The superconducting order parameter and the mass renormalization function can be solved either at discrete frequencies along the imaginary axis, or as a function of continuous real frequencies. The latter is done with a method called analytic continuation. The analytic continuation can conveniently be done by approximating a power series to the functions, the Padè approximation. Studied in this project is the difference between the Padè approximation, and a formally exact analytic continuation of the functions. As it turns out, the Padè approximant is applicable to calculate the superconducting order parameter at temperatures sufficiently below the critical temperature. However close to the critical temperature the approximation fails, while the solution presented in this report remains reliable.

In this paper, the ramified Cauchy problem in C² for operator with multiple characteristics of constant multiplicity and second member ramified around some analytic set is studied.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 76-79). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 76-79). Also available on the Internet.

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abstract: With the power system being increasingly operated near its limits, there is an increasing need for a power-flow (PF) solution devoid of convergence issues. Traditional iterative methods are extremely initial-estimate dependent and not guaranteed to converge to the required solution. Holomorphic Embedding (HE) is a novel non-iterative procedure for solving the PF problem. While the theory behind a restricted version of the method is well rooted in complex analysis, holomorphic functions and algebraic curves, the practical implementation of the method requires going beyond the published details and involves numerical issues related to Taylor's series expansion, Padé approximants, convolution and solving linear matrix equations. The HE power flow was developed by a non-electrical engineer with language that is foreign to most engineers. One purpose of this document to describe the approach using electric-power engineering parlance and provide an understanding rooted in electric power concepts. This understanding of the methodology is gained by applying the approach to a two-bus dc PF problem and then gradually from moving from this simple two-bus dc PF problem to the general ac PF case. Software to implement the HE method was developed using MATLAB and numerical tests were carried out on small and medium sized systems to validate the approach. Implementation of different analytic continuation techniques is included and their relevance in applications such as evaluating the voltage solution and estimating the bifurcation point (BP) is discussed. The ability of the HE method to trace the PV curve of the system is identified. / Dissertation/Thesis / Masters Thesis Electrical Engineering 2014

Analytic functions in a domain Ω are uniquely determined by their values on any curve Γ ⊂ Ω. We provide sharp quantitative version of this statement. Namely, let f be of order E on Γ relative to its global size in Ω (measured in some Hilbert space norm). How large can f be at a point z away from the curve? We give a sharp upper bound on |f(z)| in terms of a solution of a linear integral equation of Fredholm type and demonstrate that the bound behaves like a power law: E^γ(z). In special geometries, such as the upper halfplane, annulus or ellipse the integral equation can be solved explicitly, giving exact formulas for the optimal exponent γ(z). Our methods can be applied to non-Hilbertian settings as well. Further, we apply the developed theory to study the degree of reliability of extrapolation of the complex electromagentic permittivity function based on its analyticity properties. Given two analytic functions, representing extrapolants of the same experimental data, we quantify how much they can differ at an extrapolation point outside of the experimentally accessible frequency band.

In this thesis we will consider and investigate the properties of analytic functions via their behavior near the boundary of the domain on which they are defined. To do that we introduce the notion of the hyperbolic distortion and the hyperbolic derivative. Classical results state that the hyperbolic derivative is bounded from above by 1, and we will consider the case when it is bounded from below by some positive constant. Boundedness from below of the hyperbolic derivative implies some nice properties of the function near the boundary. For instance Krauss & all in 2007 proved that, if the function is defined on a domain bounded by analytic curve, then boundedness from below of the hyperbolic derivative implies that the function has an analytic continuation across the boundary. We extend this result for the domains with slightly more general boundary, namely for smooth Jordan domains, and get that in this case the function and its derivative will have only continuous extensions to the boundary.

In this thesis we will consider and investigate the properties of analytic functions via their behavior near the boundary of the domain on which they are defined. To do that we introduce the notion of the hyperbolic distortion and the hyperbolic derivative. Classical results state that the hyperbolic derivative is bounded from above by 1, and we will consider the case when it is bounded from below by some positive constant. Boundedness from below of the hyperbolic derivative implies some nice properties of the function near the boundary. For instance Krauss & all in 2007 proved that, if the function is defined on a domain bounded by analytic curve, then boundedness from below of the hyperbolic derivative implies that the function has an analytic continuation across the boundary. We extend this result for the domains with slightly more general boundary, namely for smooth Jordan domains, and get that in this case the function and its derivative will have only continuous extensions to the boundary.

L'objet de cette thèse porte sur le problème de Cauchy singulier dans le domaine complexe. Il s'agit d'étudier les singularités de la solution du problème pour trois classes d'équations aux dérivées partielles. Cette thèse s'inscrit dans la continuité des travaux initiés par Jean Leray et son école. Pour décrire les singularités de la solution, on cherche la solution sous la forme d'un développement asymptotique de fonctions hypergéométriques de Gauss. Comme les singularités sont portées par les fonctions hypergéométriques, l'étude de la ramification de la solution se ramène à celle de ces fonctions. / This thesis deals with the singular Cauchy problem in the complex domain. We study the singularities of the solution of the problem for three classes of partial differential equations. This thesis is a continuation of the work initiated by Jean Leray and his school. To describe the singularities of the solution, we seek the solution in the form of asymptotic an expansion of Gauss hypergeometric functions. As the singularities are carried by the hypergeometric functions, the study of the ramification of the solution reduces to that of these functions.