The theory of equations: with an introduction to the theory of binary algebraic forms [NOOK Book]

Overview

This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections ...
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The theory of equations: with an introduction to the theory of binary algebraic forms

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Overview

This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book.
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Product Details

  • BN ID: 2940019360161
  • Publisher: Dublin, Hodges,Figgis, & co. (ltd.); London, Longmans, Green, & co.
  • Sold by: Barnes & Noble
  • Format: eBook
  • Edition description: Digitized from 1892 volume
  • File size: 823 KB

Table of Contents

Introduction
Art. 1 Definitions 1
Art. 2 Numerical and algebraical equations 2
Art. 3 Polynomials 4
Chapter I General Properties of Polynomials
Art. 4 Theorem relating to polynomials when the variable receives large values 5
Art. 5 Similar theorem when the variable receives small values 6
Art. 6 Change of form of a polynomial corresponding to an increase or diminution of the variable. Derived functions 8
Art. 7 Continuity of a rational integral function 9
Art. 8 Form of the quotient and remainder when a polynomial is divided by a binomial 10
Art. 9 Tabulation of functions 12
Art. 10 Graphic representation of a polynomial 13
Art. 11 Maximum and minimum values of polynomials 17
Chapter II General Properties of Equations
Art. 12, 13, 14 Theorems relating to the real roots of equations 19
Art. 15 Existence of a root in the general equation. Imaginary roots 21
Art. 16 Theorem determining the number of roots of an equation 22
Art. 17 Equal roots 25
Art. 18 Imaginary roots enter equations in pairs 26
Art. 19 Descartes' rule of signs for positive roots 28
Art. 20 Descartes' rule of signs for negative roots 30
Art. 21 Use of Deseartes' rule in proving the existence of imaginary roots 30
Art. 22 Theorem relating to the substitution of two given numbers for the variable 31
Examples 32
Chapter III Relations Between the Roots and Coefficients of Equations, with Applications to Symmetric Functions of the Roots
Art. 23 Relations between the roots and coefficients. Theorem 35
Art. 24 Applications of the theorem 36
Art. 25 Depression of an equation when a relation exists between two of its roots 42
Art. 26 The cube roots of unity 43
Art. 27 Symmetric functions of the roots 46
Examples 48
Art. 28 Theorems relating to symmetric functions 53
Examples 54
Chapter IV Transformation of Equations
Art. 29 Transformation of equations 60
Art. 30 Roots with signs changed 60
Art. 31 Roots multiplied by a given quantity 61
Art. 32 Reciprocal roots and reciprocal equations 62
Art. 33 To increase or diminish the roots by a given quantity 64
Art. 34 Removal of terms 67
Art. 35 Binomial coefficients 68
Art. 36 The cubic 71
Art. 37 The biquadratic 73
Art. 38 Homographic transformation 75
Art. 39 Transformation by symmetric functions 76
Art. 40 Formation of the equation whose roots are any powers of the roots of the proposed equation 78
Art. 41 Transformation in general 80
Art. 42 Equation of squared differences of a cubic 81
Art. 43 Criterion of the nature of the roots of a cubic 84
Art. 44 Equation of differences in general 84
Examples 86
Chapter V Solution of Reciprocal and Binomial Equations
Art. 45 Reciprocal equations 90
Art. 46-52 Binomial equations. Propositions embracing their leading general properties 92
Art. 53 The special roots of the equation x[superscript n] - 1 = 0 95
Art. 54 Solution of binomial equations by circular functions 98
Examples 100
Chapter VI Algebraic Solution of the Cubic and Biquadratic
Art. 55 On the algebraic solution of equations 105
Art. 56 The algebraic solution of the cubic equation 108
Art. 57 Application to numerical equations 109
Art. 58 Expression of the cubic as the difference of two cubes 111
Art. 59 Solution of the cubic by symmetric functions of the roots 113
Examples 114
Art. 60 Homographic relation between two roots of a cubic 120
Art. 61 First solution by radicals of the biquadratic. Euler's assumption 121
Examples 125
Art. 62 Second solution by radicals of the biquadratic 127
Art. 63 Resolution of the quartic into its quadratic factors. Ferrari's solution 129
Art. 64 Resolution of the quartic into its quadratic factors. Descartes' solution 133
Art. 65 Transformation of the biquadratic into the reciprocal form 135
Art. 66 Solution of the biquadratic by symmetric functions of the roots 139
Art. 67 Equation of squared differences of a biquadratic 142
Art. 68 Criterion of the nature of the roots of a biquadratic 144
Examples 146
Chapter VII Properties of the Derived Functions
Art. 69 Graphic representation of the derived function 154
Art. 70 Theorem relating to the maxima and minima of a polynomial 155
Art. 71 Rolle's Theorem. Corollary 157
Art. 72 Constitution of the derived functions 157
Art. 73 Theorem relating to multiple roots 158
Art. 74 Determination of multiple roots 159
Art. 75, 76 Theorems relating to the passage of the variable through a root of the equation 161
Examples 163
Chapter VIII Symmetric Functions of the Roots
Art. 77 Newton's theorem on the sums of powers of roots. Prop. I 165
Art. 78 Expression of a rational symmetric function of the roots in terms of the coefficients. Prop. II 167
Art. 79 Further proposition relating to the expression of sums of powers of roots in terms of the coefficients. Prop. III 169
Art. 80 Expression of the coefficients in terms of sums of powers of roots 170
Art. 81 Definitions of order and weight of symmetric functions, and theorem relating to the former 173
Art. 82 Calculation of symmetric functions of the roots 174
Art. 83 Homogeneous products 178
Chapter IX Limits of the Roots of Equations
Art. 84 Definition of limits 180
Art. 85 Limits of roots. Prop. I 180
Art. 86 Limits of roots. Prop. II 181
Art. 87 Practical applications 183
Art. 88 Newton's method of finding limits. Prop. III 185
Art. 89 Inferior limits, and limits of the negative roots 186
Art. 90 Limiting equations 187
Examples 188
Chapter X Separation of the Roots of Equations
Art. 91 General explanation 189
Art. 92 Theorem of Fourier and Budan 189
Art. 93 Application of the theorem 192
Art. 94 Application of the theorem to imaginary roots 194
Art. 95 Corollaries from the theorem of Fourier and Budan 197
Art. 96 Sturm's theorem 198
Art. 97 Sturm's theorem. Equal roots 203
Art. 98 Application of Sturm's theorem 206
Art. 99 Conditions for the reality of the roots of an equation 210
Art. 100 Conditions for the reality of the roots of a biquadratic 211
Examples 212
Chapter XI Solution of Numerical Equations
Art. 101 Algebraical and numerical equations 215
Art. 102 Theorem relating to commensurable roots 216
Art. 103 Newton's method of divisors 217
Art. 104 Application of the method of divisors 218
Art. 105 Method of limiting the number of trial-divisors 221
Art. 106 Determination of multiple roots 222
Art. 107 Newton's method of approximation 225
Art. 108 Horner's method of solving numerical equations 227
Art. 109 Principle of the trial-divisor in Horner's method 231
Art. 110 Contraction of Horner's process 235
Art. 111 Application of Horner's method to cases where roots are nearly equal 238
Art. 112 Lagrange's method of approximation 241
Art. 113 Numerical solution of the biquadratic by Descartes' method 242
Miscellaneous examples 245
Chapter XII Complex Numbers and the Complex Variable
Art. 114 Complex numbers-Graphic representation 249
Art. 115 Complex numbers-Addition and subtraction 250
Art. 116 Multiplication and division 251
Art. 117 Other operations on complex numbers 252
Art. 118 The complex variable 253
Art. 119 Continuity of a function of the complex variable 255
Art. 120 Variation of the amplitude of the function corresponding to the description of a small closed curve by the complex variable 256
Art. 121 Cauchy's theorem relating to the number of roots comprised within a plane area 258
Art. 122 Proof of the fundamental theorem relating to the number of roots of the general equation 259
Art. 123 Second proof of fundamental theorem 260
Art. 124 Determination of complex numerical roots: solution of the cubic 261
Art. 125 Solution of the biquadratic 265
Art. 126 Solution of biquadratic continued 267
Notes
A Algebraic solution of equations 271
B Solution of numerical equations 275
C The proposition that every equation has a root 279
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