Ebook: Complex Analysis in One Variable

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.

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Content:
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

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Content:
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
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