A new method of convergence acceleration of series expansion for analytic functions in the complex domain

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Abstract

This paper proposes a new method of convergence acceleration of series expansion of complex functions which are analytic on and inside the unit circle in the complex plane. This class of complex functions may have some singularities outside the unit circle, which dominate convergence of series expansion. In the proposed method, the singular points are moved away from the origin using conformal mapping, and the function is expanded using a sequence of polynomials orthogonalized on the boundary of the mapped complex domain. The decay rate of coefficients of the orthogonal polynomial expansion can be related to the convergence region in a similar form to the Cauchy–Hadamard formula for power series. Using this relation, we quantitatively evaluate and maximize the convergence rate of the improved series. Numerical examples demonstrate that the proposed method is effective for slow convergent series, and may converge faster than Padé approximants.

Keywords

Convergence acceleration Approximation in the complex domain Conformal mapping Orthogonal polynomials 

Mathematics Subject Classification

30B10 30E10 42C05 65B99 

Copyright information

© The JJIAM Publishing Committee and Springer Japan 2015

Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsIbaraki UniversityMitoJapan
  2. 2.School of Systems Information ScienceFuture University HakodateHakodateJapan

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