Ramanujan’s Notebooks, Partie 2
Springer 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.
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analytic Apply Entry assume asymptotic expansion asymptotic formula Chapter 14 complete the proof continued fraction converges corollary of Entry cosh defined desired result easily Eisenstein series employ Entry 14 Entry 21 Erdelyi's established Euler Euler's constant evaluation Example gamma functions given Gradshteyn and Ryzhik Hence hypergeometric functions hypergeometric series hypotheses identity Indian Mathematical induction Jacobsen left side Lemma letter to Hardy letting N tend Log(l modular equation multiply both sides nonnegative integer notation obtain order of summation paper Perron's Poisson summation formula polynomial positive integer power series proof of Entry proposed formula prove Entry R. J. Evans Ramanujan Ramanujan claims readily follows recursion formula replace residue theorem respectively right side second notebook Second Proof Section simple poles simplification Stirling's formula summation formula Suppose tends to oo Theorem 6.1 transformation valid Watson
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...