Ramanujan summation of divergent series

Description

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. Front Matter ....Pages i-xxiii Ramanujan Summation (Bernard Candelpergher)....Pages 1-29 Properties of the Ramanujan Summation (Bernard Candelpergher)....Pages 31-60 Dependence on a Parameter (Bernard Candelpergher)....Pages 61-111 Transformation Formulas (Bernard Candelpergher)....Pages 113-155 An Algebraic View on the Summation of Series (Bernard Candelpergher)....Pages 157-173 Back Matter ....Pages 175-195