7. Uspensky, J. V.: Theory of Equations, McGraw-Hill, New York, 1948.  1.10-2  ELEMENTARY ALGEBRA  28  Additional Background Material  1.8. Cohen, L. W., et al.: The Structure of the Real Number System, Van Nostrand, Princeton, N.J., 1963. 1.9. Feferman, S.: The Number Systems: Foundations of Algebra and Analysis, Addison-Wesley, Reading, Mass., 1964. 1.10. Landin, J., and N. T. Hamilton: Set Theory: The Structure of Arithmetic, Allyn and Bacon, Boston, 1961. 1.11. Struik, D. J.: A Concise History of Mathematics, 2d ed., Dover, New York, 1948. (See also Sees. 12.9-2 and 13.7-2.)  CHAPTE  r2  PLANE ANALYTIC GEOMETRY  2.1. Introduction and Basic  Concepts 2.1-1. Introduction  2.1-2. Cartesian Coordinate Systems 2.1-3. Right-handed Rectangular Car tesian Coordinate Systems 2.1-4. Basic Relations in Terms of Rec  tangular Cartesian Coordinates 2.1-5. Translation of the Coordinate Axes  2.1-6. Rotation  of  the  Coordinate  Axes  2.1-7. Simultaneous Translation and Rotation of Coordinate Axes 2.1-8. Polar Coordinates  2.1-9. Representation of Curves  2.4. Second-order Curves (Conic Sections)  2.4-1. General Second-degree Equation 2.4-2. Invariants  2.4-3. Classification of Conies  2.4-4. Similarity of Proper Conies 2.4-5. Characteristic Quadratic Form and Characteristic Equation 2.4-6. Diameters and Conic Sections  Centers  of  2.4-7. Principal Axes 2.4-8. Transformation of the Equa tion of a Conic to Standard or  Type Form 2.4-9. Definitions of Proper Conies in Terms of Loci  2.2. The Straight Line  2.2-1. The Equation of the Straight Line  2.2-2. Other Representations of Straight Lines 2.3. Relations Involving Points and Straight Lines  2.3-1. Points and Straight Lines 2.3-2. Two or More Straight Lines 2.3-3. Line Coordinates  2.4-10. Tangents Conic Poles  and  Sections.  Normals Polars  of and  2.4-11. Other Representations of Conies  2.5. Properties of Circles, Ellipses, Hyperbolas, and Parabolas 2.5-1. Special Formulas and Theorems Relating to Circles 2.5-2. Special Formulas and Theorems Relating to Ellipses and Hyperbolas 29  2.1-1  PLANE ANALYTIC GEOMETRY  2.5-3. Construction of Ellipses and Hyperbolas and Their Tangents  30  2.6-2. Examples of Transcendental Curves  and Normals  2.5-4. Construction of Parabolas and  Their Tangents and Normals 2.6. Higher Plane Curves  2.6-1. Examples of Algebraic Curves  2.7. Related Topics, References, and Bibliography  2.7-1. Related Topics 2.7-2. References and Bibliography  2.1. INTRODUCTION AND BASIC CONCEPTS  2.1-1. Introduction (see also Sec. 12.1-1). A geometry is a mathemati cal model involving relations between objects referred to as points. Each geometry is defined by a self-consistent set of defining postulates; the latter may or may not be chosen so as to make the properties of the model correspond to physical space relationships. The study of such models is also called geometry. Analytic geometry represents each point by an ordered set of numbers (coordinates), so that relations between points are represented by relations between coordinates.  Chapters 2 (Plane Analytic Geometry) and 3 (Solid Analytic Geometry) introduce their subject matter in the manner of most elementary courses: the concepts of Euclidean geometry are assumed to be known and are simply translated into analytical language. A more flexible approach, involving actual construction of various geometries from postulates, is  briefly discussed in Chap. 17. The differential geometry of plane curves, including the definition of tangents, normals, and curvature, is outlined in Sees. 17.1-1 to 17.1-6.  2.1-2. Cartesian Coordinate Sys tems.  A  cartesian coordinate  system (cartesian reference system, see also Sec. 17.4-66) associates a unique ordered pair of real numbers (cartesian coordinates), the ab scissa x and the ordinate y, with  every point P = (x, y) in the finite portion of the Euclidean plane by reference to a pair of directed straight  HI  Fig. 2.1-1. Right-handed oblique car tesian coordinate system. The points marked "1" scales used.  define  the  coordinate  lines (coordinate axes) OX, OY intersecting at the origin 0 (Fig. 2.1-1). The parallel to OY through P intersects the x axis OX at the point P'. Similarly, the parallel to OX through P intersects the y axis OY at P".  RECTANGULAR CARTESIAN COORDINATES  31  2.1-4  The directed distances OP' = x (positive in the positive x axis direction) and OP" = y (positive in the positive y axis direction) are the cartesian coordinates of the point P = (x, y). x and y may or may not be measured with equal scales. In a general (oblique) cartesian coordinate system, the angle XOY = a> between the coordinate axes may be between 0 and 180 deg (right-handed car tesian coordinate systems) or between 0 and —180 deg (left-handed cartesian coordinate systems). A system of cartesian reference axes divides the plane into four quadrants (Fig. 2.1-1). The abscissa x is positive for points (x, y) in quadrants I and IV, negative for points in quadrants II and III, and zero for points on the y axis. The ordinate y is positive in quadrants I and II, negative in quadrants III and IV, and zero on the x axis. The origin is the point (0, 0). Note: Euclidean analytic geometry postulates a reciprocal one-to-one correspond ence between the points of a straight line and the real numbers (coordinate axiom, axiom of continuity, see also Sec. 4.3-1).  2.1-3. Right-handed Rectangular Cartesian Coordinate Systems. In a right-handed rectangular cartesian coordinate system, the directions of the coordinate axes are chosen so that a rotation of 90  deg in the positive (counterclock wise) sense would make the posi tive x axis OX coincide with the  positive y axis 0 Y (Fig. 2.1-2). The coordinates x and y are thus equal to the respective directed distances Fig. 2.1-2. Right-handed rectangular cartesian coordinate system and polarcoordinate system.  between the y axis and the point P, and between the x axis and the point P.  Throughout the remainder of this chapter, all cartesian coordi nates x, y refer to right-handed rectangular cartesian coordinate systems, and equal scale units of unit length are used to measure x and y, unless the contrary is specifically stated. 2.1-4. Basic Relations in Terms of Rectangular Cartesian Coordi  nates.  In terms of rectangular cartesian coordinates (x, y), the following  relations hold:  1. The distance d between the points Pi s= (xi, yx) andP2 = (x2, y2) is d -+  -*l)2 + (2/2 -2/i)2  (2.1-1)  2.1-5  PLANE ANALYTIC GEOMETRY  32  2. The oblique angle y between two directed straight-line segments P1P2 and P3P4 is given by COS 7  =  (x2 ~ Xi)(xA - x3) + (t/2 - 7/Q(;/4 - 7/3)  (2.1-2)  V(x2 - xtf + (ij2 - 7/1)2 V(*i - *,)' + (2/4 - y*)2  where the coordinates of the points Pi, P2, P3, P4 are denoted by the respective corresponding subscripts. The direction cosines cos ax and cos ay of a directed line segment F1P2 are the cosines of the angles ax and ay = 90 deg —ax, respectively. Xi — Xi  COS OLx =  V(x2 - si)2 + (2/2 - 2/O2 +  cos oty =  2/2 ~  sin a*  (2.1-3)  2/i  V(*2 - zi)2 + (2/2 - 2/i)2  3. The coordinates a;, 2/ of the point P dividing the directed line segment between the points Pi = (xh y{) and P2 = (x2, y2) in the ratioP\P\PP2 = m:n = ii'.l are x  =  mx2 + nzi  xi + \xx2  m + n  l+n  = m?/2 + nyx = 2/1 + ny2 m + n  1 + m  (— 00 < ix < 00)  (2.1-4)  Specifically, the coordinates of the mid-point of P1P2 are X\ + x2 x  =  V =  2/i + 2/2  (2.1-5)  4. The area S of the triangle with the vertices Pi = (xlf 2/1), P2 S3 (£2, 2/2), P3 S5 (X3, 2/3) iS #1  S = M #2 xz  2/1 2/2 2/3  1 1 1  = }4[x\(y2 - 2/3) + #2(2/3 - 2/1) + #3(2/1 - 2/2)] (2.1-6)  This expression is positive if the circumference P1P2P3 runs around the inside of the triangle in a positive (counterclockwise) direction. Specifically, if #3 = 2/3 = 0, s = y2  X\  x2  2/i  2/2  = M(#i2/2 - #22/1)  2.1-5. Translation of the Coordinate Axes.  (2.1-7)  Let x, y be the coordi  nates of any point P with respect to a right-handed rectangular cartesian  ROTATION OF COORDINATE AXES  33  2.1-7  reference system. Let x, y be the coordinates of the same point P with respect to a second right-handed rectangular cartesian reference system whose axes have the same directions as those of the x, y system, and whose origin has the coordinates x = Xo and y = 2/0 in the x, y system. If equal scales are used to measure the coordinates in both systems, the coordi nates x, y are related to the coordinates x3 y by the transformation equa tions (Fig. 2.1-3a; see also Chap. 14) x = x -  xQ  x = x + xo  or  y =  y -  yQ  y = 2/ + 2/o  (2.1-8)  The equations (8) permit a second interpretation. If x, y are considered as coor dinates referred to the x, y system of axes, then the point defined by x, y is translated by a directed amount —x0 in the x axis direction and by a directed amount —yo in the y axis direction with respect to the point (x, y). Transformations of this type applied to each point x, y of a plane curve may be used to indicate the translation of the entire curve.  2.1-6. Rotation of the Coordinate Axes. Let x, y be the coordinates  of any point P with respect to a right-handed rectangular cartesian reference system. Let x, y be the coordinates of the same point P with respect to a second right-handed rectangular cartesian reference system having the same origin 0 and rotated with respect to the x, y system so that the angle XOX between the x axis OX and x axis OX is equal to #  measured in radians in the positive (counterclockwise) sense (Fig. 2.1-36). If equal scales are used to measure all four coordinates x, y, x, y, the coordinates x, y are related to the coordinates x, y by the trans formation equations x = x cos & + y sin # or x = x cos & — y sin #  y = —x sin & + y cos # 2/ = x sin # + 2/ cos &  (2.1-9)  A second interpretation of the transformation (9) is the definition of a point (x, y) rotated about the origin by an angle —t? with respect to the point (x, y). 2.1-7. Simultaneous Translation and Rotation of Coordinate Axes. If the origin of the x, y system in Sec. 2.1-6 is not the same as  the origin of the x, y system but has the coordinates x = x0 and y = 2/0 in the x, y system, the transformation equations become x = (x — xo) cos & + (2/ — 2/0) sin # y = - (x — xo) sin # + (y - y0) cos & or  x = xo + x cos # — 2/ sin # V = 2/o + x sin # + y cos #  (2.1-10)  2.1-7  PLANE ANALYTIC GEOMETRY  4y  34  Ay  hJP  -*>*  11  -•*  Fig. 2.1-3a. Translation of coordinate axes.  x sin #  Fig. 2.1-36. Rotation of coordinate axes.  *>x  35  POLAR COORDINATES  2.1-8  The relations (10) permit one to relate the coordinates of a point in any two right-handed rectangular cartesian reference systems if the same scales are used for all coordinate measurements.  The transformation (10) may also be considered as the definition of a point (£, y) translated and rotated with respect to the point (x, y). Note: The transformations (8), (9), and (10) do not affect the value of the distance (1) between two points or the value of the angle given by Eq. (2). The relations constituting Euclidean geometry are unaffected by (invariant with respect to) translations and rotations of the coordinate system (see also Sees. 12.1-5, 14.1-4, and 14.4-5).  2.1-8. Polar Coordinates.  A plane polar-coordinate system associ  ates ordered pairs of numbers r, <p (polar coordinates) with each point P of the plane by reference to a directed straight line OX (Fig. 2.1-2), the polar axis. Each point P has the polar coordinates r, defined as the directed distance OP, and <p, defined as the angle XOP measured in radians in the counterclockwise sense between OX and OP. The point 0 is called the pole of the polar-coordinate system; r is the radius vector of the point P. Negative values of the angle v are measured in the clockwise sense from  the polar axis OX. Points (r, <p) are by definition identical to the points (—r, <p ± 180 deg); this convention associates points of the plane with pairs of numbers (r, <p) with negative as well as positive radius vectors r. Note: Unlike a cartesian coordinate system, a polar-coordinate system does not establish a reciprocal one-to-one correspondence between the pairs of numbers (r, <p) and the points of the plane. The ambiguities involved may, however, be properly taken into account in most applications.  If the pole and the polar axis of a polar-coordinate system coincide with the origin and the x axis, respectively, of a right-handed rectangular cartesian coordinate system (Fig. 2.1-2), then the following transforma tion equations relate the polar coordinates (r, <?) and the rectangular cartesian coordinates (x, y) of corresponding points if equal scales are used for the measurement of r, x, and y: x = r cos (p  \r\ = y/x2 + y2 +  y = r am<p  <p = arctan -  (2.1-11)  x  In terms of polar coordinates (r,<p), the following relations hold: 1. The distance d between the points (rh <pi) and (r2, <p2) is  d = Vn2 + r22 - 2rir2 cos (<p2 —<p\)  (2.1-12)  2. The area S of the triangle with the vertices Pi = (rh <pi), P2 = (r2, <p2), and ^t = (rz, <pt) is  S = H[rir2 sin (<p2 — <px) -f- rtr9 sin (*>8 — <p2) + nu sin (<pi —<pz)]  (2.1-13)  2.1-9  PLANE ANALYTIC GEOMETRY  36  This expression is positive if the circumference P1P2P3 runs around the inside of the triangle in the positive (counterclockwise) direction. Specifically, if r3 = 0, 8 = Jinn sin (<p2 - <pi)  (2.1-14)  See Chap. 6 for other curvilinear coordinate systems.  2.1-9. Representation of Curves (see also Sees. 3.1-13 and 17.1-1). (a) Equation of a Curve. A relation of the form <p(x, y) = 0  or  y = f(x)  (2.1-15)  is, in general, satisfied only by the coordinates x, y of points belonging to a special set defined by the given relation. In most cases of interest, the point set will be a curve (seealsoSec. 3.1-13). Conversely, a given curve will be represented by a suitable equation (15), which must be satisfiedby the coordinates of all points (x, y) on the curve. Note that a curve may have more than one branch.  The curves corresponding to the equations <p(x, y) = 0  \<p(x, y) = 0  where X is a constant different from zero, are identical.  (b) Parametric Representation of Curves.  A plane curve can  also be represented by two equations  x = x(t)  y = y(t)  (2.1-16)  where t is a variable parameter.  (c) Intersection of Two Curves. Pairs of coordinates x, y which simultaneously satisfy the equations of two curves  <Pi(x, y) = 0  <p2(x, y) = 0  (2.1-17)  represent the points of intersection of the two curves. In particular, if the equation v(x, 0) = 0 has one or more real roots x, the latter are the abscissas of the intersections of the curve <p(x, y) = 0 with the x axis.  If <P&, y) is a polynomial of degree n (Sec. 1.4-3), the curve <p(x, y) = 0 intersects the x axis (and any straight line, Sec. 2.2-1) in n points (nth-order curve); but some of these points of intersection may coincide and/or be imaginary. For any real number X, the equation  <pi(x, y) + \<p*(x, y) = 0  (2.1-18)  describes a curve passing through all points of intersection (real and imaginary) of the two curves (17).  (d) Given two curves corresponding to the two equations (17), the equation <pi(x, y)<p2(x, y) = 0  is satisfiedby all points of both original curves, and by no other points.  (2.1-19)  37  THE EQUATION OF THE STRAIGHT LINE  2.2-1  2.2. THE STRAIGHT LINE  2.2-1. The Equation of the Straight Line. Given a right-handed rectangular cartesian coordinate system, every equation linear in x and y, i.e., an equation of the form  Ax + By + C = 0  (2.2-1)  where A and B must not vanish simultaneously, represents a straight line (Fig. 2.2-1). Conversely, every straight line in the finite portion of the plane can be represented by a linear equation (1). The special case C = 0 corresponds to a straight line through the origin.  Fig. 2.2-1. The equation of the straight line.  The following special forms of the equation of a straight line are of particular interest:  1. Slope-intercept form. Straight line at an angle $ with the positive x axis and intercepting the y axis at y = b: y = mx + b  m = tan $>  m is the slope of the straight line.  2. Two-intercept form.  Straight line intercepting the  x axis at x = a and the y axis at y = b: 5-4.2 a  b  1 = 0  2.2-2  PLANE ANALYTIC GEOMETRY  3. Normal form.  38.  Let p be the length of the directed  perpendicular from the origin onto a straight line; let # be the angle between the positive x axis and the directed perpendicular, measured in the positive (counterclockwise) sense. Then the equation of the straight line is x cos & + y sin # — p = 0  4. Point-slope form. Straight line through the point (xi, yi) and having the slope m: y - 2/i = rn(x - Xi)  5. Two-point form. Straight line through the (noncoincident) points Pi = (xh yx) and P2 = (x2, y2): y - 2/1 = s - si 2/2 — 2/i  or  x2 — xx  x xi x2  y 2/1 2/2  1 1 1  0  When the equation of a straight line is given in the general form (1), the intercepts a and b, the slope m, the perpendicular distance p from the  origin, cos #, and sin # are related to the parameters A, B, and C as follows:  a=- ^ *>=-5 m=tan *=- ^ 3> =t>-90deg (2.2-2)  V=~ +y/l2 +B2 C°**= ±VA2 +B2 S[n*= ±VA2 +B2  (2.2-3)  In order to avoid ambiguity, one chooses the sign of + y/A2 + B2 in Eq. (3) so that p > 0.  In this case, the directed line segment between origin and straight line defines the direction of the positive normal of the straight line; cos t> and sin # are the direction cosines of the positive normal (Sec. 17.1-2).  2.2-2. Other Representations of Straight Lines. In terms of a variable parameter t, the rectangular cartesian coordinates x and y of a point on any straight line may be expressed in the form x = cit + c2  lth  fc,  m=-  a-  (2.2-4)  y = kit + k2  kic2 - k2Ci  ^  r  b  k2ci — kxc2  Ci  (2.2-5)  39  TWO OR MORE STRAIGHT LINES  and p =  kxc2 — k2Ci  _  0  ki  cos # =  ± y/ci2 + W  ± y/ci2 + ki2  2.3-2 -Ci  sin # =  ± y/ci2 + A;!2 (2.2-6)  If the sign of ± y/ci2 + kx2 is chosen so that p > 0, theorigin will always lie ontheright ofthedirection ofmotion as tincreases, i.e., inthedirection of the negative normal (Sec. 17.1-2).  In terms of polar coordinates r, <p, the equation ofany straight line may be expressed in the form  r(A cos <p + B sin <p) + C = 0 r cos (<p — #) = p  or  (2.2-7) (2.2-8)  where A, B, C, p, and # are defined as in Sec. 2.2-1. 2.3. RELATIONS INVOLVING POINTS AND STRAIGHT LINES  2.3-1. Points and Straight Lines. The directed distance d from the straight line (2.2-1) to a point (x0, 2/0) is  d = Axp + Byp + C ± y/A2 + B2  (2.3-1)  where the sign of ± y/A2 + B2 ischosen to beopposite to that ofC. dis positive if the straight line lies between the origin and the point (x0, 2/0). Three points (xlf yx), (x2, y2), and (xz, yz) are on a straight line if and only if (see also Sec. 2.2-1) Xi  2/i  1  x2  2/2  1  Xz  2/3  1  = 0  (2.3-2)  2.3-2. Two or More Straight Lines, (a) Two straight lines and  Axx + Biy + Ci = 0 A2x + B2y + C2 = 0  or or  2/ = mxx + bi y = m2x + b2  (2.3-3a) (2.3-36)  intersect in the point x  —  BXC2 - B2Ci  AXB2 - A2BX  = bx-b  m2- mi  y  C\A2 —C2Ai = m2bi —mj)2 AXB2 - A2BX m2 — mi (2.^-4)  (b) Either angle 712 from the straight line (3a) to the line (36) isgiven by tan 712 =  AiB2 — A2Bi 4i^2 + BiB2  m2 — mi mm2 + 1  (2.3-5)  2.3-2  PLANE ANALYTIC GEOMETRY  40  where 712 is measured counterclockwise from the line (3a) to the line (36).  (c) The straight lines (3a) and (36) are parallel if AXB2 - A2Bi = 0  or  mi = m2  (2.3-6)  m2 = - —  (2.3-7)  and perpendicular to each other if  AiA2 + BXB2 =0  or  (d) The equation of a straight line through a point (x1} 2/1) and at an angle 712 (or 180 deg - 712) to the straight line (3a) is  »-*-?:£!»: »>-*> Ai tan 712 — #1  (2-3-8)  Specifically, theequation of thenormal to the straight line (3a) through >e the point (xh 2/1) is  2/ - 2/i =|j (x - Xl) =- ^ (* - *i)  (2.3-9)  (e) Theequation ofany straight line parallel to (3a) may beexpressed in the form  Axx + Bxy + C2 = 0  (2.3-10)  The distance d between the parallel straight lines (3a) and (10) is  d=  Cl ~ °2  + y/A 12 + Bi2  (2.3-11)  If the sign of ± y/AX2 + Bx2 in Eq. (11) is chosen to be the opposite to that of Ci, d will be positive if the straight line (3a) is between the origin and the straight line (10).  (f) The equation of every straight line passing through point of inter section of two straight lines (3a) and (36) will be of the form \1(A1x + Biy + Ci) + \2(A2x + B2y + C2) = 0  (2.3-12)  with Xi, X2 not both equal to zero. Conversely, every equation of the form (12) describes a straight line passing through the point of intersection.  If the straight lines (3a) and (36) are parallel, Eq. (12)  represents a straight line parallel to the two. If the straight lines are  given in the normal form (Sec. 2.2-1), -X is the ratio of the distances (1) between the first and second straight line and any one point on the third  straight line, and the straight lines corresponding to X= 1 and X= -1 bisect the angles between the given straight lines.  41  INVARIANTS  2.4-2  (g) Three straight lines  Aix + Biy + Ci = 0  A2x + B2y + C2 = 0 Azx + Bzy + Cz = 0 intersect in a point or are parallel if and only if Ax A2 Az  Bx B2 Bz  d C2 = 0 Cz  (2.3-13)  (2.3-14)  i.e., if the three equations (13) are linearly dependent (Sec. 1 9-3a). 2.3-3. Line Coordinates.  The equation £c + vy + 1 = 0  (2.3-15)  describes a straight line (£, i,) "labeled" by the line coordinates £, v. If the point coordinates x, y are considered as constant parameters and the line coordinates £, 17 as variables, Eq. (15) may be interpreted as the equation ofthe point (x, y) [point of  intersection of all straight lines (15)]. The symmetry of Eq. (15) in the pairs (x, £)  and (y, rj) results in a correspondence (duality) between theorems dealing with the positions ofpoints and straight lines (Sees. 2.3-1 and 2.3-2). An equation F(£, v) = 0  represents a set of straight lines which will, in general, envelop a curve determined by the nature of the function F(£, rj) (see also Table 2.4-2).  2.4. SECOND-ORDER CURVES (CONIC SECTIONS)  2.4-1. General Second-degree Equation. The second-order curves or conic sections (conies) are represented by the general seconddegree equation  anx2 + 2a12xy + a22y2 + 2anx + 2a2zy + a33 = 0  or (axxx + a12y + alz)x + (a2Xx + a22y + a2z)y + (azxx + aZ2y + a33) = 0  with  aik = aki  (2.4-1)  (i, k = 1, 2, 3)  2.4-2. Invariants. For any equation (1), the three quantities I = axx + a22  D = A33 =  axx a21  ax2 a22  A =  Q>u  aX2  axz  a2x  a22  a2Z  azx  aZ2  a33  (2.4-2)  and the sign of the quantity A' =  a22  a2Z  aZ2  a33  +  an  alz  Q>n  a33  (2.4-3)  are invariants with respect to the translation and rotation transforma-  42  PLANE ANALYTIC GEOMETRY  2.4-3  tions (2.1-8), (2.1-9), and (2.1-10). Such invariants define properties of the conic which do not depend on its position. Either A or A = SA is sometimes called the discriminant of Eq. (1).  2.4-3. Classification of Conies.  Table 2.4-1 shows the classification  of conies in terms of the invariants defined in Sec. 2.4-2. Table 2.4-1. Classification of Conic Sections (Conies) Proper conies  Improper (degenerate) conies A  A 5*0  = 0  Real ellipse  D >0  f<>  (circle if I2 = 4D or dll = «22, Gl2 = 0)  f>.  (imaginary ellipse)  No real locus  Central conies  Point in finite portion of plane  D 5*0  (point ellipse; real intersection of two imaginary straight lines)  *-  Two real straight lines inter D  Hyperbola  <0  secting in finite portion of plane (degenerate hyperbola) No real locus  (imaginary parallels)  A' >0 Noncentral conies  D  ==  A' <0  Parabola  Two real parallel straight lines  u  A' =0  One real straight line (coincident parallels)  2.4-4. Similarity of Proper Conies. Two proper conies (A 5* 0) given by equa tionsof the form (1) are similar if either D = 0 for both equations (i.e., if both conies  are parabolas) orif D 5* 0 for bothequations and the ratios an:a12:a22 are thesame for both conies.  2.4-5. Characteristic Quadratic Form and Characteristic Equation. Impor  tant properties of conies maybe studied in terms of the (symmetric) characteristic quadratic form  FQ(x, y) « anx* + 2a12xy + a22y*  (2.4-4)  corresponding to Eq. (1). In particular, a proper conic (A 5* 0) is a real ellipse, imaginary ellipse, hyperbola, or parabola if F0(x, y) is, respectively, positive definite,  negative definite, indefinite, or semidefinite as determined by the (necessarily real)  43  PRINCIPAL AXES  2.4-7  roots Xi, X2 of the characteristic equation flu — X  O12  #21  d22 — X  0  or  X* - I\ + D - 0  (2.4-5)  Xi and X2 are the eigenvalues of the matrix at* (Sees. 13.4-5 and 13.5-2).  2.4-6. Diameters and Centers of Conic Sections, (a) A diameter ofa conic described by Eq. (1) is the locus of the centers of parallel chords.  The diameter conjugate to the chords inclined at an angle # with respect to the positive x axis bisects these chords and is a straight line with the equation  (axxx + aX2y + axz) cos # + (a2Xx + a22y + a2Z) sin &= 0 (2.4-6) (b) All diameters of a conic (1) either intersect at a unique point, the center of the conic (see Sec. 2.4-10 for an alternative definition), or they are parallel, according to whether D 5* 0 or D = 0. In theformer case, the conic is a central conic.  The coordinates x0, 2/0 of the center are given by  anxo + Gi22/o + ai3 = 0  sothat  Xo=-±\ai* ai2| DI a23 a221  «0BS yo  a21x0 + a22y0 + a28 = 0  l|a» <*i»|  (D*0)  (2.4-7)  (24R)  D\ a21 a231 ( ; {ZA 8) Given the equation (1) of a central conic, a translation (2.1-8) of the coordinate  origin to the center (8) of the conic results in the new equation anx2 + 2a12xy -j- a22y2 + - = 0  (2.4-9)  in terms of the new coordinates £, y.  (c) Two conjugate diameters of a central conic each bisect the  chords parallel to the other diameter (see also Sec. 2.5-2e). 2.4-7. Principal Axes. A diameter perpendicular to its associated  (conjugate) chords (principal chords) is a symmetry axis or principal axis of the conic. Every (real) noncentral conic has one principal axis; every (real) central conic has either two mutually perpendicular principal axes, or every diameter is a principal axis (circle). The principal axes are directed along eigenvectors of thematrix [aik] (Sec. 14.8-6). More specifically, the direction cosines cos #, sin 1? ofthe normal to a principal axis (Sec. 2.2-1) satisfy the conditions (an - X) cos t? + ai2 sin # =01 a2\ cos  # + (a22 - X) sin t? = 0 /  (2.4-10)  where Xis a nonvanishing root of the characteristic equation (5). The angle *  between the positive xaxis andanyprincipal axis oftheconic (1) satisfies thecondition  tan 2* = tan 2tf = an2ai2 - a22  (2 4-in ^' 11;  2.4-8  PLANE ANALYTIC GEOMETRY  44  2.4-8. Transformation of the Equation of a Conic to Standard or  Type Form. If one introduces a new reference system by combining a rotation of the coordinate axes throughan angle satisfying Eq. (11) with a suitable translation of the origin (Sec. 2.1-7), the equation (1) of any  proper conic reduces toone of the standard or type forms (sometimes called canonical forms) listed below. The parameters a2, b2, p appearing in the standard forms are simply related to the invariants A,D,I, and to the roots Xi > X2 of the characteristic equation (5).  g+|!=1(ELLIPSE)  I A  ^ ^ fe2~  X-2 - g = 1(HYPEBBOLA)  _^f (2.4.12a) Xi2X2  a2= _I^= _ A  ^J  0  y2 = 4pz (parabola)  XxZ)  A  ^\2D  ^ = 2/ V" 7 = 2X"i \  ^  (2-«»)  ^XiX22  XT > ° X2 = 0  (2.4-12c)  The equations of the improper (degenerate) conies are similarly transformed to the standard or type forms  2? -I- yl = 0 (point) a2 b2 K '  ^s —fs = 0 (intersecting straight lines) I a2 62 I (2.4-13)  —= 1(parallel straight lines) a2  x2 =0 (one straight line) J '  Note: Every rotation (2.1-9) through an angle # satisfying Eq. (11) diagonalizes the matrix [a.-*] ofthe characteristic quadratic form (4) (principal-axis transformation, see also Sec. 14.8-6). The values of 0 satisfying Eq. (11) differ by multiples of 90  deg, corresponding to interchanges of x, -x, y, and -y. The standard forms (12a, b, c) correspond to choices of&such that the foci (Sec. 2.4-9) ofthe conies lie on the x axis. Equation (11) becomes indeterminate for (real and imaginary) circles and points, which have no definite principal axes.  2.4-9. Definitions of Proper Conies in Terms of Loci. Once the  equation of any proper conic is reduced to its standard form (12), a simple translation (Sec. 2.1-5) may be used tointroduce a new system of coordinates x, y such that the equation of the conic appears in the form y2 = 4px - (1 ~ e2)x2  (2.4-14)  The conic passes through theorigin of thenew x, ysystem; thex axis is a symmetry axis (principal axis) of the conic.  45  TANGENTS AND NORMALS OF CONIC SECTIONS  2.4-10  Equation (14) describes a proper conic as the locus of a point which moves so that the ratio e > 0 (eccentricity) of its distances from a fixed point (focus) and from a fixed line (directrix) is a constant.  The conic  will be an ellipse if e < 1 and, specifically, a circle if e = 0. The conic will be a hyperbola if e > 1, and a parabola if e = 1.  The equation of the directrix of the conic represented by Eq. (14) is  • = " W+7)  <2-4-15)  The coordinates x and y of the focus are  * = Y^T€  V= °  (2.4-16)  The directrix is perpendicular to the symmetry axis. The latter passes through the focus and also through the vertex x = y = 0 of the conic. The distance between the focus and the directrix is equal to 2p/c.  In the case of a central conic (ellipse or hyperbola), the straight line 2p  x = j-^fp = a  (2.4-17)  is a symmetry axis (principal axis) of the conic, so that two foci and two directrices can be defined.  The latus rectum of a proper conic is defined as the length of a chord through the focus and perpendicular to the symmetry axis and is equal to |4p|. Note: All types of improper as well as of proper conies may be obtained as the inter sections of a right circular cone with a plane for various