The convergence of Padé approximants to functions with branch points
TLDR
In this paper, the convergence of Pade approximants is studied under two types of assumptions: in the first case the function f to be approximated has to have all its singularities in a compact set E ⊆ C of capacity zero (the function may be multi-valued in C \ E ), and in the second case f has to be analytic in a domain possessing a certain symmetry property (this notion is defined and discussed below).About:
This article is published in Journal of Approximation Theory.The article was published on 1997-11-01 and is currently open access. It has received 212 citations till now. The article focuses on the topics: Rate of convergence & Function (mathematics).read more
Citations
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Proceedings ArticleDOI
The Holomorphic Embedding Load Flow method
TL;DR: This paper reviews the embedded load flow method and highlights the technological breakthroughs that it enables: reliable real-time applications based on unsupervised exploratory load flows, such as Contingency Analysis, OPF, Limit-Violations solvers, and Restoration plan builders.
Journal ArticleDOI
The Holomorphic Embedding Method Applied to the Power-Flow Problem
TL;DR: The Holomorphic Embedding Load-Flow Method (HELM) as discussed by the authors solves the power-flow problem to obtain the bus voltages as rational approximants, that is, a ratio of complex-valued polynomials of the embedding parameter.
Journal ArticleDOI
Robust Padé Approximation via SVD
TL;DR: The success of this algorithm suggests that there might be variants of Pade approximation that are pointwise convergent as the degrees of the numerator and denominator increase to $\infty$, unlike traditional Pade approximants, which converge only in measure or capacity.
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The Holomorphic Embedding Loadflow Method for DC Power Systems and Nonlinear DC Circuits
Antonio Trias,J. L. Marín +1 more
TL;DR: The Holomorphic Embedding Loadflow Method is shown to extend naturally to DC power transmission systems, preserving all the constructive and deterministic properties that allow it to obtain the white branch solution in an unequivocal way.
Journal ArticleDOI
Padé approximants, continued fractions, and orthogonal polynomials
Alexander Ivanovich Aptekarev,Viktor Ivanovich Buslaev,Andrei Martínez-Finkelshtein,Sergey Pavlovich Suetin +3 more
TL;DR: A survey of results constituting the foundations of the modern convergence theory of Pade approximants can be found in this paper, where the authors present a collection of 204 titles.
References
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Book
Foundations of Modern Potential Theory
TL;DR: In this paper, the authors define the notion of potentials and their basic properties, including the capacity and capacity of a compact set, the properties of a set of irregular points, and the stability of the Dirichlet problem.
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Asymptotics of diagonal Hermite-Padé polynomials
TL;DR: In this article, a relation etroite entre polynomes orthogonaux and polynomials utilised dans the definition of Pade is discussed, and a conjecture which generalizes the result of Beinstein et Szego is presented.
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Orthogonal polynomials with complex-valued weight function, II
TL;DR: In this paper, the authors studied the asymptotic behavior of polynomials with degree ≥ n satisfying the orthogonal relation and proved that ωm+n(z) is a polynomial of degreem+n+1 with all its zeros contained inV, and the path of integrationC separatesV from the setE.
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Extremal domains associated with an analytic function I
TL;DR: In this article, it was shown that there exist extremal domains D 0 with minimal condenser capacity, and that these domains are uniquely determined up to a boundary set of capacity zero.