Handbook of Complex Analysis

In spite of being nearly 500 years old, the subject of complex analysis is still today a vital and active part of mathematics. There are important applications in physics, engineering, and other aspects of technology. This Handbook presents contributed chapters by prominent mathematicians, including the new generation of researchers. More than a compilation of recent results, this book offers students an essential stepping-stone to gain an entry into the research life of complex analysis. Classes and seminars play a role in this process. More, though, is needed for further study. This Handbook will play that role. This book is also a reference and a source of inspiration for more seasoned mathematicians—both specialists in complex analysis and others who want to acquaint themselves with current modes of thought. The chapters in this volume are authored by leading experts and gifted expositors. They are carefully crafted presentations of diverse aspects of the field, formulated for a broad and diverse audience. This volume is a touchstone for current ideas in the broadly construed subject area of complex analysis. It should enrich the literature and point in some new directions.

Preface1.Something about poisson and dirichlet2.The Cauchy-Leray operator for convex domains3.Fractional linear maps and some applications. An “Augenblick” 4.Biholomorphic transformations5.Positivity in the @-Neumann Problem6.Symmetry and art 7.A glimpse into invariant distances in complex analysis8.Variations on the (eternal) theme of analytic continuation9.Complex convexity10.Reproducing kernels in complex analysis11.The Green’s function method on the Riemann mapping theorem12.Polynomial trace identities in quaternion algebras and two-generator Kleinian groups13.Boundary value problems on klein surfacesBibliographyIndex

Steven G. Krantz is a professor of mathematics at Washington University in St. Louis. He has previously taught at UCLA, Princeton University, and Pennsylvania State University. He has written more than 130 books and more than 250 scholarly papers and is the founding editor of the Journal of Geometric Analysis. An AMS Fellow, Dr. Krantz has been a recipient of the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize. He received a Ph.D. from Princeton University.