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Ramanujan Summation of Divergent Series

  • Book
  • © 2017

Overview

Authors:
  1. Bernard Candelpergher

    1. Laboratoire J.A. Dieudonné. CNRS, Université de Nice, Côte d’Azur, Nice, France

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  • Provides a clear and rigorous exposition of Ramanujan's theory of divergent series
  • A special chapter is devoted to an algebraic formalism unifying the most important summation processes
  • Only little basic knowledge in analysis is required to read this monograph

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2185)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-xxiii
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  2. Ramanujan Summation

    • Bernard Candelpergher
    Pages 1-29

  3. Properties of the Ramanujan Summation

    • Bernard Candelpergher
    Pages 31-60

  4. Dependence on a Parameter

    • Bernard Candelpergher
    Pages 61-111

  5. Transformation Formulas

    • Bernard Candelpergher
    Pages 113-155

  6. An Algebraic View on the Summation of Series

    • Bernard Candelpergher
    Pages 157-173

  7. Back Matter

    Pages 175-195
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About this book

The aim of this monograph is to give a detailed exposition of the summation method that Ramanujan uses in Chapter VI of his second Notebook. This method, presented by Ramanujan as an application of the Euler-MacLaurin formula, is here extended using a difference equation in a space of analytic functions. This provides simple proofs of theorems on the summation of some divergent series. Several examples and applications are given. For numerical evaluation, a formula in terms of convergent series is provided by the use of Newton interpolation. The relation with other summation processes such as those of Borel and Euler is also studied. Finally, in the last chapter, a purely algebraic theory is developed that unifies all these summation processes. This monograph is aimed at graduate students and researchers who have a basic knowledge of analytic function theory.

Authors and Affiliations

  • Laboratoire J.A. Dieudonné. CNRS, Université de Nice, Côte d’Azur, Nice, France

    Bernard Candelpergher

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