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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 |
Table of Contents provided by Ingram. All Rights Reserved. |
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Celebrating 100 years in print with Cambridge, this newly updated edition includes a foreword by T. W. K_rner, describing the huge influence the book has had on the teaching and development of mathematics worldwide. There are few textbooks in 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 rigor 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. 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.
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