Ramanujan Summation of Divergent Series

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.

List of contents

Introduction: The Summation of Series.- 1 Ramanujan Summation.- 3 Properties of the Ramanujan Summation.- 3 Dependence on a Parameter.- 4 Transformation Formulas.- 5 An Algebraic View on the Summation of Series.- 6 Appendix.- 7 Bibliography.- 8 Chapter VI of the Second Ramanujan's Notebook.

Summary

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

Product details

• Authors: Bernard Candelpergher
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 01.01.2017
• EAN: 9783319636290
• ISBN: 978-3-31-963629-0
• No. of pages: 195
• Dimensions: 160 mm x 14 mm x 237 mm
• Weight: 356 g
• Illustrations: XXIII, 195 p. 7 illus.
• Series: Lecture Notes in Mathematics; Lecture Notes in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; Zahlentheorie, B, Komplexe Analysis, komplexe Variablen, Funktionentheorie, Mathematics and Statistics, Number Theory, Complex analysis, complex variables, Sequences, Series, Summability, Sequences (Mathematics), Functions of a Complex Variable, Functions of complex variables