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Generalized Euler transformation in extracting useful information from divergent (asymptotic) perturbation series and the construction of Padé approximants
Author(s) -
Bhattacharyya Kamal
Publication year - 1982
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560220210
Subject(s) - euler's formula , perturbation (astronomy) , mathematics , series (stratigraphy) , conjecture , convergent series , context (archaeology) , transformation (genetics) , convergence (economics) , parametrization (atmospheric modeling) , mathematical analysis , pure mathematics , physics , power series , quantum mechanics , paleontology , biochemistry , chemistry , radiative transfer , gene , economics , biology , economic growth
Euler transformation for accelerating convergence of a series is considered in the context of handling divergent (asymptotically convergent) perturbation series. A generalized (parametrized) version of this transformation is developed, based on the conjecture of Dalgarno and Stewart, which works better. Viewed from this standpoint, the Padé approximants follow as a special case of the parametrized Euler transformation ( PET ), as is the case with the μ transformation procedure of Feenberg in a perturbative context. The PET is shown to serve as a more general method of handling a divergent series and is able to appreciate the construction and convergence behavior of specific sequences of Padé approximants. The role of parametrization in the context of the Z −1 perturbation theory of atoms is also noted and the workability of the adopted strategy is demonstrated by choosing some specific test cases.