Ramanujan's Lost Notebook, Part 1

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Springer Science & Business Media, May 6, 2005 - Mathematics - 441 pages
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This volume is the first of approximately four volumes devoted to providing statements, proofs, and discussions of all the claims made by Srinivasa Ramanujan in his lost notebook and all his other manuscripts and letters published with the lost notebook. In addition to the lost notebook, this publication contains copies of unpublished manuscripts in the Oxford library, in particular, his famous unpublished manuscript on the partition and tau-functions; fragments of both published and unpublished papers; miscellaneous sheets; and Ramanujan's letters to G. H. Hardy, written from nursing homes during Ramanujan's final two years in England. This volume contains accounts of 442 entries (counting multiplicities) made by Ramanujan in the aforementioned publication. The present authors have organized these claims into eighteen chapters, containing anywhere from two entries in Chapter 13 to sixty-one entries in Chapter 17.

Most of the results contained in Ramanujan's Lost Notebook fall under the purview of q-series. These include mock theta functions, theta functions, partial theta function expansions, false theta functions, identities connected with the Rogers-Fine identity, several results in the theory of partitions, Eisenstein series, modular equations, the Rogers-Ramanujan continued fraction, other q-continued fractions, asymptotic expansions of q-series and q-continued fractions, integrals of theta functions, integrals of q-products, and incomplete elliptic integrals. Other continued fractions, other integrals, infinite series identities, Dirichlet series, approximations, arithmetic functions, numerical calculations, diophantine equations, and elementary mathematics are some of the further topics examined by Ramanujan in his lost notebook.

 

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Contents

II
1
III
9
IV
13
V
18
VI
21
VII
24
VIII
26
IX
28
LIII
237
LIV
241
LV
247
LVII
248
LVIII
251
LIX
252
LX
253
LXI
256

X
33
XI
39
XII
44
XIII
57
XIV
59
XV
66
XVI
71
XVII
75
XVIII
79
XIX
85
XX
86
XXI
94
XXII
100
XXIII
107
XXIV
108
XXV
114
XXVI
116
XXVII
121
XXVIII
125
XXIX
126
XXX
133
XXXI
137
XXXII
140
XXXIII
143
XXXIV
144
XXXV
158
XXXVI
159
XXXVII
162
XXXVIII
165
XXXIX
169
XL
172
XLI
179
XLII
181
XLIII
187
XLIV
193
XLV
197
XLVI
199
XLVII
210
XLVIII
213
XLIX
214
L
223
LI
227
LII
232
LXIII
261
LXIV
262
LXV
265
LXVI
272
LXVII
279
LXVIII
284
LXIX
285
LXX
286
LXXI
288
LXXII
289
LXXIII
291
LXXIV
295
LXXV
297
LXXVII
302
LXXVIII
304
LXXIX
305
LXXX
309
LXXXI
310
LXXXII
314
LXXXIII
323
LXXXIV
327
LXXXV
328
LXXXVI
330
LXXXVII
333
LXXXVIII
339
LXXXIX
342
XC
349
XCI
356
XCII
361
XCIII
365
XCIV
367
XCV
368
XCVI
373
XCVII
375
XCVIII
384
XCIX
392
C
395
CI
396
CII
409
CIII
415
CIV
419
CV
433
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About the author (2005)

George E. Andrews is Evan Pugh Professor of Mathematics at the Pennsylvania State University. He has been a Guggenheim Fellow, the Principal Lecturer at a Conference Board for the Mathematical Sciences meeting, and a Hedrick Lecturer for the MAA. Having published extensively on the theory of partitions and related areas, he has been formally recognized for his contribution to pure mathematics by several prestigious universities and is a member of the National Academy of Sciences (USA).

University of Illinois, Urbana.

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