1908 · Book · Treatise · Cambridge University Press
A Course of Pure Mathematics
Description
Hardy's introduction to analysis for students in their first university year: real numbers, functions of real variables, complex numbers, limits, continuity, derivatives and integrals, the convergence of infinite series and integrals, and the logarithmic, exponential and circular functions, with over a thousand examples. This is the third edition (Cambridge University Press, 1921), in 237 numbered sections with Hardy's examples, miscellaneous examples at the end of each chapter and four appendices.
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Contents
- Preface to the Third Edition
- Extract from the Preface to the Second Edition
- Extract from the Preface to the First Edition
- Appendix IIA Note on Double Limit Problems
- Appendix IIIThe circular functions
- Appendix IVThe infinite in analysis and geometry
Chapter I — Real Variables
- 1Rational numbers
- 2The representation of rational numbers by points on a line
- 3Irrational numbers
- 4Irrational numbers (continued)
- 5Irrational numbers (continued)
- 6Irrational numbers (continued)
- 7Irrational numbers (continued)
- 8Real numbers
- 9Relations of magnitude between real numbers
- 10Algebraical operations with real numbers
- 11Algebraical operations with real numbers (continued)
- 12The number
- 13Quadratic surds
- 14Some theorems concerning quadratic surds
- 15The continuum
- 16The continuous real variable
- 17Sections of the real numbers
- 18Points of accumulation
- 19Weierstrass’s Theorem
- Misc.Miscellaneous Examples on Chapter I
Chapter II — Functions of Real Variables
- 20The idea of a function
- 21The graphical representation of functions
- 22Polar coordinates
- 23Further examples of functions and their graphical representation
- 24B. Rational Functions
- 25
- 26C. Explicit Algebraical Functions
- 27D. Implicit Algebraical Functions
- 28Transcendental functions
- 29F. Other classes of transcendental functions
- 30Graphical solution of equations containing a single unknown number
- 31Functions of two variables and their graphical representation
- 32Curves in a plane
- 33Loci in space
- Misc.Miscellaneous Examples on Chapter II
Chapter III — Complex Numbers
- 34Displacements along a line and in a plane
- 35Equivalence of displacements. Multiplication of displacements by numbers
- 36Addition of displacements
- 37Multiplication of displacements
- 38
- 39Complex numbers
- 40
- 41The equation
- 42The geometrical interpretation of multiplication by
- 43The equations ,
- 44Argand’s diagram
- 45De Moivre’s Theorem
- 46
- 47Roots of complex numbers
- 48Solution of the equation
- 49The general form of De Moivre’s Theorem
- Misc.Miscellaneous Examples on Chapter III
Chapter IV — Limits of Functions of a Positive Integral Variable
- 50Functions of a positive integral variable
- 51Interpolation
- 52Finite and infinite classes
- 53Properties possessed by a function of for large values of
- 54
- 55The phrase ‘ tends to infinity’
- 56The behaviour of a function of as tends to infinity
- 57
- 58Definition of a limit
- 59
- 60
- 61Some points concerning the definitions
- 62Oscillating Functions
- 63Some general theorems with regard to limits
- 64Results subsidiary to Theorem I
- 65B. The behaviour of the product of two functions whose behaviour is known
- 66C. The behaviour of the difference or quotient of two functions whose behaviour is known
- 67
- 68
- 69Functions of which increase steadily with
- 70
- 71Alternative proof of Weierstrass’s Theorem of § 19
- 72The limit of as tends to
- 73The limit of
- 74Some algebraical lemmas
- 75The limit of
- 76Infinite Series
- 77General theorems concerning infinite series
- 78The infinite geometrical series
- 79The representation of functions of a continuous real variable by means of limits
- 80The bounds of a bounded aggregate
- 81The bounds of a bounded function
- 82The limits of indetermination of a bounded function
- 83The general principle of convergence for a bounded function
- 84Unbounded functions
- 85Limits of complex functions and series of complex terms
- 86
- 87The limit of as , being any complex number
- 88The geometric series when is complex
- Misc.Miscellaneous Examples on Chapter IV
Chapter V — Limits of Functions of a Continuous Variable. Continuous and Discontinuous Functions
- 89Limits as tends to
- 90Limits as tends to
- 91Theorems corresponding to those of Ch. IV, §§ 63–67
- 92Steadily increasing or decreasing functions
- 93Limits as tends to
- 94Limits as tends to
- 95Steadily increasing or decreasing functions
- 96Limits of indetermination and the principle of convergence
- 97
- 98Continuous functions of a real variable
- 99
- 100The fundamental property of a continuous function
- 101
- 102The range of values of a continuous function
- 103The oscillation of a function in an interval
- 104Alternative proofs of Theorem 2 of § 102
- 105Sets of intervals on a line. The Heine-Borel Theorem
- 106
- 107Continuous functions of several variables
- 108Implicit functions
- 109Inverse Functions
- Misc.Miscellaneous Examples on Chapter V
Chapter VI — Derivatives and Integrals
- 110Derivatives or Differential Coefficients
- 111Some general remarks
- 112
- 113Some general rules for differentiation
- 114Derivatives of complex functions
- 115The notation of the differential calculus
- 116Standard forms
- 117B. Rational Functions
- 118C. Algebraical Functions
- 119D. Transcendental Functions
- 120Repeated differentiation
- 121Some general theorems concerning derived functions
- 122Maxima and Minima
- 123
- 124
- 125The Mean Value Theorem
- 126
- 127Integration
- 128The practical problem of integration
- 129Polynomials
- 130Rational Functions
- 131Note on the practical integration of rational functions
- 132Algebraical Functions
- 133Integration by substitution and rationalisation
- 134Integrals connected with conics
- 135The integral
- 136The integral
- 137The integral
- 138Integration by parts
- 139The general integral , where
- 140Transcendental Functions
- 141Polynomials in cosines and sines of multiples of
- 142The integrals , and associated integrals
- 143Rational Functions of and
- 144Integrals involving , , and
- 145Areas of plane curves
- 146Lengths of plane curves
- Misc.Miscellaneous Examples on Chapter VI
Chapter VII — Additional Theorems in the Differential and Integral Calculus
- 147Higher Mean Value Theorems
- 148Taylor’s Series
- 149Applications of Taylor’s Theorem
- 150B. The calculation of certain limits
- 151C. The contact of plane curves
- 152Differentiation of functions of several variables
- 153Differentiation of a function of two functions
- 154The Mean Value Theorem for functions of two variables
- 155Differentials
- 156Definite Integrals and Areas
- 157The definite integral
- 158Area of a sector of a circle. The circular functions
- 159Calculation of the definite integral from its definition as the limit of a sum
- 160General properties of the definite integral
- 161Integration by parts and by substitution
- 162Proof of Taylor’s Theorem by Integration by Parts
- 163Application of Cauchy’s form to the Binomial Series
- 164Integrals of complex functions of a real variable
- Misc.Miscellaneous Examples on Chapter VII
Chapter VIII — The Convergence of Infinite Series and Infinite Integrals
- 165
- 166Series of Positive Terms
- 167
- 168First applications of these tests
- 169
- 170Multiplication of Series of Positive Terms
- 171Further tests for convergence and divergence
- 172
- 173Abel’s (or Pringsheim’s) Theorem
- 174Maclaurin’s (or Cauchy’s) Integral Test
- 175The series
- 176Cauchy’s Condensation Test
- 177Infinite Integrals
- 178The case in which is positive
- 179Application to infinite integrals of the rules for substitution and integration by parts
- 180Other types of infinite integrals
- 181
- 182
- 183Series of positive and negative terms
- 184Absolutely Convergent Series
- 185Extension of Dirichlet’s Theorem to absolutely convergent series
- 186Conditionally convergent series
- 187Tests of convergence for conditionally convergent series
- 188Alternating Series
- 189Abel’s and Dirichlet’s Tests of Convergence
- 190Series of complex terms
- 191Power Series
- 192
- 193The region of convergence of a power series. The circle of convergence
- 194Uniqueness of a power series
- 195Multiplication of Series
- Misc.Miscellaneous Examples on Chapter VIII
Chapter IX — The Logarithmic and Exponential Functions of a Real Variable
- 196
- 197Definition of
- 198The functional equation satisfied by
- 199The manner in which tends to infinity with
- 200Proof that as
- 201The behaviour of as
- 202Scales of infinity. The logarithmic scale
- 203The number
- 204The exponential function
- 205The principal properties of the exponential function
- 206
- 207The general power
- 208The representation of as a limit
- 209The representation of as a limit
- 210Common logarithms
- 211Logarithmic tests of convergence for series and integrals
- 212Series connected with the exponential and logarithmic functions. Expansion of by Taylor’s Theorem
- 213The logarithmic series
- 214The series for the inverse tangent
- 215The Binomial Series
- 216An alternative method of development of the theory of the exponential and logarithmic functions
- Misc.Miscellaneous Examples on Chapter IX
Chapter X — The General Theory of the Logarithmic, Exponential, and Circular Functions
- 217Functions of a complex variable
- 218
- 219Real and complex curvilinear integrals
- 220The definition of
- 221The values of
- 222The exponential function
- 223The value of
- 224The functional equation satisfied by
- 225The general power
- 226The general value of
- 227The exponential values of the sine and cosine
- 228Definition of and for all values of
- 229The generalised hyperbolic functions
- 230Formulae for , , etc
- 231The connection between the logarithmic and the inverse trigonometrical functions
- 232The power series for
- 233The power series for and
- 234The logarithmic series
- 235
- 236Some applications of the logarithmic series. The exponential limit
- 237The general form of the Binomial Theorem
- Misc.Miscellaneous Examples on Chapter X
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