04788nam a22005415i 4500001001800000003000900018005001700027007001500044008004100059020003700100024003500137050001600172072001600188072002300204072001500227100015100242245010700393250001800500264007500518300003400593336002600627337002600653338003600679347002400715490011900739505059200858520203701450650004303487650003303530650002203563650004303585650002403628650002903652650004203681650002203723650004403745650002403789700008303813710003403896773002603930776003603956776003603992830012604028856004604154912001404200912001504214999001704229978-3-031-65133-5DE-He21320251030113126.0cr nn 008mamaa241004s2024    sz |    s    |||| 0|eng d  a97830316513359978-3-031-65133-57 a10.1007/978-3-031-65133-52doi 4aQA404.7-405 7aPBK2bicssc 7aMAT0330002bisacsh 7aPBK2thema1 aSaff, Edward B.eauthor.0(orcid)0000-0003-1079-90741https://orcid.org/0000-0003-1079-90744aut4http://id.loc.gov/vocabulary/relators/aut98560110aLogarithmic Potentials with External Fieldsh[electronic resource] /cby Edward B. Saff, Vilmos Totik.  a2nd ed. 2024. 1aCham :bSpringer International Publishing :bImprint: Springer,c2024.  aXVI, 594 p.bonline resource.  atextbtxt2rdacontent  acomputerbc2rdamedia  aonline resourcebcr2rdacarrier  atext filebPDF2rda1 aGrundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics,x2196-9701 ;v3160 aPart 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.  aThis 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. 0aPotential theory (Mathematics).972223 0aApproximation theory.974825 0aFourier analysis. 0aFunctions of complex variables.972222 0aNumerical analysis.14aPotential Theory.97222624aApproximations and Expansions.97482724aFourier Analysis.24aFunctions of a Complex Variable.97222424aNumerical Analysis.1 aTotik, Vilmos.eauthor.4aut4http://id.loc.gov/vocabulary/relators/aut9856022 aSpringerLink (Online service)0 tSpringer Nature eBook08iPrinted edition:z978303165132808iPrinted edition:z9783031651342 0aGrundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics,x2196-9701 ;v31698007440uhttps://doi.org/10.1007/978-3-031-65133-5  aZDB-2-SMA  aZDB-2-SXMS  c63991d63991