Ramanujan’s Notebooks, Partie 2Springer Science & Business Media, 18 janv. 1999 - 360 pages During the years 1903-1914, Ramanujan recorded many of his mathematical discoveries in notebooks without providing proofs. Although many of his results were already in the literature, more were not. Almost a decade after Ramanujan's death in 1920, G.N. Watson and B.M. Wilson began to edit his notebooks but 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 second of four volumes devoted to the editing of Ramanujan's Notebooks. Part I, published in 1985, contains an account of Chapters 1-9 in the second notebook as well as a description of Ramanujan's quarterly reports. In this volume, we examine Chapters 10-15 in Ramanujan's second notebook. If a result is known, we provide references in the literature where proofs may be found; if a result is not known, we attempt to prove it. Not only are the results fascinating, but, for the most part, Ramanujan's methods remain a mystery. Much work still needs to be done. We hope readers will strive to discover Ramanujan's thoughts and further develop his beautiful ideas. |
Table des matières
Hypergeometric Series I | 7 |
Hypergeometric Series II | 48 |
Continued Fractions | 103 |
Integrals and Asymptotic Expansions | 185 |
Infinite Series | 240 |
Asymptotic Expansions and Modular Forms | 300 |
| 339 | |
| 355 | |
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Expressions et termes fréquents
a₁ Apply Entry aẞ asymptotic expansion asymptotic formula b₁ b₂ c₁ calculations Chapter 14 coefficient complete the proof complex numbers continued fraction converges corollary of Entry cosh deduce defined desired result easily Entry 14 Entry 21 Erdélyi's established Euler Example F₁ follows gamma functions given Gradshteyn and Ryzhik Hence hypergeometric series Indian Mathematical induction inversion formula left side Lemma letter to Hardy letting N tend modular equation nonnegative integer notation obtain P₁ paper Perron's Poisson summation formula polynomial positive integer power series proof of Entry prove Entry R. J. Evans Ramanujan replace residue theorem right side S₁ SECOND PROOF Section simple poles simplification Stirling's formula Suppose tends to oo Theorem 6.1 transformation Watson Σ Σ
Fréquemment cités
Page 1 - Whichever way we turn in trying to understand things in their reality, if we analyse far enough, we find that at last we come to a peculiar state of things, seemingly a contradiction: something which our reason cannot grasp and yet is a fact. We take up something—we know it is finite ; but as soon as we begin to...
Page 1 - In a certain sense, mathematics has been advanced most by those who are distinguished more for intuition than for rigorous methods of proof.
Page 6 - This book could not have been completed without the support of the Vaughn Foundation.