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A Course of Pure Mathematics Centenary Edition

Book Description

There are few textbooks of mathematics as well-known as Hardy's Pure Mathematics. Since its publication in 1908, this classic book has inspired successive generations of budding mathematicians at the beginning of their undergraduate courses. In its pages, Hardy combines the enthusiasm of the missionary with the rigour of the purist in his exposition of the fundamental ideas of the differential and integral calculus, of the properties of infinite series and of other topics involving the notion of limit. Celebrating 100 years in print with Cambridge, this edition includes a Foreword by T. W. Körner, describing the huge influence the book has had on the teaching and development of mathematics worldwide. Hardy's presentation of mathematical analysis is as valid today as when first written: students will find that his economical and energetic style of presentation is one that modern authors rarely come close to.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Chapter I. Real variables
    1. 1–2. Rational numbers
    2. 3–7. Irrational numbers
    3. 8. Real numbers
    4. 9. Relations of magnitude between real numbers
    5. 10–11. Algebraical operations with real numbers
    6. 12. The number √2
    7. 13–14. Quadratic surds
    8. 15. The continuum
    9. 16. The continuous real variable
    10. 17. Sections of the real numbers. Dedekind’s theorem
    11. 18. Points of accumulation
    12. 19. Weierstrass’s theorem
    13. Miscellaneous examples
  7. Chapter II. Functions of real variables
    1. 20. The idea of a function
    2. 21. The graphical representation of functions. Coordinates
    3. 22. Polar coordinates
    4. 23. Polynomials
    5. 24–25. Rational functions
    6. 26–27. Algebraical functions
    7. 28–29. Transcendental functions
    8. 30. Graphical solution of equations
    9. 31. Functions of two variables and their graphical representation
    10. 32. Curves in a plane
    11. 33. Loci in space
    12. Miscellaneous examples
  8. Chapter III. Complex numbers
    1. 34–38. Displacements
    2. 39–42. Complex numbers
    3. 43. The quadratic equation with real coefficients
    4. 44. Argand’s diagram
    5. 45. De Moivre’s theorem
    6. 46. Rational functions of a complex variable
    7. 47–49. Roots of complex numbers
    8. Miscellaneous examples
  9. Chapter IV. Limits of functions of a positive integral variable
    1. 50. Functions of a positive integral variable
    2. 51. Interpolation
    3. 52. Finite and infinite classes
    4. 53–57. Properties possessed by a function of n for large values of n
    5. 58–61. Definition of a limit and other definitions
    6. 62. Oscillating functions
    7. 63–68. General theorems concerning limits
    8. 69–70. Steadily increasing or decreasing functions
    9. 71. Alternative proof of Weierstrass’s theorem
    10. 72. The limit of xn
    11. 73. The limit of (1 + 1/n)n
    12. 74. Some algebraical lemmas
    13. 75. The limit of n(n√x − 1)
    14. 76–77. Infinite series
    15. 78. The infinite geometrical series
    16. 79. The representation of functions of a continuous real variable by means of limits
    17. 80. The bounds of a bounded aggregate
    18. 81. The bounds of a bounded function
    19. 82. The limits of indetermination of a bounded function
    20. 83–84. The general principle of convergence
    21. 85–86. Limits of complex functions and series of complex terms
    22. 87–88. Applications to zn and the geometrical series
    23. 89. The symbols O, o, ~
    24. Miscellaneous examples
  10. Chapter V. Limits of functions of a continuous variable. Continuous and discontinuous functions
    1. 90–92. Limits as x → ∞ or x → − ∞
    2. 93–97. Limits as x → a
    3. 98. The symbols O, o, ~: orders of smallness and greatness
    4. 99–100. Continuous functions of a real variable
    5. 101–105. Properties of continuous functions. Bounded functions. The oscillation of a function in an interval
    6. 106–107. Sets of intervals on a line. The Heine-Borel theorem
    7. 108. Continuous functions of several variables
    8. 109–110. Implicit and inverse functions
    9. Miscellaneous examples
  11. Chapter VI. Derivatives and integrals
    1. 111–113. Derivatives
    2. 114. General rules for differentiation
    3. 115. Derivatives of complex functions
    4. 116. The notation of the differential calculus
    5. 117. Differentiation of polynomials
    6. 118. Differentiation of rational functions
    7. 119. Differentiation of algebraical functions
    8. 120. Differentiation of transcendental functions
    9. 121. Repeated differentiation
    10. 122. General theorems concerning derivatives. Rolle’s theorem
    11. 123–125. Maxima and minima
    12. 126–127. The mean value theorem
    13. 128. Cauchy’s mean value theorem
    14. 129. A theorem of Darboux
    15. 130–131. Integration. The logarithmic function
    16. 132. Integration of polynomials
    17. 133–134. Integration of rational functions
    18. 135–142. Integration of algebraical functions. Integration by rationalisation. Integration by parts
    19. 143–147. Integration of transcendental functions
    20. 148. Areas of plane curves
    21. 149. Lengths of plane curves
    22. Miscellaneous examples
  12. Chapter VII. Additional theorems in the differential and integral calculus
    1. 150–151. Taylor’s theorem
    2. 152. Taylor’s series
    3. 153. Applications of Taylor’s theorem to maxima and minima
    4. 154. The calculation of certain limits
    5. 155. The contact of plane curves
    6. 156–158. Differentiation of functions of several variables
    7. 159. The mean value theorem for functions of two variables
    8. 160. Differentials
    9. 161–162. Definite integrals
    10. 163. The circular functions
    11. 164. Calculation of the definite integral as the limit of a sum
    12. 165. General properties of the definite integral
    13. 166. Integration by parts and by substitution
    14. 167. Alternative proof of Taylor’s theorem
    15. 168. Application to the binomial series
    16. 169. Approximate formulae for definite integrals. Simpson’s rule
    17. 170. Integrals of complex functions
    18. Miscellaneous examples
  13. Chapter VIII. The convergence of infinite series and infinite integrals
    1. 171–174. Series of positive terms. Cauchy’s and d’Alembert’s tests of convergence
    2. 175. Ratio tests
    3. 176. Dirichlet’s theorem
    4. 177. Multiplication of series of positive terms
    5. 178–180. Further tests for convergence. Abel’s theorem. Maclaurin’s integral test
    6. 181. The series Σn−s
    7. 182. Cauchy’s condensation test
    8. 183. Further ratio tests
    9. 184–189. Infinite integrals
    10. 190. Series of positive and negative terms
    11. 191–192. Absolutely convergent series
    12. 193–194. Conditionally convergent series
    13. 195. Alternating series
    14. 196. Abel’s and Dirichlet’s tests of convergence
    15. 197. Series of complex terms
    16. 198–201. Power series
    17. 202. Multiplication of series
    18. 203. Absolutely and conditionally convergent infinite integrals
    19. Miscellaneous examples
  14. Chapter IX. The logarithmic, exponential, and circular functions of a real variable
    1. 204–205. The logarithmic function
    2. 206. The functional equation satisfied by log x
    3. 207–209. The behaviour of log x as x tends to infinity or to zero
    4. 210. The logarithmic scale of infinity
    5. 211. The number e
    6. 212–213. The exponential function
    7. 214. The general power ax
    8. 215. The exponential limit
    9. 216. The logarithmic limit
    10. 217. Common logarithms
    11. 218. Logarithmic tests of convergence
    12. 219. The exponential series
    13. 220. The logarithmic series
    14. 221. The series for arc tan x
    15. 222. The binomial series
    16. 223. Alternative development of the theory
    17. 224–226. The analytical theory of the circular functions
    18. Miscellaneous examples
  15. Chapter X. The general theory of the logarithmic, exponential, and circular functions
    1. 227–228. Functions of a complex variable
    2. 229. Curvilinear integrals
    3. 230. Definition of the logarithmic function
    4. 231. The values of the logarithmic function
    5. 232–234. The exponential function
    6. 235–236. The general power aζ
    7. 237–240. The trigonometrical and hyperbolic functions
    8. 241. The connection between the logarithmic and inverse trigonometrical functions
    9. 242. The exponential series
    10. 243. The series for cos z and sin z
    11. 244–245. The logarithmic series
    12. 246. The exponential limit
    13. 247. The binomial series
    14. Miscellaneous examples
  16. Appendix I. The inequalities of Hölder and Minkowski
  17. Appendix II. The proof that every equation has a root
  18. Appendix III. A note on double limit problems
  19. Appendix IV. The infinite in analysis and geometry
  20. Index