A Course of Pure Mathematics Centenary edition

, by G. H. Hardy , Foreword by T. W. KÃ¶rner

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ISBN: 9780521720557 | 0521720559
Cover: Paperback
Copyright: 3/24/2008

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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.

Real variables
Rational numbers	p. 1
Irrational numbers	p. 3
Real numbers	p. 14
Relations of magnitude between real numbers	p. 16
Algebraical operations with real numbers	p. 17
The number [radical]2	p. 20
Quadratic surds	p. 20
The continuum	p. 24
The continuous real variable	p. 27
Sections of the real numbers. Dedekind's theorem	p. 28
Points of accumulation	p. 30
Weierstrass's theorem	p. 31
Miscellaneous examples	p. 32
Decimals	p. 1
Gauss's theorem	p. 7
Graphical solution of quadratic equations	p. 21
Important inequalities	p. 33
Arithmetical and geometrical means	p. 34
Cauchy's inequality	p. 34
Cubic and other surds	p. 36
Algebraical numbers	p. 38
Functions of real variables
The idea of a function	p. 40
The graphical representation of functions. Coordinates	p. 43
Polar coordinates	p. 45
Polynomials	p. 46
Rational functions	p. 49
Algebraical functions	p. 52
Transcendental functions	p. 55
Graphical solution of equations	p. 60
Functions of two variables and their graphical representation	p. 61
Curves in a plane	p. 62
Loci in space	p. 63
Miscellaneous examples	p. 67
Trigonometrical functions	p. 55
Arithmetical functions	p. 58
Cylinders	p. 64
Contour maps	p. 64
Cones	p. 65
Surfaces of revolution	p. 65
Ruled surfaces	p. 66
Geometrical constructions for irrational numbers	p. 68
Quadrature of the circle	p. 70
Complex numbers
Displacements	p. 72
Complex numbers	p. 80
The quadratic equation with real coefficients	p. 84
Argand's diagram	p. 87
De Moivre's theorem	p. 88
Rational functions of a complex variable	p. 90
Roots of complex numbers	p. 101
Miscellaneous examples	p. 104
Properties of a triangle	p. 92
Equations with complex coefficients	p. 94
Coaxal circles	p. 96
Bilinear and other transformations	p. 97
Cross ratios	p. 99
Condition that four points should be concyclic	p. 100
Complex functions of a real variable	p. 100
Construction of regular polygons by Euclidean methods	p. 103
Imaginary points and lines	p. 106
Limits of functions of a positive integral variable
Functions of a positive integral variable	p. 110
Interpolation	p. 111
Finite and infinite classes	p. 112
Properties possessed by a function of n for large values of n	p. 113
Definition of a limit and other definitions	p. 120
Oscillating functions	p. 126
General theorems concerning limits	p. 129
Steadily increasing or decreasing functions	p. 136
Alternative proof of Weierstrass's theorem	p. 138
The limit of x[superscript n]	p. 139
The limit of [characters not reproducible]	p. 142
Some algebraical lemmas	p. 143
The limit of [characters not reproducible]	p. 144
Infinite series	p. 145
The infinite geometrical series	p. 149
The representation of functions of a continuous real variable by means of limits	p. 153
The bounds of a bounded aggregate	p. 155
The bounds of a bounded function	p. 156
The limits of indetermination of a bounded function	p. 156
The general principle of convergence	p. 158
Limits of complex functions and series of complex terms	p. 160
Applications to z[superscript n] and the geometrical series	p. 162
The symbols O, o, [tilde]	p. 164
Miscellaneous examples	p. 166
Oscillation of sin n[theta pi]	p. 125
Limits of [characters not reproducible]	p. 141
Decimals	p. 149
Arithmetic series	p. 152
Harmonic series	p. 153
Equation x[subscript n+1]=f(x[subscript n])	p. 166
Limit of a mean value	p. 167
Expansions of rational functions	p. 170
Limits of functions of a continuous variable. Continuous and discontinuous functions
Limits as x to [infinity] or x to - [infinity]	p. 172
Limits as x to a	p. 175
The symbols O, o, [tilde]: orders of smallness and greatness	p. 183
Continuous functions of a real variable	p. 185
Properties of continuous functions. Bounded functions. The oscillation of a function in an interval	p. 190
Sets of intervals on a line. The Heine-Borel theorem	p. 196
Continuous functions of several variables	p. 201
Implicit and inverse functions	p. 203
Miscellaneous examples	p. 206
Limits and continuity of polynomials and rational functions	p. 179
Limit of [characters not reproducible]	p. 181
Limit of [characters not reproducible]	p. 182
Infinity of a function	p. 188
Continuity of cos x and sin x	p. 188
Classification of discontinuities	p. 188
Semicontinuity	p. 209
Derivatives and integrals
Derivatives	p. 210
General rules for differentiation	p. 216
Derivatives of complex functions	p. 218
The notation of the differential calculus	p. 218
Differentiation of polynomials	p. 220
Differentiation of rational functions	p. 223
Differentiation of algebraical functions	p. 224
Differentiation of transcendental functions	p. 225
Repeated differentiation	p. 228
General theorems concerning derivatives. Rolle's theorem	p. 231
Maxima and minima	p. 234
The mean value theorem	p. 242
Cauchy's mean value theorem	p. 244
A theorem of Darboux	p. 245
Integration. The logarithmic function	p. 245
Integration of polynomials	p. 249
Integration of rational functions	p. 250
Integration of algebraical functions. Integration by rationalisation. Integration by parts	p. 254
Integration of transcendental functions	p. 264
Areas of plane curves	p. 268
Lengths of plane curves	p. 270
Miscellaneous examples	p. 273
Derivative of x[superscript m]	p. 214
Derivatives of cos x and sin x	p. 214
Tangent and normal to a curve	p. 214
Multiple roots of equations	p. 221
Rolle's theorem for polynomials	p. 222
Leibniz's theorem	p. 229
Maxima and minima of the quotient of two quadratics	p. 238
Axes of a conic	p. 241
Lengths and areas in polar coordinates	p. 273
Differentiation of a determinant	p. 274
Formulae of reduction	p. 282
Additional theorems in the differential and integral calculus
Taylor's theorem	p. 285
Taylor's series	p. 291
Applications of Taylor's theorem to maxima and minima	p. 293
The calculation of certain limits	p. 293
The contact of plane curves	p. 296
Differentiation of functions of several variables	p. 300
The mean value theorem for functions of two variables	p. 305
Differentials	p. 307
Definite integrals	p. 311
The circular functions	p. 316
Calculation of the definite integral as the limit of a sum	p. 319
General properties of the definite integral	p. 320
Integration by parts and by substitution	p. 324
Alternative proof of Taylor's theorem	p. 327
Application to the binomial series	p. 328
Approximate formulae for definite integrals. Simpson's rule	p. 328
Integrals of complex functions	p. 331
Miscellaneous examples	p. 332
Newton's method of approximation to the roots of equations	p. 288
Series for cos x and sin x	p. 292
Binomial series	p. 292
Tangent to a curve	p. 298
Points of inflexion	p. 298
Curvature	p. 299
Osculating conics	p. 299
Differentiation of implicit functions	p. 310
Maxima and minima of functions of two variables	p. 311
Fourier's integrals	p. 318
The second mean value theorem	p. 325
Homogeneous functions	p. 334
Euler's theorem	p. 334
Jacobians	p. 335
Schwarz's inequality	p. 340
The convergence of infinite series and infinite integrals
Series of positive terms. Cauchy's and d'Alembert's tests of convergence	p. 341
Ratio tests	p. 343
Dirichlet's theorem	p. 347
Multiplication of series of positive terms	p. 347
Further tests for convergence. Abel's theorem. Maclaurin's integral test	p. 349
The series [Sigma]n[superscript -3]	p. 352
Cauchy's condensation test	p. 354
Further ratio tests	p. 355
Infinite integrals	p. 356
Series of positive and negative terms	p. 371
Absolutely convergent series	p. 373
Conditionally convergent series	p. 375
Alternating series	p. 376
Abel's and Dirichlet's tests of convergence	p. 379
Series of complex terms	p. 381
Power series	p. 382
Multiplication of series	p. 386
Absolutely and conditionally convergent infinite integrals	p. 388
Miscellaneous examples	p. 390
The series [Sigma]n[superscript k]r[superscript n] and allied series	p. 345
Hypergeometric series	p. 355
Binomial series	p. 356
Transformation of infinite integrals by substitution and integration by parts	p. 361
The series [Sigma]a[subscript n] cos n[theta], [Sigma]a[subscript n] sin n[theta]	p. 374
Alteration of the sum of a series by rearrangement	p. 378
Logarithmic series	p. 385
Multiplication of conditionally convergent series	p. 388
Recurring series	p. 392
Difference equations	p. 393
Definite integrals	p. 395
The logarithmic, exponential, and circular functions of a real variable
The logarithmic function	p. 398
The functional equation satisfied by log x	p. 401
The behaviour of log x as x tends to infinity or to zero	p. 402
The logarithmic scale of infinity	p. 403
The number e	p. 405
The exponential function	p. 406
The general power a[superscript x]	p. 409
The exponential limit	p. 410
The logarithmic limit	p. 411
Common logarithms	p. 412
Logarithmic tests of convergence	p. 417
The exponential series	p. 422
The logarithmic series	p. 425
The series for arc tan x	p. 426
The binomial series	p. 429
Alternative development of the theory	p. 431
The analytical theory of the circular functions	p. 432
Miscellaneous examples	p. 438
Integrals containing the exponential function	p. 413
The hyperbolic functions	p. 415
Integrals of certain algebraical functions	p. 416
Euler's constant	p. 420
Irrationality of e	p. 423
Approximation to surds by the binomial theorem	p. 430
Irrationality of log[subscript 10] n	p. 438
Definite integrals	p. 445
The general theory of the logarithmic, exponential, and circular functions
Functions of a complex variable	p. 447
Curvilinear integrals	p. 448
Definition of the logarithmic function	p. 449
The values of the logarithmic function	p. 451
The exponential function	p. 456
The general power a[superscript zeta]	p. 457
The trigonometrical and hyperbolic functions	p. 462
The connection between the logarithmic and inverse trigonometrical functions	p. 466
The exponential series	p. 468
The series for cos z and sin z	p. 469
The logarithmic series	p. 471
The exponential limit	p. 474
The binomial series	p. 476
Miscellaneous examples	p. 479
The functional equation satisfied by Log z	p. 454
The function e[superscript zeta]	p. 460
Logarithms to any base	p. 461
The inverse cosine, sine, and tangent of a complex number	p. 464
Trigonometrical series	p. 470
Roots of transcendental equations	p. 479
Transformations	p. 480
Stereographic projection	p. 482
Mercator's projection	p. 482
Level curves	p. 484
Definite integrals	p. 486
The proof that every equation has a root	p. 487
A note on double limit problems	p. 493
The infinite in analysis and geometry	p. 497
The infinite in analysis and geometry	p. 502
Index	p. 505
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