This text provides a masterful and systematic treatment of all the basic analytic and geometric aspects of Bergman's classic theory of the kernel and its invariance properties. These include calculation, invariance properties, boundary asymptotics, and asymptotic expansion of the Bergman kernel and metric. Moreover, it presents a unique compendium of results with applications to function theory, geometry, partial differential equations, and interpretations in the language of functional analysis, with emphasis on the several complex variables context. Several of these topics appear here for the first time in book form. Each chapter includes illustrative examples and a collection of exercises which will be of interest to both graduate students and experienced mathematicians.
Graduate students who have taken courses in complex variables
and have a basic background in real and functional analysis will find this textbook appealing. Applicable courses for either main or supplementary usage include those in complex variables, several complex variables, complex differential geometry, and partial differential equations. Researchers in complex analysis, harmonic analysis, PDEs, and complex differential geometry will also benefit from the thorough treatment of the many exciting aspects of Bergman's theory.

Preface.- 1. Introductory Ideas.- 2. The Bergman Metric.- 3. Geometric and Analytic Ideas.- 4. Partial Differential Equations.- 5. Further Geometric Explorations.- 6. Additional Analytic Topics.- 7. Curvature of the Bergman Metric.- 8. Concluding Remarks.- Table of Notation.- Bibliography.- Index.

Steven G. Krantz is one of Springer's and Birkhäuser's most prolific and popular authors in the field of functional analysis, geometric analysis, and partial differential equations. Krantz is the series editor of Birkhäuser's ' Cornerstones' graduate text series and the founder and editor-in-chief of "The Journal of Geometric Analysis", considered a society journal previously published by the AMS and often acts as an advisor to several senior editors at Springer/ Birkhäuser . He is also editor-in-chief of the "Journal of Mathematical Analysis and Applications". Professor Krantz is currently the editor-in-chief of the AMS Notices and also edits for "The American Mathematical Monthly", "Complex Analysis and Elliptical Equations", and "The Bulletin of the American Mathematical Monthly". Krantz is also known for his wide breadth of expertise in several areas of mathematics such as harmonica analysis, differential geometry, and Lie groups, to name a few. Notable awards include Chauvenet Prize (1992), Beckenbach Book Award (1994), Kemper Prize (1994), Outstanding Academic Book Award (1998), Washington University Faculty Mentor Award (2007).