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Hardy G.H. — A course of pure mathematics
Hardy G.H. — A course of pure mathematics



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Название: A course of pure mathematics

Автор: Hardy G.H.

Аннотация:

This book has been designed primarily for the use of first year students at the Universities whose abilities reach or approach something like what is usually described as 'scholarship standard'. I hope that it may be useful to other classes of readers, but it is this class whose wants I have considered first. It is in any case a book for mathematicians: I have nowhere made any attempt to meet the needs of students of engineering or indeed any class of students whose interests are not primarily mathematical.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1908

Количество страниц: 428

Добавлена в каталог: 03.03.2006

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Предметный указатель
Abel's test for convergence      329
Abel's theorem on the multiplication of series      339
Algebraical functions: differentiation of      200 217
Algebraical functions: explicit      41
Algebraical functions: implicit      42
Algebraical functions: integration of      226 et seq. 246
Algebraical numbers: see Numbers, algebraical Amplitude of a complex number      82
Areas of curves      236 et seq. 239 249
Areas of curves in polar coordinates      241
Areas of curves, proof of existence of      271 et seq.
Argand diagram      81
Argument of a function      377
Argument of a point      82
Axes of a conic      63 212
Axes of coordinates      29 51
Axes of coordinates, change of      29 et seq.
Binomial theorem      256 288 334 365 368 405
Cauchy's tests for convergence: condensation test      308
Cauchy's tests for convergence: general test      297
Cauchy's tests for convergence: integral test      303 305
Cis $\theta$, defined      83
Classes, finite and infinite      110
Coaxal circles      89
Comparison theorems for integrals      311 319
Comparison theorems for series      297 325
Condition that four points lie on a circle      92 93
Condition that three points lie on a straight line      88
Cone      58
Conjugate complex numbers      77
Contact of plane curves      259 et seq.
Continuity      171 et seq.
Continuity of cos x and sin x      174
Continuity of functions of two variables      182
Continuity of polynomials and rational functions      173
Continuity of sums and products      173
Continuity, geometrical illustration of      173
Continuous functions without derivatives      187 et seq.
Continuum, the      15 et seq.
Contour maps      57
Convergence, circle and radius of      333
Convergent and divergent      309 et seq.
coordinates      29 51
Coordinates, polar      34
Cos $n \theta$ and sin $n \theta$, applications      97 98 104
Cos $n \theta$ and sin $n \theta$, formulae for      95 et seq. 103
Cos x, sin x, series for      255 400
Cross ratios      92
Curvature      261 291
Cylinder      56
d'Alembert's test for convergence      298
De Moivre's theorem      82 et seq. 101 390
De Moivre's theorem, applications      96 et seq. 103
Decimals, terminating      3
Derivative      185 et seq.
Derivative of $a^x$      351
Derivative of $e^x$      349
Derivative of $x^m$      190 199 351
Derivative of a complex function      194
Derivative of a constant      190
Derivative of a determinant      242
Derivative of a function of a function      216
Derivative of a function of two functions      264 et seq.
Derivative of a polynomial      196 et seq.
Derivative of a rational function      198 et seq.
Derivative of a sum, a product, etc.      192 et seq. 195
Derivative of a transcendental function      200 et seq. 217
Derivative of an algebraical function      200 217
Derivative of an inverse function      194 195
Derivative of arc sin x, etc.      196 201
Derivative of cos x and sin x      190
Derivative of exp{(a+ib)x}      40824
Derivative of f(ax+b)      193
Derivative of log x      343
Derivative of tan x, etc.      200
Derivative, meaning of sign of      205
Derivatives of $a/(a^2+x^2)$ and $x/(a^2+x^2)$      204
Derivatives of $e^{ax}$, etc.      353
Derivatives of functions of several variables      262 et seq.
Derivatives, higher      202 et seq.
Derivatives: discontinuous      213
Derivatives: general theorems concerning      205 et seq.
Difference-equations      337
Differential coefficient      187
Differential equations: $\phi'' = a^2 \phi$      249 371
Differential equations: of straight lines, circles, etc.      242 243
Differential, partial      263
Differential, repeated      202 et seq.
Differential, total      264; see also “Derivative”
Dirichlet's test for convergence      328
Dirichlet's theorem on the rearrangement of series      301 324 330
Discontinuity, types of      175
Displacements      66 et seq.
Double limit problems      420 et seq.
e, number      347 et seq.
Equation of a circle      32 61
Equation of a cone      58
Equation of a conic      63
Equation of a cylinder      56
Equation of a locus in a plane      30 et seq. 53
Equation of a locus in space      51 et seq.
Equation of a plane      52
Equation of a ruled surface      59
Equation of a sphere      53
Equation of a straight line      30
Equation of a surface in general      55
Equation of a surface of revolution      58
Equation with complex coefficients      89 102
Equation x=f(x)      155
Equation, complete      384
Equation, cubic or biquadratic      15 et seq. 42 60 102
Equation, general: of first degree      31 52
Equation, general: of second degree      63
Equation, quadratic: complex roots of      79
Equation, quadratic: geometrical construction for roots of      8
Equation, quadratic: graphical solution of      50
Equation, quadratic: with complex coefficients      88 102
Equation, repeated roots of      197 243
Equations, algebraical: approximation to roots of, by Newton's method      253
Equations, algebraical: graphical solution of      51 60
Equations, algebraical: number of roots of      80 415
Equations, algebraical: proof of existence of roots of      415 et seq.
Equations, algebraical: rational roots of      7 20
Equations, algebraical: repeated roots of      197 209 243
Equations, algebraical: with complex coefficients, cubic      89
Equations, algebraical: with complex coefficients, linear      88
Equations, algebraical: with complex coefficients, quadratic      88 102
Equations, algebraical: with rational coefficients, irrational roots occur in pairs      11 86
Equations, algebraical: with real coefficients, complex roots occur in pairs      85
Equations, differential      see “Differential equations”
Equations, functional      292 344 344 349 350 371 384 388
Equations, transcendental      51 61 201 209 244 254 354 371 396 et 408
Equations: of a line or curve in space      55
Equations: of a section of a cone      62
Equations: of the tangent and normal to a curve      190 218
Euler's      359 369
Euler's constant      359
Euler's theorem on homogeneous functions      291
Expansions      see “Power-series”
Exponential function      348 et seq. 386
Exponential function, continuity of      349 367 387
Exponential function, derivative of      349 367
Exponential function, functional equation satisfied by      349 388
Exponential function, graph of      349
Exponential function, integrals involving      353 et seq.
Exponential function, order of infinity of, as $x \rightarrow +\infty$      350 356 369 372
Exponential function, power-series for      360 et seq. 399
Exponential function, representation of, as a limit      351 et seq. 393
Exponential limit      140 et seq. 154 351 393
Exponential series      300 331 335 360 373 399
Exponential theorem      361
Exponential values of cos x and sin x      394
Factor theorem for complex numbers      77
Formulae of reduction      247 et seq. 316 354
Fourier's integrals      280
Function of several variables      182
Function, attains its upper and lower limits      181
Function, continuous      171 et seq.
Function, fundamental property of      175
Function, range of values of      180 et seq.
Functions      see also “Algebraical functions” “Exponential “Function continuous” “Logarithmic “Oscillation” “Polynomials” “Rational
Functions in stricter sense      179
Functions of a continuous real variable.      25 et seq.
Functions of a positive integral variable      108 et seq.
Functions of several variables      51 et seq.
Functions, $a^x$      350 et seq. 388
Functions, algebraical, explicit      41
Functions, algebraical, implicit      42
Functions, algebraical, transcendental      44 et seq.
Functions, complex, of a real variable      94
Functions, exponential      348 et seq. 386
Functions, expression of as limits      149 150 155
Functions, graphical representation of      28 et seq. 51
Functions, homogeneous      291
Functions, hyperbolic      355 395
Functions, hyperbolic, inverse      356
Functions, increasing and decreasing      135 et seq. 162 171
Functions, independent and non-independent      270
Functions, inverse      178 et seq.
Functions, logarithmic      222 et seq. 342 380 392
Functions, of a complex variable      376 et seq.
Functions, rational      38 et seq.
Functions, trigonometrical or circular      44 et seq. 394
Functions, trigonometrical or circular, inverse      45 et seq. 398
Geometrical allied series      148 et seq. 154 156 299
Geometrical series      145 et seq. 153
Geometrical series, rapidity of convergence of      303
Geometrical series, tests of convergence derived from comparison with      297 et seq.
Harmonic relation of four points      92
homogeneous functions      291
Hyperbolic functions      355 395
Hyperbolic functions, inverse      356
Inequalities $mx^{m-n}(x-a) \gtrless x^m - a^m \gtrless ma^{m-1}(x-a)$      165
Infinite      146 et seq.
infinity      114 et seq.
Infinity of a function      174 175
Infinity scales of, logarithmic and exponential      346 350 370
Inflexion, point of      260
Integral function      220
Integral function, $\int \limits_0^a x^{-s} dx$      317 et seq.
Integral function, $\int \limits_a^{\infty} x^{-s} dx$      307 312
Integral function, indefinite      278
Integral, approximation to      295
Integral, calculation of, from indefinite integral      279
Integral, curvilinear      377 et seq.
Integral, definite      277 et seq.
Integral, direct      280
Integral, finite      310
Integral, general properties of      281 et seq.
Integral, infinite, of first kind      309 et seq.
Integral, limits of      278
Integral, of second kind      316 et seq.
Integrals connected with conics      227 et seq. 246 250
Integration      219 et seq.
Integration $(Ax+B)/(ax^2+2bx+c)$, explicit      224
Integration by parts      230 234 247
Integration by parts for definite integrals, finite      284 et seq. 287
Integration by parts for definite integrals, infinite      316 320
Integration by substitution      226 et seq. 231 234 246 354
Integration by substitution for definite integrals, finite      284 et seq.
Integration by substitution for definite integrals, finite, infinite      314 et seq. 320 321
Integration of $(cos x)^m (sin x)^n$, etc.      233 248
Integration of $1 / \sqrt(x^2+a^2) $, etc.      229
Integration of $1/(a+b \cos x)$, etc.      235 249
Integration of $exp \{ (a+ib)x      \}$408
Integration of $e^{ax} x^n$, etc.      354
Integration of $e^{ax} \cos bx$, etc.      353 et seq.
Integration of $R \{ ,x \sqrt(ax^2+2bx+c) \}$      228 et seq. 246
Integration of $x^n \cos x$, etc.      233 248
Integration of $x^p(1+x)^q$      232 248
Integration of $\sqrt(x^2 + a^2)$, etc.      231
Integration of algebraical functions      226 et seq. 246
Integration of arc sin, etc.      235 et seq.
Integration of log x, etc.      235 et seq. 249
Integration of polynomials      222
Integration of R (cos x, sin x)      234 et seq.
Integration of rational functions      222 et seq. 247
Integration of tan x, etc.      234 248
Integration of transcendental functions      233 et seq.
Integration of trigonometrical functions      233 et seq.
Integration, explicit      225
Integration, path of      378
Integration, range of      278
Integration, subject of      278
Integration: arbitrary constant of      221
Interpolation, functional      109
Irrational numbers      see “Numbers” “irrational”
Jacobians      270
Leibniz's theorem      202
Lengths of curves      238 et seq.
Lengths of curves in polar coordinates      241
Lengths of curves, existence of      276
Level curves      412 et seq.
Limit and value      121 166
Limit as $n \rightarrow 0$ of $(a^x-1)/x$      350 351
Limit as $n \rightarrow 0$ of (sin x)/x, etc.      169 et seq.
Limit as $n \rightarrow 0$ of {log(l+x)}/x      344 392
Limit as $n \rightarrow a$ of $(x^m- a^m)/(x - a)$      168
Limit as $n \rightarrow a$ of $x^m$      165 166 173
Limit as $n \rightarrow a$ of P(x), R(x)      165
Limit as $n \rightarrow \infty$ $1 + {1 \over 2 } + \ldots + { 1 \over n } - \log n$, etc.      359 369
Limit as $n \rightarrow \infty$ of $(n!)^{1/n}$      140
Limit as $n \rightarrow \infty$ of $(s_1+s_2+ \ldots s_n)/n$      158
Limit as $n \rightarrow \infty$ of $n! (a/n)^n$      370
Limit as $n \rightarrow \infty$ of $n(x^{1/n}-1)$      141 et seq. 352
Limit as $n \rightarrow \infty$ of $n^k$      122
Limit as $n \rightarrow \infty$ of $n^r x^n$, $n^{-r} x^n$      140
Limit as $n \rightarrow \infty$ of $n^{1/n}$      140
Limit as $n \rightarrow \infty$ of $x^n$      138 153
Limit as $n \rightarrow \infty$ of $x^n/{n!}$      140
Limit as $n \rightarrow \infty$ of $x^{1/n}$      140
Limit as $n \rightarrow \infty$ of $x_n$, where $x_{n+1} = f(x_n)$      155
Limit as $n \rightarrow \infty$ of $\sqrt [n] {(n!)/n}$      356
Limit as $n \rightarrow \infty$ of ${ \{ 1 +(1/n) \} }^n$      140 et seq. 154
Limit as $n \rightarrow \infty$ of ${ \{ 1 +(x/n) \} }^n$      351 et seq. 393
Limit as $n \rightarrow \infty$ of ${\binom m n} x^n$      155
Limit as $n \rightarrow \infty$ of f(n+1)-f(n) and {f(n)}/n      158
Limit as $n \rightarrow \infty$ of R (n) and $R \{ \phi (n) , \psi (n) \}$      134 et seq.
Limit as $n \rightarrow \infty$ of {f(n+1)}/{f(n)} and $\sqrt[n] {\{ f(n) \}} $      356
Limit as $x \rightarrow 0$ of $x^m$      165 168
Limit of a complex function      151 et seq.
Limit of a function of a complex variable      393
Limit of a sum, a product, etc.      127 et seq. 161 164
Limit of an increasing or decreasing function      135 et seq. 162 171
Limit of f(x), as $x \rightarrow + \infty$      159 et seq.
Limit of f(x), as $x \rightarrow - \infty$      160
Limit of f(x), as $x \rightarrow 0$      162
Limit of f(x), as $x \rightarrow a$      163
Limit, calculation of by differentiation      257 et seq.
Limit, exponential      140 et seq. 154 351 393
Limit, logarithmic      141 et seq. 352
Limit, upper and lower, of a function      180 et seq.
Limit, upper and lower, of an integral      278
Limit: of f(n), as $n \rightarrow \infty$      116 et seq.
Limits, geometrical illustrations of definitions of      119 161
Locus in a plane      31 et seq.
Locus in space      54 et seq.
Logarithm      342 et seq.; see also “Logarithmic function” “Logarithmic “Logarithmic
Logarithm of a complex number      380 et seq.
Logarithm of a complex number to any base      392
Logarithm of a negative number      384 385
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