Ramanujan's Lost Notebook, Part 1

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Springer Science & Business Media, May 6, 2005 - Mathematics - 441 pages
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In the spring of 1976, George Andrews of Pennsylvania State University visited the library at Trinity College, Cambridge, to examine the papers of the late G.N. Watson. Among these papers, Andrews discovered a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan. This manuscript was soon designated, "Ramanujan's lost notebook." Its discovery has frequently been deemed the mathematical equivalent of finding Beethoven's tenth symphony.

The "lost notebook" contains considerable material on mock theta functions and so undoubtedly emanates from the last year of Ramanujan's life. It should be emphasized that the material on mock theta functions is perhaps Ramanujan's deepest work. Mathematicians are probably several decades away from a complete understanding of those functions. More than half of the material in the book is on q-series, including mock theta functions; the remaining part deals with theta function identities, modular equations, incomplete elliptic integrals of the first kind and other integrals of theta functions, Eisenstein series, particular values of theta functions, the Rogers-Ramanujan continued fraction, other q-continued fractions, other integrals, and parts of Hecke's theory of modular forms.

 

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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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