05004nam a22005895i 4500 978-3-031-65133-5 DE-He213 20251030113126.0 cr nn 008mamaa 241004s2024 sz | s |||| 0|eng d 9783031651335 978-3-031-65133-5 10.1007/978-3-031-65133-5 doi QA404.7-405 PBK bicssc MAT033000 bisacsh PBK thema Saff, Edward B. author. (orcid)0000-0003-1079-9074 https://orcid.org/0000-0003-1079-9074 aut http://id.loc.gov/vocabulary/relators/aut 85601 Logarithmic Potentials with External Fields [electronic resource] / by Edward B. Saff, Vilmos Totik. 2nd ed. 2024. Cham : Springer International Publishing : Imprint: Springer, 2024. XVI, 594 p. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics, 2196-9701 ; 316 Part 1 Fundamentals. I Weighted Potentials -- II Recovery of Measures, Green Functions and Balayage -- III Weighted Polynomials -- IV Determination of the Extremal Measure -- Part 2 Applications and Generalizations -- V Extremal Point Methods -- VI Weights on the Real Line -- VII Applications Concerning Orthogonal Polynomials -- VIII Signed Measures -- IX Some Problems from Physics -- X Generalizations -- Part 3 Appendices -- A.I Basic Tools -- A.II The Dirichlet Problem and Harmonic Measures -- A.III Weighted approximation in ℂᴺ -- A.IV Classical Logarithmic Potential Theory. This is the second edition of an influential monograph on logarithmic potentials with external fields, incorporating some of the numerous advancements made since the initial publication. As the title implies, the book expands the classical theory of logarithmic potentials to encompass scenarios involving an external field. This external field manifests as a weight function in problems dealing with energy minimization and its associated equilibria. These weighted energies arise in diverse applications such as the study of electrostatics problems, orthogonal polynomials, approximation by polynomials and rational functions, as well as tools for analyzing the asymptotic behavior of eigenvalues for random matrices, all of which are explored in the book. The theory delves into diverse properties of the extremal measure and its logarithmic potentials, paving the way for various numerical methods. This new, updated edition has been thoroughly revised and is reorganized into three parts, Fundamentals, Applications and Generalizations, followed by the Appendices. Additions to the new edition include: new material on the following topics: analytic and C² weights, differential and integral formulae for equilibrium measures, constrained energy problems, vector equilibrium problems, and a probabilistic approach to balayage and harmonic measures; a new chapter entitled Classical Logarithmic Potential Theory, which conveniently summarizes the main results for logarithmic potentials without external fields; several new proofs and sharpened forms of some main theorems; expanded bibliographic and historical notes with dozens of additional references. Aimed at researchers and students studying extremal problems and their applications, particularly those arising from minimizing specific integrals in the presence of an external field, this book assumes a firm grasp of fundamental real and complex analysis. It meticulously develops classical logarithmic potential theory alongside the more comprehensive weighted theory. Potential theory (Mathematics). 72223 Approximation theory. 74825 Fourier analysis. Functions of complex variables. 72222 Numerical analysis. Potential Theory. 72226 Approximations and Expansions. 74827 Fourier Analysis. Functions of a Complex Variable. 72224 Numerical Analysis. Totik, Vilmos. author. aut http://id.loc.gov/vocabulary/relators/aut 85602 SpringerLink (Online service) Springer Nature eBook Printed edition: 9783031651328 Printed edition: 9783031651342 Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics, 2196-9701 ; 316 80074 https://doi.org/10.1007/978-3-031-65133-5 ZDB-2-SMA ZDB-2-SXMS 63991 63991