# A Course of Pure Mathematics

Cambridge University Press, 1952 - Mathematics - 509 pages
There can be few textbooks of mathematics as well-known as Hardy's Pure Mathematics. Since its publication in 1908, it has been a classic work to which successive generations of budding mathematicians have turned at the beginning of their undergraduate courses. In its pages, Hardy combines the enthusiasm of a missionary with the rigor of a 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.

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### Contents

 CHAPTER 1 Real numbers 14 The number V 2 20 The continuous real variable 27 Gausss theorem 7 Graphical solution of quadratic equa 34 CHAPTER II 40 Polynomia s 46 Graphical solution of equations 60
 Derivative of xm 214 Derivatives of cos x and sin x 214 Tangent 274 Formulae of redu cion 282 Taylors series 291 The mean value theorem for functions of two variables 305 The circular functions 316 Integration by parts and by substitution 324 Integrals of complex functions 331 functions of two variables 311 Fouriers integrals 318 323 The second 340

 Trigonometrical functions 55 Arithmetical functions 58 Cylinders 70 COMPLEX NUMBERS 72 The quadratic equation with real coefficients 84 Rational functions of a complex variable 90 Properties of a triangle 92 104 Equations with complex coefficients 106 Interpolation 112 Oscillating functions 126 Alternative proof of Weierstrasss theorem 138 The limit of nrl 144 The representation of functions of a continuous real 153 SECT PAGE 162 Equation zn+ixB 166 Limit of a mean value 167 Expansions 170 SECT PAGE 171 orders of smallness and greatness 183 Continuous functions of several variables 201 CHAPTER VI 210 General rules for differentiation 216 Differentiation of rational functions 223 General theorems concerning derivatives Rolles 231 Cauchys mean value theorem 244 SECT PAGE 245 Areas of plane curves 268
 tests of convergence 341 Dirichlets theorem 347 Cauchys condensation test 354 Series of positive and negative terms 371 Abels and Dirichlets tests of convergence 379 Multiplication of series 386 CHAPTER IX 398 The number e 405 SECT PAGE 412 The exponential series 422 The binomial series 429 Integrals containing the exponential function 413 The hyperbolic func 445 The values of the logarithmic function 451 The general power a 409 457 The connection between the logarithmic and inverse 466 The exponential limit 410 474 The functional equation satisfied by Log z 454 The function e1 460 480 Stereographic projection 482 Mercators projection 482 Level curves 486 APPENDIX IT A note on double limit problems 493 The infinite in analysis and geometry 502 Copyright