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Sokolnikoff I.S. Advanced Calculus
Sokolnikoff I.S.  Advanced Calculus









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: Advanced Calculus

: Sokolnikoff I.S.

: en

: /

:

ed2k: ed2k stats

: 1st edition

: 1939

: 510

: 12.05.2008

: | | ID
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$\Pi$-function, Gausss      247
$\psi$-function, Gausss      248 249 485 496
Abel      491
Abels, inequality      157
Abels, test      157 159
Abels, theorem      162 187
Abels, theorem for integrals      385 437
Addition      10
Admissible values of variable      40
Aggregate      25
Analytical continuation      133
Andoyer      201
Angle, solid      366
Approximations to numbers      15
Arc to chord, ratio of      295
Archimedes      9
Area of closed curve      299
Area of curved surface      325
Area, element of      312 314 326 329 352
Area, measure of      298
Area, sign of      358
Argand diagram      173
Asymptotic expansion for log $\Gamma(x)$      484
Axes, change of      329
base e      45
Bateman      143
Bell      128 328 354 371
Bernoullis numbers      239241 475
Bertrand      456 466
beta function      388 389 431 462
Binets function $\mu(x)$      481 484
Binomial series      185
Bocher      212
Bolzano Weierstrass theorem      32
Borchardt      50
bounds      26 56 61 62 99 102 211 258 310 342
Bromwich      2 151 153 157 159 161 165 169 187 201 213 226 238 395 443 456 463 473 485 496
Cantor      2
Cauchy      39 196 201 377 481
Cauchys, condensation test      155
Cauchys, first and second theorems      39
Cauchys, integral test      403
Cauchys, test for sequences      36
Cauchys, test for series      157
Cesaro      53 165
Cesaros theorem      164 251
Change of axes      329
Change of parameters      330
Change of variables in partial derivatives      112
Chord to arc, ratio of      295
Chrystal      2 3 12 15 19 155 180 209 217 235 238 241 244
Clausen      172
Closed interval      17
Closed region      95 96
Columns      324
Complex functions of a real variable      181 397
complex numbers      173
Condensation, point of      32 95
Continuity      2 13 17 41 5862 97 100102
Continuity of integral with respect to a parameter      304 415418 440
Continuity, uniform      61 100
Continuum      17
convergence of integrals      376386 408414
Convergence of integrals, absolute      378 411
Convergence of integrals, tests for      381 382 385 386 391
Convergence of integrals, uniform      411
Convergence of integrals, uniform in general      413
Convergence of products      224236
Convergence of products, absolute      227
Convergence of products, Tannerys theorem for      234
Convergence of products, tests for      226
Convergence of products, uniform      232
Convergence of series      151164 173175 403
Convergence of series of complex terms      173
Convergence of series, Abels theorem for      162 187
Convergence of series, absolute      174
Convergence of series, conditional      153
Convergence of series, Tannerys theorem for      161 175
Convergence of series, tests for      153158 403
Convergence of series, uniform      158163 187
Correspondence of number and point      1 16 25
Curve of infinite discontinuity      408
Curve on a surface      327
Curve, auxiliary      409
Curve, composite      293 299
Curve, quadrable      300
Curve, rectifiable      291 300
Curve, tangent to      295 328
Curvilinear integral      296
Darboux      235 237
Darbouxs theorem      261 311
Decimal representation      14
Dedekind      22
Dedekinds axiom      2
Dependence of functions      142146
Dependence of functions, linear      9092
Derangement of a product      230
Derangement of a series      151 166 175
derivatives      63 64 66
Derivatives of $x^n$      46
Derivatives of complex functions      181183
Derivatives of determinants      88
Derivatives of implicit functions      134
Derivatives of inverse functions      68
Derivatives, change of variables in      112120
Derivatives, higher      7884
Derivatives, partial      103120
Determinant, derivative of      88
Determinant, derivative of, Hadamards      219
Differential equations      120122
Differentials      7577 105112
Differentials, higher      76 109
Differentials, total      107
Differentiation of Lagranges expansion      204
Differentiation of products      233
Differentiation under the integral sign      307 426 451
Digamma function      248
Dini      473
Dirichlet      492
Dirichlets, formulae      334
Dirichlets, test      158 160 187
Dirichlets, test for integrals      386 438
Dirichlets, theorem on derangement of series      151
Discontinuity      62
Discontinuity of integrand      273 303 315
Discontinuity of the first or second kind      63
Discontinuity, admissible      273 317
Discontinuity, curve of infinite      408 412
Discontinuity, line of      303
Discontinuity, normal      304
Discontinuity, planes of      344
Discontinuity, point of      273
Discontinuity, point of infinite      408 412
Discontinuity, removable      62
Division      11
Division by a series      192
Division of area of integration      312
Division of interval      257
Division, consecutive      257
DAlemberts test      157
Element of sequence      25
Element of set      25
Element of surface      329
Element of volume      343 353
Elimination of functions      120122
Elliptic coordinates      371
Equality      7
Ermakoffs tests      406
Euclid      9
Eudoxus      9
Euler      177 235 242
Eulers, constant      51 246 404 478
Eulers, numbers      244
Existence theorems      131 134 139 191 192 207
Exponent of a power      18
Exponential function      42 46 175 182
Exponential function, periodicity of      177
Extension of range of definition of function      49 400
Factorisation of function of two variables      209
Fluctuation of function      263
Forms, algebraic      209213
Forsyth      119 122
Fouriers double integral      464
Frobenius      165
Frullanis integral      395
Functions of functions      66 102
Functions of several variables      94122 130147
Functions with limited variation      263 269 Beta Gamma Logarithmic Hyperbolic Hypergeometric and
Functions, arbitrary      121
Functions, bounded      29 (see also under Bounds)
Functions, complex      181186 397
Functions, continuous      56 208
Functions, dependent      9092 142146
Functions, difierentiable      63 64
Functions, discontinuous      62 63
Functions, expansion in series of      190
Functions, extension of range of definition of      49
Functions, factorisation of implicit      209
Functions, implicit      130142 206209 219 353
Functions, independent      90 142
Functions, integrable      265276 314317 344 378 379 382386
Functions, inverse      44 67 142
Functions, limits of      40
Functions, monotonic      28 263 269
Functions, oscillation of      56
Functions, region of definition of      94 95
Functions, strictly decreasing      29
Functions, strictly increasing      29
Functions, values of      41 97
Gamma function      245254 398
Gamma function, as an integral      398
Gamma function, asymptotic expansion for logarithm of      484
Gamma function, Gausss multiplication formula for      252 483 499
Gamma function, integral for logarithm of      480 487
Gamma function, integrals expressible in terms of      489495
Gamma function, minimum value of      488
Gamma function, product formula for      246
Gauss      31
Gausss, $\Pi$-function      247
Gausss, $\Psi$-function      248 249 485 496
Gausss, multiplication formula for $\Gamma(mx)$      252 483 499
Gausss, test      154
Godefroy      489
Goursat      87 131 143 201 263 264
Greens Theorem      335
Greens theorem, general theorem      360
Hadamards determinant      219
Halphen      86
Hardy      2 171 394 395 397 455 473
Heath      9
hermite      72 201 203 488
hessian      147
Hilton      213 467
Hobson      2 6 9 12 64 180 217 238 323 408 411
Hyperbolic functions      178 236 237 239
Hypergeometric function      156 251 477 478
Index of a power      18
Inequalities      19 47
Inequality      8
Infinitesimals      75 105
Integrability of a function, condition of      265 314
Integrable functions      265276 314317
Integral of complex function      181 183 397
Integral, as function of a parameter      302309 411415 426 436444 451
Integral, as measure of an area      298
Integral, curvilinear      296 308 335 359
Integral, definite      264 265 267
Integral, definite, as a function of its limits      279
Integral, definite, change of variable in      see under Transformation of integral
Integral, definite, continuity of      279
Integral, definite, continuity with respect to a parameter of      304
Integral, definite, properties of      276
Integral, differentiation of      279 307 426 451
Integral, double      310323
Integral, double, change of variables in      332 345
Integral, double, reduction to repeated integrals of      317
Integral, finite      375
Integral, Fouriers double      464
Integral, generalised      375
Integral, improper      375391
Integral, improper double      408430 440 445 463
Integral, improper double with infinite limits      445 463
Integral, improper double, absolute convergence of      411
Integral, improper double, change of variables in      430
Integral, improper double, continuity of      416 440
Integral, improper double, evaluation of      429
Integral, improper with range finite      376
Integral, improper with range infinite      378 436
Integral, improper, Abels theorem for      385
Integral, improper, absolute convergence of      378
Integral, improper, as function of its limits      384
Integral, improper, as limit of a sum      401
Integral, improper, change of values of integrand in      380
Integral, improper, change of variable in      380 388 428
Integral, improper, comparison test for      391
Integral, improper, continuity with respect to a parameter of      415 440
Integral, improper, convergence at infinity of      381
Integral, improper, convergence of      376
Integral, improper, convergence- at a singular point of      381
Integral, improper, differentiation of      426 451
Integral, improper, Dirichlets theorem for      386
Integral, improper, limit with respect to a parameter of      441 443 444
Integral, improper, mean value theorems for      387
Integral, improper, oscillating      378
Integral, improper, principal values of      377 378
Integral, improper, singular integrals at singular points of      380
Integral, improper, singular points of      375
Integral, improper, tests for convergence of      381 382
Integral, improper, transformation to proper integral of      380
Integral, improper, uniform convergence in general of      413 436
Integral, improper, uniform convergence of      411 436
Integral, indefinite      280
Integral, infinite      375
Integral, integration of      308
Integral, multiple      342
Integral, multiple, improper      430
Integral, multiple, standard form for      349
Integral, proper      375
Integral, proper double      408
Integral, quadruple      345
Integral, repeated      317323 344 416426 445449 458463
Integral, repeated with one limit infinite      445
Integral, repeated with two limits infinite      458
Integral, surface      355 358 360
Integral, triple      343
Integral, volume      343 353 360
Integrals, upper and lower      265 312 343
Integration by parts      284
Integration of series      468475
Integration, change of order of      308 318 418426
Integration, Riemanns theory of      264
interval      17
Interval, closed      17
Interval, division of      257
Interval, method of the decreasing      32
Interval, open      17
inversion      44 142 179
jacobi      81
Jacobian      108 136147 350354
JORDAN      217 291 292
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