This Book Presents Complex Analysis In One Variable In The Context Of Modern Mathematics, With Clear Connections To Several Complex Variables, De Rham Theory, Real Analysis, And Other Branches Of Mathematics. Thus, Covering Spaces Are Used Explicitly In Dealing With Cauchy's Theorem, Real Variable Methods Are Illustrated In The Loman-menchoff Theorem And In The Corona Theorem, And The Algebraic Structure Of The Ring Of Holomorphic Functions Is Studied. Using The Unique Position Of Complex Analysis, A Field Drawing On Many Disciplines, The Book Also Illustrates Powerful Mathematical Ideas And Tools, And Requires Minimal Background Material. Cohomological Methods Are Introduced, Both In Connection With The Existence Of Primitives And In The Study Of Meromorphic Functionas On A Compact Riemann Surface. The Proof Of Picard's Theorem Given Here Illustrates The Strong Restrictions On Holomorphic Mappings Imposed By Curvature Conditions. New To This Second Edition, A Collection Of Over 100 Pages Worth Of Exercises, Problems, And Examples Gives Students An Opportunity To Consolidate Their Command Of Complex Analysis And Its Relations To Other Branches Of Mathematics, Including Advanced Calculus, Topology, And Real Applications. By Raghavan Narasimhan, Yves Nievergelt. Front Matter....Pages i-xiv Front Matter....Pages 1-1 Elementary Theory of Holomorphic Functions....Pages 3-51 Covering Spaces and the Monodromy Theorem....Pages 53-68 The Winding Number and the Residue Theorem....Pages 69-85 Picard’s Theorem....Pages 87-96 The Inhomogeneous Cauchy-Riemann Equation and Runge’s Theorem....Pages 97-114 Applications of Runge’s Theorem....Pages 115-137 The Riemann Mapping Theorem and Simple Connectedness in the Plane....Pages 139-149 Functions of Several Complex Variables....Pages 151-160 Compact Riemann Surfaces....Pages 161-185 The Corona Theorem....Pages 187-208 Subharmonic Functions and the Dirichlet Problem....Pages 209-252 Back Matter....Pages 253-253 Front Matter....Pages 255-255 Introduction....Pages 257-257 Review of Complex Numbers....Pages 259-266 Elementary Theory of Holomorphic Functions....Pages 267-295 Covering Spaces and the Monodromy Theorem....Pages 297-304 The Winding Number and the Residue Theorem....Pages 305-312 Picard’s Theorem....Pages 313-313 The Inhomogeneous Cauchy-Riemann Equation and Runge’s Theorem....Pages 315-330 Applications of Runge’s Theorem....Pages 331-335 The Riemann Mapping Theorem and Simple Connectedness in the Plane....Pages 337-342 Front Matter....Pages 255-255 Functions of Several Complex Variables....Pages 343-349 Compact Riemann Surfaces....Pages 351-359 The Corona Theorem....Pages 361-364 Subharmonic Functions and the Dirichlet Problem....Pages 365-368 Back Matter....Pages 369-381 The original edition of this book has been out of print for some years. The appear ance of the present second edition owes much to the initiative of Yves Nievergelt at Eastern Washington University, and the support of Ann Kostant, Mathematics Editor at Birkhauser. Since the book was first published, several people have remarked on the absence of exercises and expressed the opinion that the book would have been more useful had exercises been included. In 1997, Yves Nievergelt informed me that, for a decade, he had regularly taught a course at Eastern Washington based on the book, and that he had systematically compiled exercises for his course. He kindly put his work at my disposal. Thus, the present edition appears in two parts. The first is essentially just a reprint of the original edition. I have corrected the misprints of which I have become aware (including those pointed out to me by others), and have made a small number of other minor changes.
