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Holomorphic Functions (holomorphic + function)

Distribution by Scientific Domains
Mathematics and Statistics75%

Selected Abstracts

Sum rules and exact relations for quantal Coulomb systems

CONTRIBUTIONS TO PLASMA PHYSICS, Issue 5-6 2003
V.M. Adamyan
Abstract A complex response function describing a reaction of a multi-particle system to a weak alternating external field is the boundary value of a Nevanlinna class function (i.e. a holomorphic function with non-negative imaginary part in the upper half-plane). Attempts of direct calculations of response functions based on standard approximations of the kinetic theory for real Coulomb condensed systems often result in considerable discrepancies with experiments and computer simulations. At the same time a relatively simple approach using only the exact values of leading asymptotic terms of the response function permits to restrict essentially a subset of Nevanlinna class functions containing this response function, and in this way to obtain sufficient data to explain and predict experimental results. Mathematical details of this approach are demonstrated on an example with the response function being the (external) dynamic electrical conductivity of cold dense hydrogen-like plasmas. In particular, the exact values of the leading terms of asymptotic expansions of the conductivity are calculated. (© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

A solution method for the linear Chandrasekhar equation

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2006
Elias Wegert
Abstract The paper is devoted to the linearized H -equation of Chandrasekhar and Ambarzumyan. A Stieltjes-type transform reduces the equation to a boundary value problem for holomorphic functions in the upper half-plane which is solved in closed form. Additional conditions ensure that the solutions , extend holomorphically to the lower half-plane slit along a straight line segment. The solutions of the original problem are then determined from the boundary values of , on this slit. The approach gives necessary and sufficient conditions for Fredholmness and describes all Fredholm parameters in terms of zeros of two functions 1,K and G associated with the kernel and the right-hand side of the equation. Explicit formulas for the complete set of solutions are presented. Die Arbeit ist der linearisierten H -Gleichung von Chandrasekhar und Ambarzumyan gewidmet. Die Gleichung wird mit Hilfe einer modifizierten Stieltjes-Transformation auf ein Randwertproblem für holomorphe Funktionen in der oberen Halbebene zurückgeführt das in geschlossener Form gelöst wird. Unter zusätzlichen Bedingungen können die Lösungen holomorph in die längs einer Strecke aufgeschnittene untere Halbebene fortgesetzt werden. Die Lösungen , des Ausgangsproblems werden dann aus den Randwerten von , längs des Schlitzes bestimmt. Der Zugang liefert notwendige und hinreichende Bedingungen dafür, dass der Operator Fredholmsch ist und charakterisiert alle Fredholmparameter mit Hilfe der Nullstellen zweier Funktionen 1,K und G, die dem Kern und der rechten Seite der Gleichung zugeordnet sind. Es werden explizite Darstellungen für die vollständige Lösungsmenge angegeben. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Nonlinear Riemann,Hilbert problems with circular target curves

MATHEMATISCHE NACHRICHTEN, Issue 9 2008
Christer Glader
Abstract The paper gives a systematic and self-contained treatment of the nonlinear Riemann,Hilbert problem with circular target curves |w , c | = r, sometimes also called the generalized modulus problem. We assume that c and r are Hölder continuous functions on the unit circle and describe the complete set of solutions w in the disk algebra H, , C and in the Hardy space H, of bounded holomorphic functions. The approach is based on the interplay with the Nehari problem of best approximation by bounded holomorphic functions. It is shown that the considered problems fall into three classes (regular, singular, and void) and we give criteria which allow to classify a given problem. For regular problems the target manifold is covered by the traces of solutions with winding number zero in a schlicht manner. Counterexamples demonstrate that this need not be so if the boundary condition is merely continuous. Paying special attention to constructive aspects of the matter we show how the Nevanlinna parametrization of the full solution set can be obtained from one particular solution of arbitrary winding number. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

On a class of holomorphic functions representable by Carleman formulas in the disk from their values on the arc of the circle

MATHEMATISCHE NACHRICHTEN, Issue 1-2 2007
Lev Aizenberg
Abstract Let D be a unit disk andM be an open arc of the unit circle whose Lebesgue measure satisfies 0 < l (M) < 2,. Our first result characterizes the restriction of the holomorphic functions f , ,(D), which are in the Hardy class ,1 near the arcM and are denoted by ,, ,1M(,,), to the open arcM. This result is a direct consequence of the complete description of the space of holomorphic functions in the unit disk which are represented by the Carleman formulas on the open arc M. As an application of the above characterization, we present an extension theorem for a function f , L1(M) from any symmetric sub-arc L , M of the unit circle, such that , M, to a function f , ,, ,1L(,,). (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]