Full text of "Higher Algebra"
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Full text of "Higher Algeb ra"

DAMAGE BOOK 
DO 
_1 60295 > 
^ DO 
HIGHER ALGEBRA 
BY 
S. BARNARD, M.A. 
FORMERLY ASSISTANT MASTER AT RUGBY SCHOOL, LATE FELLOW AND LECTURER AT 
EMMANUEL COLLEGE, CAMBRIDGE 
AND 
J. M. CHILD, B.A., B.Sc. 
FORMERLY LECTURER IN MATHEMATICS IN THE UNIVERSITY OF' MANCHESTER 
LATE HEAD OF MATHEMATICAL DEPARTMENT, TECHNICAL COLLEGE, DERBY 
FORMERLY SCHOLAR AT JESUS COLLEGE, CAMBRIDGE 
LON-DON 
MACMILLAN fcf'CO LTD 
*v 
NEW YORK ST MARTIN *S PRESS 
1959 
This book is copyright in all countries which 
are signatories to the Berne Convention
First Edition 1936 
Reprinted 1947, ^949> I952> *955, 1959 
MACMILLAN AND COMPANY LIMITED 
London Bombay Calcutta Madras Melbourne 
THE MACMILLAN COMPANY OF CANADA LIMITED 
Toronto 
ST MARTIN'S PRESS INC 
New York 
PRINTED IN GREAT BRITAIN BY 
LOWE AND BRYDONE (PRINTERS) LIMITED, LONDON, N.W.IO 
CONTENTS ix 
IjHAPTER 
EXEKCISE XV (128). 
Minors, Expansion in Terms of Second Minors (132, 133). Product 
of Two Iteterminants (134). Rectangular Arrays (135). Reciprocal 
Deteyrrtlilnts, Two Methods of Expansion (136, 137). Use of Double 
Suffix, Symmetric and Skew-symmetric Determinants, Pfaffian (138- 
143), 
ExERtad XVI (143) 
X. SYSTEMS OF EQUATIONS. 
Definitions, Equivalent Systems (149, 150). Linear Equations in 
Two Unknowns, Line at Infinity (150-152). Linear Equations in 
Three Unknowns, Equation to a Plane, Plane at Infinity (153-157). 
EXEKCISE XVII (158). 
Systems of Equations of any Degree, Methods of Solution for 
Special Types (160-164). 
EXERCISE XVIII (164). 
XL RECIPROCAL AND BINOMIAL EQUATIONS. 
Reduction of Reciprocal Equations (168-170). The Equation 
x n - 1=0, Special Roots (170, 171). The Equation x n -A =0 (172). 
The Equation a 17 - 1 ==0, Regular 17-sided Polygon (173-176). 
EXERCISE XIX (177).
AND BIQUADRATIC EQUATIONS. 
The Cubic Equation (roots a, jS, y), Equation whose Roots are 
( -y) 2 , etc., Value of J, Character of Roots (179, 180). Cardan's 
Solution, Trigonometrical Solution, the Functions a -f eo/? -f-\>V> 
a-f a> 2 4- a>y (180, 181). Cubic as Sum of Two Cubes, the Hessftfh 
(182, 183). Tschirnhausen's Transformation (186). 
EXERCISE XX (184). 
The Biquadratic Equation (roots a, , y, 8) (186). The Functions 
A=y + aS, etc., the Functions /, J, J, Reducing Cubic, Character 
of Roots (187-189). Ferrari's Solution and Deductions (189-191). 
Descartes' Solution (191). Conditions for Four Real Roots (192-ty). 
Transformation into Reciprocal Form (194). Tschirnhausen's Trans- 
formation (195). 
EXERCISE XXI (197). 
OP IRRATIONALS. 
Sections of the System of Rationals, Dedekind's Definition (200, 
201). Equality and Inequality (202). Use of Sequences in defining 
a Real Number, Endless Decimals (203, 204). The Fundamental 
Operations of Arithmetic, Powers, Roots and Surds (204-209). 
Irrational Indices, Logarithms (209, 210). Definitions, Interval, 
Steadily Increasing Functions (210). Sections of the System of^Real 
Numbers, the Continuum (211, 212). Ratio and Proportion, Euclid's 
Definition (212, 213). 
EXERCISE XXII (214). 
x CONTENTS 
CHAPTER 
XIV/INEQUALITIES. 
Weierstrass' Inequalities (216). Elementary Methods (210, 217) 
For n Numbers a l9 a 2 
a 
> \*JACJJ 
n n n 
(a* -!)/* (a" -I)/*,, (219).
xa x ~ l (a-b)$a x -b x ^ xb x ~ l (a - 6), (219). 
(l+x) n ^l+nx, (220). 
Arithmetic and Geometric Means (221, 222). 
- - 
V ^ 
n 
and Extension (223). Maxima and Minima (223, 224). 
EXERCISE XXIII (224). 
XV. SEQUENCES AND LIMITS. 
Definitions, Theorems, Monotone Sequences (228-232). E* 
ponential Inequalities and Limits, 
l\m / i\n / l\-m / 1 \ ~n 
1) >(!+-) and (1--) <(l--) if m>n, 
m/ \ n/ \ mj \ nj 
/ 1 \ n / l\" w 
lim (1-f- =lim(l--) =e, (232,233). 
n _ >00 V nj \ nj 
EXERCISE XXIV (233). 
General Principle of Convergence (235-237). Bounds of a Sequent 
Limits of Inde termination (237-240). 
Theorems : 
(1) Increasing Sequence (u n ), where u n -u n ^ l 0 and u n+l lu n -*l, then u n n -*L 
(3) If lim u n l, then lim (Ui+u 2 + ... +u n )jn I. 
n >oo n->oo 
(4) If lim a n ~a, and lim fe w = 6, then 
n-~>oo n~>oo 
lim (a n 6 1 + a w _ 1 6 1 + ...H-a 1 6 n )/n=o6, (240-243). 
n >ao
Complex Sequences, General Principle of Convergence (243, 244). 
EXERCISE XXV (244). 
CONTENTS xi 
XVI. \CONVERGENCE OP SERIES (1). 
Definitions, Elementary Theorems, Geometric Series (247, 248). 
Series of Positive Terms. Introduction and Removal of Brackets, 
Changing Order of Terms, Comparison Tests, 271 /w p , D'Alembert's 
and Cauchy's Tests (248-254). 
EXERCISE XXVI (254). 
Series with Terms alternately Positive and Negative (256). 
Series with Terms Positive or Negative. Absolute Convergence, 
General Condition for Convergence, Pringsheim's Theorem, Intro- 
duction and Removal of Brackets, Rearrangement of Terms, Ap- 
proximate Sum, Rapidity of Convergence or Divergence (256-261). 
Series of Complex Terms. Condition of Convergence, Absolute Con- 
vergence, Geometric Series, Zr n cos nd, Sr n sin n6. 
If u n /u n+l = l+a n /n, where a n ->a>0, then u n -*Q. 
Convergence of Binomial Series (261-263). 
EXERCISE XXVII (264). 
XVII^CoNTiNuous VARIABLE. ^ 
Meaning of Continuous Variation, Limit, Tending to <*> , Theorems 
on Limits and Polynomials (266-268) . Continuous and Discontinuous 
Functions (269, 270). Continuity of Sums, Products, etc., Function 
of a Function, lirn {f(x)}, Rational Functions, x n (271). Funda- 
mental Theorems (272). Derivatives, Tangent to a Curve, Notation 
of the Calculus, Rules of Differentiation (273-277). Continuity of 
{f(x)}, Derivatives of {f(x)} and x n (278, 279). Meaning of Sign 
of J'(x) (279). Complex Functions, Higher Derivatives (2*9, 280). 
Maxima and Minima, Points of Inflexion (280-282). 
EXERCISE XXVIII (282). 
Inverse Functions, Bounds of a Function, Rolle's Theorem, Mean- 
Value Theorem (284-288). Integration (289). Taylor's Theorem, 
Lagrange's Form of Remainder (290, 291). Function of a Complex 
Variable, Continuity (291, 292). 
EXEBCISE XXIX (293). 
XVIIL, THEORY OF EQUATIONS (2), POLYNOMIALS (2), RATIONAL FRAC-
TIONS (1). 
Multiple Roots, Rolle's Theorem, Position of Real Roots of/(&)=0 
(296, 296). Newton's Theorem on Sums of Powers of the Roots of 
f(x) =0 (297). Order and Weight of Symmetric Functions (298, 299). 
Partial Derivatives, Taylor's Theorem for Polynomials in x and in 
x, y, ... . Euler's Theorem for Polynomials, x + y + ...~nu 
J dx dy 
(299-302). A Theorem on Partial Fractions (302). The Equation 
lf + ...=0, (303). 
EXERCISE XXX (304). 
xii CONTENTS 
CHAPTER 
XIX. EXPONENTIAL AND LOGARITHMIC FUNCTIONS AND SERIES. 
Continuity, Inequalities and Limits (306, 307). The Exponential 
Theorem, Series for a x (307, 308). Meaning of an Irrational Index, 
Derivatives of a x , log x and x n (309). Inequalities and Limits, the 
way in which e x and log x tend to oo , Euler's Constant y, Series for 
log 2 (310-312). The Exponential Function E(z), Complex Index 
(312, 313). Series for sinx, cos x arid Exponential Values (313). 
Use of Exponential Theorem in Summing Series (314). 
EXERCISE XXXI (315). 
Logarithmic Series and their Use in Summation of Series, Calcula- 
tion of Logarithms (315-319). The Hyperbolic Functions (319-321). 
EXERCISE XXXII (321). 
XX. CONVERGENCE (2). 
Series of Positive Terms. Cauchy's Condensation Test, Test Series 
2 - (325). Rummer's, Raabc's and Gauss's Tests (326-328). 
' 
Binomial and Hyper-geometric Series (328, 329). De Morgan and 
Bertrand's Tests % (330). 
Series with Terms Positive or Negative. Theorem, Abel's Inequality, 
Dirichlct's and Abel's Tests (330, 331). Power Series, Interval and 
Radius of Convergence, Criterion for Identity of Power Series (332- 
334). Binomial Series l-f?iz4-... when z is complex (334). Multi- 
plication of Series, Merten's and Abel's Theorems (335-338). 
EXERCISE XXXIII (338).
XXI. JBlNOMIAL AND MULTINOMIAL THEOREMS. 
Statement, Vandermonde's Theorem (340). Binomial Theorem. 
Euler's Proof, Second Proof, Particular Ins tancesj 34 1-345). Num- 
erically Greatest Term, Approximate Values of " ; 1 + x (345-348). 
EXERCISE XXXIV (349). 
Use of Binomial Theorem in Summing Series, ^Multinomia^Theorem 
(351-355). 
EXERCISE XXXV (355). 
XXII. RATIONAL FRACTIONS (2), PR,ECURRING)SERIES AND DIFFERENCE 
EQUATIONS. 
Expansion of a Rational Fraction (357-359). 
EXERCISE XXXVI (359). 
Expansions of cos nd and sin nOjsin 9 in Powers of cos 6 (360). 
Recurring Series, Scale of Relation, Convergence, Generating Func- 
tion, Sum (360-363). Linear Difference Equations with Constant 
Coefficients (363-365). 
EXERCISE XXXVII (365). 
Difference Equations, General and Particular Solutions (367-370). 
EXERCISE XXXVIII (370). 
CONTENTS xiii 
CHAPTER 
XXIII. THE OPERATORS 4> E, D. INTERPOLATION. 
The Operators J, E, Series for A r u x , u x+r ; U 1 + u 2 + u 3 + ... in 
Terms of u l9 Au^ A 2 u l9 ... (373-379). Interpolation, Lagrange's and 
/ d \ n 
Bessel's Formulae (379-382). The Operator f) 9 Value of ( j- ) (uv) 
(382, 383). Vcte/ 
EXERCISE XXXIX (384). 
XXIV. CONTINUED FRACTIONS (1). 
Definitions, ForniationjoConvergents, Infinite Continued Fractions 
(388-391). Simple and ^eeuTrirSg; Continued Fractions (391-394). 
EXERCISE XL (394). 
Simple Continued Fractions, Properties of the Convergents, an 
Irrational as a Simple Continued Fraction (396-401). Approxima- 
tions, Miscellaneous Theorems (402-406). Symmetric Continued 
Fractions, Application to Theory of Numbers (406-409). Simple
Solution for Higher Algebra

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