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Ramanujan's lost notebook: part IV

By: Andrews, George E.
Contributor(s): Berndt, Bruce C [Co author].
Material type: materialTypeLabelBookPublisher: New York Springer 2013Description: xvii, 439p. With index.ISBN: 9781493976270.Subject(s): Ramanujan Aiyangar, Srinivasa, 1887-1920 | Mathematicians -- India -- biography | Fourier analysis | Mathematical analysis | Number theoryDDC classification: 510 Summary: 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 analysis and 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 circle and divisor problems. https://www.springer.com/in/book/9781461440802
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Table of contents
Introduction

Double Series of Bessel Functions and the Circle and Divisor Problems

Koshliakov’s Formula and Guinand’s Formula

Theorems Featuring the Gamma Function

Hypergeometric Series

Two Partial Manuscripts on Euler’s Constant γ

Problems in Diophantine Approximation

Number Theory

Divisor Sums

Identities Related to the Riemann Zeta Function and Periodic Zeta Functions

Two Partial Unpublished Manuscripts on Sums Involving Primes

An Unpublished Manuscript of Ramanujan on Infinite Series Identities

A Partial Manuscript on Fourier and Laplace Transforms

Integral Analogues of Theta Functions and Gauss Sums

Functional Equations for Products of Mellin Transforms

A Preliminary Version of Ramanujan’s Paper “On the Product”

A Preliminary Version of Ramanujan’s Paper “On the Integral”

A Partial Manuscript Connected with Ramanujan’s Paper “Some Definite Integrals”

Miscellaneous Results in Analysis

Elementary Results

A Strange, Enigmatic Partial Manuscript

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 analysis and 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 circle and divisor problems.

https://www.springer.com/in/book/9781461440802

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