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Bruce C.Berndt Ч Ramanujan's Notebooks (part 4)
Bruce C.Berndt Ч Ramanujan's Notebooks (part 4)

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Ќазвание: Ramanujan's Notebooks (part 4)

јвтор: Bruce C.Berndt

јннотаци€:

During the years 1903-1914, Ramanujan worked in almost complete isolation in India. During this time, he recorded most of his mathematical discoveries without proofs in notebooks. Although many of his results were already found in the literature, most were not. Almost a decade after Ramanujan's death in 1920, G.N. Watson and B.M. Wilson began to edit Ramanujan's notebooks, but they never completed the task. A photostat edition, with no editing, was published by the Tata Institute of Fundamental Research in Bombay in 1957. This book is the fourth of five volumes devoted to the editing of Ramanujan's notebooks. Parts I, II, and III, published in 1985, 1989, and 1991, contain accounts of Chapters 1-21 in Ramanujan's second notebook as well as a description of his quarterly reports. This is the first of two volumes devoted to proving the results found in the unorganized portions of the second notebook and in the third notebook. The author also proves those results in the first notebook that are not found in the second or third notebooks. For those results that are known, references in the literature are provided. Otherwise, complete proofs are given. Over 1/2 of the results in the notebooks are new. Many of them are so startling and different that there are no results akin to them in the literature.


язык: en

–убрика: ћатематика/јнализ/ѕродвинутый анализ/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1994

 оличество страниц: 451

ƒобавлена в каталог: 30.10.2004

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ѕредметный указатель
Adiga, C.      93
Ahlfors, L.V.      390
Aiyar, S.N.      112
American Mathematical Monthly      312
Andree, R.V.      54
Andrews, G.E.      3 6 74 139 155 158 170 261Ч262 265 286
Anthoniszoon, A.      49
Askey, R.A.      6 345Ч346 349
Avenhaus, R.      285
Ayoub, R.      245Ч246
Bailey pairs      139 155 158Ч163
Bailey's lemma      158
Bailey's q-analogue of Whipple's theorem      156
Bailey, W.N.      247Ч248 345Ч346
Balachandran, K.      35
Balasubramanian, G.      35
Barbette, E.      95Ч96
Basic hypergeometric series      155Ч157
Bateman, P.T.      6 75
Bays, C.      136
Beckmann, P.      49
Bell numbers and polynomials      402
Bernoulli numbers      53 75Ч76 81Ч85 124 127 262 273 296 403
Bernoulli numbers (extended)      129 355
Bernoulli polynomials      273
Bernoulli, James      245
Bertrand's postulate      135
Bessel functions      342Ч344
Beyer, W.      285
Bhargava, S.      3 5 6 30 91Ч92 97 99Ч100 102 246 354
Biagioli, A.J.F.      239 244
Binary form      30
Binet's formula for log $\Gamma$(a)      335Ч336
Binet, J.P.M.      107
Borwein, J.M.      48Ч49 90 354
Borwein, P.B.      49 90 354
Bradley, D.      3 6 334 346
Brams, S.J.      285
Brauer, A.      135
Brent, R.P.      134
Buckholtz, T.J.      76
Bundy, M.      285
Carr, G.S.      135
Castellanos, D.      49
Cauchy, A.      262
Cayley, A.      39
Chan, H.H.      176
Chandrasekharan, K.      249Ч250 259Ч260
Chebyshev, P.L.      136
Chellali, M      85
Chen, M.P.      272
Chudnovsky, D.V.      354
Chudnovsky, G.V.      354
Cipra, B.      48
Cohn, H.      244
Composition of functions      384Ч389
Continued fractions      412Ч417
Copson, E.T.      293 322
Cox, D.A.      71
Davenport, H.      310
de Barra, G.      48
de Bruijn, N.G.      276
Dedekind eta-function      139 175Ч184 237Ч244 252
Dedekind eta-function multiplier system      179
Delange, H.      61 396
Dharmarajan, T.      8
Diamond, H.G.      6 64
Dickson, L.E.      12 70 95Ч96 107Ч108
Dieter, U.      93
Dilogarithm      262 265 283 285 287Ч288 322Ч329
Diophantus      108
Divisor function      57Ч58 70 74 78Ч79 87
Dougall's theorem      350Ч351
Eisenstcin series      200Ч202 249 259Ч260 351Ч352
Elkies, N.D.      108
Elliptic integral      245Ч260 420Ч421
Elliptic integral of the first kind      200 246Ч247 256 259
Elliptic integral of the second kind      247
Equations cubic      22Ч23 32 37Ч38 40Ч41
Equations quartic      31Ч32
Equations quintic      38Ч39
Equations systems      10Ч22 27Ч32 42Ч48
Erdos, P.      53
Eta-function identities      186Ч187 192 203Ч244
Euler numbers      403
Euler Ч Maclaurin summation formula      273
Euler's diophantine equation      52 54Ч56 107Ч108
Euler's second identity      265 286
Euler's transformation      247 253
Eulerian polynomials and numbers      403
Evans, R.J.      3 6 124 139 174Ч175 177Ч179 182 286 288 291 312 318 327
Fagnano, G.      245
Fermat, P.      70
Ferrari, F.      96Ч97
Fichtenholz, G.M.      306
Fichtner, J.      285
Fine, N.J.      313
Ford, D.      244
Ford, K.      6 100
Fourier series of      247
Frullani's theorem      316 319
Galway, W.      6 113
Gamma function      305Ч306 322 330 334Ч342 374Ч377
Garvan, F.G.      354
Gauss's theorem      345
Gelfond, A.O.      91
Geometry      8 9
George, A.      69
Gerardin, A.      96
Girard, A.      70
Gleason, A.M.      78
Gloden, A.      107
Gordon, B.      244
Gosper, R.W.      6 19 354
Gram, J.P.      124Ч125 127 129 135
Gray son, D.      6 19
Greatest integer function      91Ч94
Gronwall, T.H.      94
Grosjean, C.C.      313
Grosswald, E.      75
Guy, R.K.      73
Hafner.J.L.      75
Halberstam, H.      6
Haldeman, C.B.      95Ч96 101 107
Halter-Koch, F.      71
Hansen, E.R.      74
Hardy, G.H.      2 3 51Ч52 56Ч58 60Ч61 108 111 118Ч124 129 132 136Ч137 288 291 378 383
Hardy-Ramanujan asymptotic formula for p(n)      3
Hart, D.S.      95Ч96
Hausman, M.      67
Haussner, R.      85
Hecke, E.      244
Heine's theorem      262 12
Heine's transformation      263
Henrici, P.      351
Hermite, C.      90
Heupel, W.      62
Highly composite numbers      52Ч54 79 86Ч87
Hildebrand, A.      6
Honsberger, R.      54
Hooley, C.      56
Hudson, R.H.      135Ч136
Hurwitz zeta-function      412
Hypergeometric functions      193Ч194 247 259Ч260 344Ч354 404Ч405 407Ч412 417Ч422
Hypergeometric functions, balanced      345Ч348
Ikehara, S.      396
Illegitimate convergent series      395
Illegitimate divergent series      395
Integrals asymptotic expansions of      291Ч294
Invariant theory      31
Inversion formulas for elliptic integrals      245Ч260
Ismail, M.E.H.      346
Ivic, A.      58
Iwaniec, H.      62
Jackson's theorem      156
Jacobi, C.G.J.      155
Jacobian elliptic functions      246Ч248 250 253Ч254 256Ч257
Jacobsen, L.      261
James, R.D.      62
Journal of the Indian mathematical society      2 39 42 56 78 112 288 298 327 341
Keiper, J.      6
Kilgour, M.      285
Klein's absolute invariant      90Ч91
Knopp, M.I.      244
Knuth, D.E.      76
Kohler, G,      244
Kratzel, E.      58 75 94
Kronecker, L.      90
Kung, J.P.S.      31
L-function      309Ч310
Lagarias, J.C.      135
Lagrange inversion      402
Lagrange, J.L.      70
Lambert series      152Ч171 197Ч202
Lamphere, R.L.      261
Landau, E.      52Ч53 61Ч62 73Ч75 94 127Ч128
Landau, S.      48
Lander, L.J.      108 134
Laplace transform      303 322 330 332 344
Laplace's method      311Ч312
Lattice points      52Ч53 58 66Ч69 73Ч75 93Ч94
Lawden, D.F.      251
Legendre, A.M.      71Ч72
Legitimate convergent series      395
Legitimate divergent series      395
Lehmer, D.H.      83Ч85
Lehmer, D.N.      135
Lemniscate integral      245Ч255
Lemniscate integral, doubling the arc length of      246
Letters of Ramanujan to Hardy      2 4 51Ч52 66 111Ч112 118Ч119 121Ч124 131Ч132 136Ч137 291
Lewin, L.      285 329
Littlewood, J.E.      67 118 136 396
Lost notebook      53 118Ч119 121Ч122 261 269 272Ч273 284
Lou, S.      135
MacMahon, P.A.      159 161
Macsyma      18 124
Magic squares      401Ч402
Maple      284
Martin, A.      95Ч96 101 107
Master theorem      313
Mathematica      7 15 17Ч18 43 47 74 89 102 113 131Ч132 140 227 229Ч230 240 244 413Ч417
McKay, J.      244
Mclntosh, R.      3 6 262 268Ч269 272 284Ч285
Meinardus, G.      269Ч270 272
Meissel Ч Lehmer method      61
Meissel, E.D.F.      134Ч135
Mellin transforms      330Ч332
Mermin, N.D.      48
Mick, D.K.      6 45
Miller, V.S.      135
Mittag Ч Leffler theorem      355 367Ч368 390
Modular equations      139Ч140 186Ч187 193Ч198 201Ч244
Modular equations, P-Q      203Ч244
Modular forms      174Ч184 237Ч244
Myerson, G.      6
Nagell, T.      71Ч72
Nanjundiah, T.S.      102
Narkiewicz, W.      396
Nested radicals      14Ч20 44Ч48
Newman, M.      6 239 244
Nicolas, J.-L.      53 94
Nielsen, N.      124 127
Norfolk, T.S.      312
Nuclear war      285Ч286
Numerical calculations      41 48Ч49
Odlyzko, A.M.      6 135
Olver, F.W.J.      125Ч126 273 293 11Ч312
Order of a function      385
Parkin, T.R.      108
Partial fraction expansions      111 300Ч301 355Ч381 390
Partition function      108 176
Patrick, M.L.      74
Pi, approximations to      48Ч49 87Ч91
Pi, series for      352Ч354
Pillai, S.S.      69
Poisson summation formula      294 298Ч299 427
Poisson summation formula for Fourier sine transforms      295
Pole      63
Potler, A.      134
Preece, C.T.      291
prime number theory      58Ч59 111Ч137
Putnam Exam      78
q-series      261Ч286
q-series, asymptotic formulas      265Ч286
q-series, identities      262Ч265
Quadratic forms      70Ч73 86
Quarterly Reports      313 316 389
Quintuple product identity      188 426
Ramanathan, K.G.      6
Ramanujaha      27
Ramanujam, S.      27
Ramanujan's $\psi_1$ summation      152Ч153
Ramanujan's formula for $\zeta$(2n + 1)      432
Rankin, R.A.      3 6 60 90 118Ч119 122 132 239 249 259
Rearrangement of series      393Ч394
Reznick, B.      6 34 92 107
Ribenboim, P.      137
Richert, H.-E.      75
Riemann hypothesis      94
Riemann zeta-function      111 115Ч116 127 129 355Ч358 377Ч380 403 432
Riemann Ч Lebesgue lemma      288 302 316
Riemann Ч Stieltjes integrals      51 112
Riemann's series for $\pi$(x)      117 123Ч124 127 129 131 136
Robin, G.      53 94
Rogers Ч Ramanujan identity      261 284Ч285
Rola, G.-C.      31
Ruderman, H.      73
Rummer's theorem      419
Schierwagen, A.      75
Schmid, L.P.      61
Schoeneberg, B.      239 242Ч243
Schwarz, W.      61 396
Schwering, K.      107
Sears, D.B.      156
Selfridge, J.L.      6 73 108
Seven interval formula      81
Shanks, D.      6 61 90 136
Shapiro, H.N.      67
Shiu, P.      61
Siegel, C.L.      245Ч246
Silverman, J.H.      90
Sinor, D.      244
Six interval formula      82Ч85
Slater, L.J.      156
Smith, H.J.S.      90
Soldner      124
Somashekara, D.D.      93
Somasundaram, D.      9 49 79 81 334 399
Squares in algebraic number fields      35
Srinivasan, P.K.      8
Stanley, G.K.      61
Stark, H.M.      244
Steelman, J.H.      312
Steinig, J.      6
Stirling's formula      268 276 339 422
Sum of divisors function      94 108
Sums of powers      94Ч108
Sums of squares      52 60Ч66 70Ч73
Sylvester, J.J.      30Ч31
Systems of equations      See Ђequations systemsї
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