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Titlebook: Ramanujan‘s Lost Notebook; Part IV George E. Andrews,Bruce C. Berndt Book 2013 Springer Science+Business Media New York 2013 Bessel Functio

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书目名称Ramanujan‘s Lost Notebook
副标题Part IV
编辑George E. Andrews,Bruce C. Berndt
视频video
概述Fourth volume of a series of five volumes including some of Ramanujan‘s deepest work in the last year of his life.Contains material of which mathematicians currently lack a complete understanding.Focu
图书封面Titlebook: Ramanujan‘s Lost Notebook; Part IV George E. Andrews,Bruce C. Berndt Book 2013 Springer Science+Business Media New York 2013 Bessel Functio
描述.​​​​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..This volume is the fourth of five volumes that the authors plan to write on Ramanujan’s lost notebook.​ In contrast to the first three books on Ramanujan‘s Lost Notebook, the fourth book does not focus on q-series.  Most of the entries examined in this volume fall under the purviews of number theory and classical analysis.  Several incomplete manuscripts of Ramanujan published by Narosa with the lost notebook are discussed.  Three of the partial manuscripts are on diophantine approximation, and others are in classical Fourier analysisand prime number theory.   Most of the entries in number theory fall under the umbrella of classical analytic number theory.   Perhaps the most intriguing entries are connected with the classical, unsolved cir
出版日期Book 2013
关键词Bessel Functions; Euler‘s constant; Fourier transforms; Guinand‘s Formula; Koshliakov‘s Formula; Mellin t
版次1
doihttps://doi.org/10.1007/978-1-4614-4081-9
isbn_softcover978-1-4899-9175-1
isbn_ebook978-1-4614-4081-9
copyrightSpringer Science+Business Media New York 2013
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Problems in Diophantine Approximation,related to Ramanujan’s only publication in the subject, a problem that he published in the .. In this partial manuscript, Ramanujan provides the best Diophantine approximation to e, approximately 60 years before the theorem was rediscovered and proved in print.
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A Partial Manuscript on Fourier and Laplace Transforms, one of the highlights of the book, a beautiful new transformation formula involving the logarithmic derivative of the gamma function. An extremely clever device used to prove this transformation formula harkens back to Ramanujan’s paper, ..
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Integral Analogues of Theta Functions and Gauss Sums,sfies a transformation formula similar to that satisfied by the classical theta functions. The integral can also be thought of as an analogue of Gauss sums or as an analogue of the classical Weierstrass σ-function.
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,Koshliakov’s Formula and Guinand’s Formula,In this chapter, we relate two well-known identities, with the names of N.S. Koshliakov and A.P. Guinand attached to them, which were proved by Ramanujan and recorded in his lost notebook before their discoveries by the aforementioned mathematicians. Ramanujan also derived some related formulas that have not been rediscovered by others.
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Theorems Featuring the Gamma Function,Some integrals of gamma functions are evaluated. A remarkable approximation to the gamma function, with a slightly less precise approximation submitted as a problem by Ramanujan to the ., is examined in detail.
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Hypergeometric Series,Two fascinating formulas for bilateral hypergeometric series are proved. The second portion of the chapter is devoted to a beautiful continued fraction related to hypergeometric polynomials.
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