CONTENTS. (Entries in small print at the end of the contents of each chapter refer to subiects discussed incidentally in the examples) CHAPTER I

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1 CONTENTS (Entries in small print at the end of the contents of each chapter refer to subiects discussed incidentally in the examples) CHAPTER I REAL VARIABLES Rational numbers Irrational numbers Real numbers Relations of magnitude between real numbers Algebraical operations with real numbers The number '\1"2 Quadratio surds. The continuum The continuous real variable Sections of the real numbers. Dedekind's theorem Points of accumulation Weierstrass's theorem. Decimals, 1. Gauss's theorem, 7. Graphical solution of quadratic equa tions,21. Important inequalities, 3:3. Arithmetical and geometrical means, 34. Cauchy's inequality, 34. Cubic and other surds, 30. Algebraio num bers, 38. CHAPTER II FUNCTIONS OF REAL VARIABLES The idea of a fuilction. The graphical representation of functions. Coordinates Polar coordinates Polynomials Rational functions Algebraical functions Transcendental functions Graphical solution of equations Functions of two variables and their graphical representation Curves in a plane Loci in space Trigonometrical functions, 55. Arithmetical functions, 58. Cylinders, 64. Contour maps, 64. Cones, 65. Surfaces of revolution, 65. Ruled surfaces, 66. Geometrical constructions for irrational numbers, 68. Quadrature of the circle,

2 viii CONTENTS CHAPTER III COMPLEX NUMBERS Displacements Complex numbers The quadratic equation with real coefficients Argand's diagram De Moivre's theorem. Rational functions of a complex variable Roots of complex numbers. Properties of a triangle, 92, 104. Equations with complex coefficients, 94. Coaxal circles, 96. Bilinear and other transformations, 97, 100, 107. CrosB ratios, 99. Condition that four points should be concyclic, 100. Complex functions of a real variable, 100. Construction of regular polygons by Euclidean methods, 103. Imaginary points and lines, CHAPTER IV LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE ' Functions of a positive integral variable 110 Interpolation III Finite and infinite classes 112 Properties possessed by a function of n for large values of n 113 Definition of a limit and other definitions 120 Oscillating functions 126 General theorems concerning limits 129 Steadily increasing or decreasing functions 136 Alternative proof of Weierstrass's theorem 138 The limit of XR 139 The limit of (1 + ~r. 142 Some algebraical lemmas 143 The limit of n(~x-l) 144 Infinite series 145 The infinite geometrical series 149 The representation of functions of a continuous real variable by means of limits. 153 The bounds of a bounded aggregate 155 The bounds of a bounded function 156 The limits of indetermination of a bounded function 156 The general principle of convergence 158 Limits of complex functions and series of complex terms 160

3 OONTENTS Applications to z" and the geometrical series The symbols 0, 0,"'" ix Oscillation of sin nott, 125, 127, 158. Limits of nkx", fix, fin, ~, f/(n i). (:) x" Decimals Arithmetic series narm~~ic series Equation x"+1 =1(x,,) Limit of a mean value Expansions of rational functions, 170. CHAPTER V LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS Limits as x or x Limits as x -+ a The symbols 0, 0,,...,: orders of smallness and greatness Continuous functions of a real variable Properties of continuous functions. Bounded functions. The oscillation of a function in an interval Sets of intervals on a line. The Heine-Borel theorem Continuous functions of several variables Implicit and inverse functions Limits and continuity of polynomials and rational functions, m.... Limit of x-=:_tj., 181. Limit of ~~~ Infinity of a function, 188. Conx-a x tinuity of cos x and sin x, 188. Classification of discontinuities, 188. Semicontinuity, 209. CHAPTER VI DERIVATIVES AND INTEGRALS Derivatives _ General rules for differentiation Derivatives of complex functions The notation of the differential calculus Differentiation of polynomials Differentiation of rational functions Differentiation of algebraical functions Differentiation of transcendental functions Repeated differentiation General theorems concerning derivatives. Rolle's theorem Maxima and minima The mean value theorem _ Cauchy's mean value theorem 244

4 x OONTENTS A theorem of Darboux Integration. The logarithmio funotion Integration of polynomials. Integration of rational funotions. Integration of algebraical functions. Integration rationalisation. Integration by parts Integration of transcendental functions Areas of plane curves. Lengths of plane curves Derivative of x"', 214. Derivatives of cos x and sin x, 214. Tangent and normal to a curve, 214, 228. Multiple roots of equations, 221, 277. Rolle's theorem for polynomials, 222. Leibniz's theorem, 229. Maxima and minima of the quotient of two quadratics, 238, 277. Axes of a conic, 241. Length. and areas in polar coordinates, 273. Differentiation of a determinant, 274. Formulae of reduction, 282. by PAGB CHAPTER VII ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND INTEGRAL CALCULUS Taylor's theorem. Taylor's series Applications of Taylor's minima theorem to maxima and The calculation of certain limits The contact of plane curves Differentiation of functions of several variables The mean value theorem for functions of two variables Differentials Definite integrals. The circular functions. Caloulation of the definite integral as the limit of a sum General properties of the definite integral Integration by parts and by substitution Alternative proof of Taylor's theorem. Application to the binomial series Approximate formulae for definite integrals. Simpson's rule Integrals of complex functions Misoellaneous examples Newton's method of approximation to the roots of equations, 288. Series for cos x and sin x, 292. Binomial series, 292. Tangent to a curve, 298, 310, 335. Points of inflexion, 298. Curvature, 299, 334. Osculating eonic., 299, 334. Differentiation of implicit functions, 310. Maxima and minima of functions of two variables, 311. Fourier's integral., 318,323. The second mean value theorem, 325. Homogeneous functions, 334. Euler'. theorem, 334. Jacobians, 335. Schwarz's inequality,

5 CONTENTS xi CHAPTER VIII THE CONVERGENCE OF INFINITE SERIES AND INFINITE INTEGRALS Series of positive tenns. Cauchy's and d'alembert's tests of convergence Ratio tests Dirichlet's theorem Multiplication of series of positive terms Further tests for convergence. Abel's theorem. Maclaurin's integral test The series En-' Cauchy'S condensation tost Further ratio tests Infinite integrals Series of positive and negative terms Absolutely convergent series Conditionally convergent series Alternating sories Abel's and Dirichlet's tests of convergence Series of complex terms Power series Multiplication of series Absolutely and conditionally convergent infinite integrals The series Enkr" and allied series, 345. Hypergeometric series, 355. Binomial series, 356, 386, 387. Transformation of infinite integrals by substitution and integration by parts, 361,363, 369. The series Ea,,. cosn/}, Ea" sin nlj, 374, 380, 381. Alteration of the sum of a series by rearrangement, 378. Logarithmic series, 385. Multiplication of conditionally convergent series, 388, 394. Recurring series, 392. Differenoe equations, 393. Definite integral~, 395. CHAPTER IX l THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS OF A REAL VARIABLE The logarithmic function The functional equation satisfied by log x The behaviour of log x as x tends to infinity or to zero The logarithmic scale of infinity The number /l The exponential function The general power a Z The exponential limit The logarithmio limit

6 xii CONTENTS Common logarithms Logarithmic tests of convergence The exponential series. The logarithmic series The series for arc tan x The binomial series Alternative development of the theory The analytical theory of the circular functions Integrals containing the exponontial function, 413. The hyperbolic funo tions,415. Integrals of certain algebraical functions, 416. Euler's constant, 420. Irrationality of e, 423. Approximation to lurds by the binomial theorem, 430. Irrationality of logi. n, 438. Definite integrals, 445, <146. PAGl!! CHAPTER X THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS Functions of a complex variable. Curvilinear integrals Definition of the logarithmic function. The values of the logarithmic function The exponential function The general power a'. The trigonometrical and hyperbolic functions The connection between the logarithmic and inverse trigonometrical functions The exponential series. The series for cos z and sin z The logarithmic series The exponential limit The binomial series The functional equation satisfied by Log z, 454. The function e', 460. Logarithms to any base, 461. The inverse cosine, Bine, and tangent of a complex number, 464. Trigonometrical leries. 470, , 484, 485. Roots of transcendental equations, 479, 480. Transformations, Stereographic projection, 482. Mercator's projection, 482. Level curve8, Definite integrals, 486. ApPENDIX I. APPENDIX II. APPENDIX III. APPENDIX IV. INDEX. The inequalities of Holder and Minkowski The proof that every equation has a root A note on double limit problems The infinite in analysis and geometry