G.H. Hardy | Category: MathematicsBook Details

ISBN: 9789386677334

YOP: 2018

Pages: 587

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G.H. Hardy’s text is a good single volume refresher course for work in analysis and more advanced algebra, including number theory. Not quite as modern as Birkhoff and MacLane’s text, or Manes’ work, this volume forms the underpinnings of both works. If you have a good understanding of the preliminary work required in algebra and geometry, Hardy can be read directly and with pleasure. If you have a desire to understand the basis of what is presented in most first-year calculus texts, then Hardy’s text is for you.

The book contains a large number of descriptive and study materials together with a number of difficult problems with regards to number theory analysis. The book is organized into the following chapters, with each chapter further divided.

1. Real variables

2. Functions of real variables

3 complex numbers

4. limits of functions of a positive integral variable

5. limits of functions of a continuous variable. Continuous and discontinuous functions

6. derivatives and integrals

7. additional theorems in the differential and integral calculus

8. the convergence of infinite series and infinite integrals

9. the logarithmic, exponential and circular functions of a real variable

10. the general theory of the logarithmic, exponential and circular functions

1–2. Rational numbers

3–7. Irrational numbers

8. Real numbers

9. Relations of magnitude between real numbers

10–11. Algebraical operations with real numbers

12. The number

13–14. Quadratic surds

15. The continuum

16. The continuous real variable

17. Sections of the real numbers. Dedekind’s Theorem

18. Points of condensation

19. Weierstrass’s Theorem

Miscellaneous Examples

20. The idea of a function

21. The graphical representation of functions. Coordinates

22. Polar coordinates

23. Polynomials

24–25. Rational functions

26–27. Algebraical functions

28–29. Transcendental functions

30. Graphical solution of equations

31. Functions of two variables and their graphical representation

32. Curves in a plane

33. Loci in space

34–38. Displacements

39–42. Complex numbers

43. The quadratic equation with real coefficients

44. Argand’s diagram

45. De Moivre’s Theorem

46. Rational functions of a complex variable

47–49. Roots of complex numbers

Miscellaneous Examples

50. Functions of a positive integral variable

51. Interpolation

52. Finite and infinite classes

53–57. Properties possessed by a function of n for large values of n

58–61. Definition of a limit and other definitions

62. Oscillating functions

63–68. General theorems concerning limits

69–70. Steadily increasing or decreasing functions

71. Alternative proof of Weierstrass’s Theorem

72. The limit of xn

73. The limit of_1 +1n_n

74. Some algebraical lemmas

75. The limit of n( n√x − 1)

76–77. Infinite series

78. The infinite geometrical series

79. The representation of functions of a continuous real variable by means of limits

80. The bounds of a bounded aggregate

81. The bounds of a bounded function

82. The limits of indetermination of a bounded function

83–84. The general principle of convergence

85–86. Limits of complex functions and series of complex terms

87–88. Applications to zn and the geometrical series

Miscellaneous Examples

89–92. Limits as x→∞ or x→−∞

93–97. Limits as x → a

98–99. Continuous functions of a real variable

100–104. Properties of continuous functions. Bounded functions. The oscillation of a function in an interval

105–106. Sets of intervals on a line. The Heine-Borel Theorem

107. Continuous functions of several variables

108–109. Implicit and inverse functions

Miscellaneous Examples

110–112. Derivatives

113. General rules for differentiation

114. Derivatives of complex functions

115. The notation of the differential calculus

116. Differentiation of polynomials

117. Differentiation of rational functions

118. Differentiation of algebraically functions

119. Differentiation of transcendental functions

120. Repeated differentiation

121. General theorems concerning derivatives. Rolle’s Theorem

122–124. Maxima and minima

125–126. The Mean Value Theorem

127–128. Integration. The logarithmic function

129. Integration of polynomials

130–131. Integration of rational functions

CONTENTS xi

SECT. PAGE

132–139. Integration of algebraical functions. Integration by rationalization. Integration by parts

140–144. Integration of transcendental functions

145. Areas of plane curves

146. Lengths of plane curves

Miscellaneous Examples

147. Taylor’s Theorem

148. Taylor’s Series

149. Applications of Taylor’s Theorem to maxima and minima

150. Applications of Taylor’s Theorem to the calculation of limits

151. The contact of plane curves

152–154. Differentiation of functions of several variables

155. Differentials

156–161. Definite Integrals. Areas of curves

162. Alternative proof of Taylor’s Theorem

163. Application to the binomial series

164. Integrals of complex functions

Miscellaneous Examples

165–168. Series of positive terms. Cauchy’s and d’Alembert’s tests of convergence

169. Dirichlet’s Theorem

170. Multiplication of series of positive terms

171–174. Further tests of convergence. Abel’s Theorem. Maclaurin’s integral

test

175. The series

176. Cauchy’s condensation test

177–182. Infinite integrals

183. Series of positive and negative terms

184–185. Absolutely convergent series

186–187. Conditionally convergent series

188. Alternating series

189. Abel’s and Dirichlet’s tests of convergence

190. Series of complex terms

191–194. Power series

195. Multiplication of series in general

Miscellaneous Examples

196–197. The logarithmic function

198. The functional equation satisfied by log x

199–201. The behaviour of log x as x tends to infinity or to zero

202. The logarithmic scale of infinity

203. The number e

204–206. The exponential function

207. The general power ax

208. The exponential limit

209. The logarithmic limit

210. Common logarithms

211. Logarithmic tests of convergence

212. The exponential series

213. The logarithmic series

214. The series for arc tan x

215. The binomial series

216. Alternative development of the theory

Miscellaneous Examples

217–218. Functions of a complex variable

219. Curvilinear integrals

220. Definition of the logarithmic function

221. The values of the logarithmic function

222–224. The exponential function

225–226. The general power az

227–230. The trigonometrical and hyperbolic functions

231. The connection between the logarithmic and inverse trigonometrical functions

232. The exponential series

233. The series for cos z and sin z

234–235. The logarithmic series

236. The exponential limit

237. The binomial series

Miscellaneous Example

G.H. Hardy’s text is a good single volume refresher course for work in analysis and more advanced algebra, including number theory. Not quite as modern as Birkhoff and MacLane’s text, or Manes’ work, this volume forms the underpinnings of both works. If you have a good understanding of the preliminary work required in algebra and geometry, Hardy can be read directly and with pleasure. If you have a desire to understand the basis of what is presented in most first-year calculus texts, then Hardy’s text is for you.

The book contains a large number of descriptive and study materials together with a number of difficult problems with regards to number theory analysis. The book is organized into the following chapters, with each chapter further divided.

1. Real variables

2. Functions of real variables

3 complex numbers

4. limits of functions of a positive integral variable

5. limits of functions of a continuous variable. Continuous and discontinuous functions

6. derivatives and integrals

7. additional theorems in the differential and integral calculus

8. the convergence of infinite series and infinite integrals

9. the logarithmic, exponential and circular functions of a real variable

10. the general theory of the logarithmic, exponential and circular functions

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