Logarithmic Potentials with External Fields

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.

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 C?.- A.IV Classical Logarithmic Potential Theory.

Edward B. Saff received his B.S. in mathematics from the Georgia Institute of Technology and his Ph.D. from the University of Maryland, where he was a student of the renowned analyst Joseph L. Walsh. Saff’s research areas include approximation theory, numerical analysis, and potential theory. He has published more than 290 mathematical research articles, co-authored 9 books, and co-edited 11 volumes.  Recognitions of his research include his election as a SIAM Fellow (Society for Industrial and Applied Mathematics) in 2023, as a Foreign Member of the Bulgarian Academy of Sciences in 2013, as a Fellow of the American Mathematical Society in 2013, as well as a Guggenheim Fellowship in 1978. Saff is co-Editor-in-Chief and Managing Editor of the research journal Constructive Approximation and serves on the editorial boards of Computational Methods and Function Theory and the Journal of Approximation Theory. He has mentored 18 Ph.D.’s as well as 13 post-docs. Saff is currently Distinguished Professor of Mathematics at Vanderbilt University.

Vilmos Totik was educated in Hungary and was a professor of mathematics at the University of Szeged and the University of South Florida until his retirement. His main research interest is classical mathematical analysis, approximation theory, orthogonal polynomials and potential theory. He has published (partially with co-authors) 5 monographs, one problem book in set theory and about 220 research papers in various disciplines.