Complex Analysis in One Variable

by Narasimhan, Raghavan; Nievergelt, Yves

• ISBN13: 9780817641641
• ISBN10: 0817641645
• Edition: 2nd
• Format: Hardcover
• Copyright: 2001-01-01
• Publisher: Birkhauser

Summary

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.

Table of Contents

Preface to the Second Edition	ix
Preface to the First Edition	xi
Notation and Terminology	xiii
I Complex Analysis in One Variable	1	(254)
Raghavan Narasimhan
Elementary Theory of Holomorphic Functions	3	(50)
Some basic properties of C-differentiable and holomorphic functions	4	(6)
Integration along curves	10	(12)
Fundamental properties of holomorphic functions	22	(10)
The theorems of Weierstrass and Montel	32	(4)
Meromorphic functions	36	(7)
The Looman-Menchoff theorem	43	(10)
Covering Spaces and the Monodromy Theorem	53	(16)
Covering spaces and the lifting of curves	53	(2)
The sheaf of germs of holomorphic functions	55	(2)
Covering spaces and integration along curves	57	(3)
The monodromy theorem and the homotopy form of Cauchy's theorem	60	(3)
Applications of the monodromy theorem	63	(6)
The Winding Number and the Residue Theorem	69	(18)
The winding number	69	(4)
The residue theorem	73	(6)
Applications of the residue theorem	79	(8)
Picard's Theorem	87	(10)
Inhomogeneous Cauchy-Riemann Equation and Runge's Theorem	97	(18)
Partitions of unity	97	(2)
The equation ∂u/∂z = phis;	99	(4)
Runge's theorem	103	(8)
The homology form of Cauchy's theorem	111	(4)
Applications of Runge's Theorem	115	(24)
The Mittag-Leffler theorem	115	(4)
The cohomology form of Cauchy's theorem	119	(2)
The theorem of Weierstrass	121	(6)
Ideals in H(Ω)	127	(12)
Riemann Mapping Theorem and Simple Connectedness in the Plane	139	(12)
Analytic automorphisms of the disc and of the annulus	139	(4)
The Riemann mapping theorem	143	(2)
Simply connected plane domains	145	(6)
Functions of Several Complex Variables	151	(10)
Compact Riemann Surfaces	161	(26)
Definitions and basic theorems	161	(5)
Meromorphic functions	166	(1)
The cohomology group H1(U, O)	167	(4)
A theorem from functional analysis	171	(5)
The finiteness theorem	176	(3)
Meromorphic functions on a compact Riemann surface	179	(8)
The Corona Theorem	187	(22)
The Poisson integral and the theorem of F. and M. Riesz	188	(9)
The corona theorem	197	(12)
Subharmonic Functions and the Dirichlet Problem	209	(46)
Semi-continuous functions	209	(3)
Harmonic functions and Harnack's principle	212	(3)
Convex functions	215	(4)
Subharmonic functions: Definition and basic properties	219	(8)
Subharmonic functions: Further properties and application to convexity theorems	227	(10)
Harmonic and subharmonic functions on Riemann surfaces	237	(1)
The Dirichlet problem	237	(7)
The Rado-Cartan theorem	244	(11)
Appendix: Baire's Theorem	253	(2)
II Exercises	255	(114)
Yves Nievergelt
Introduction	257	(2)
Review of Complex Numbers	259	(8)
Algebraic properties of the complex numbers	259	(2)
Complex equations of generalized circles	261	(1)
Complex fractional linear transformations	262	(3)
Topological concepts	265	(2)
Elementary Theory of Holomorphic Functions	267	(30)
Some basic properties of C-differentiable and holomorphic functions	267	(11)
Complex derivatives and Cauchy-Riemann equations	267	(2)
Differentials and conformal maps	269	(1)
Conformal maps	270	(5)
Radius of convergence of power series	275	(2)
Exponential, trigonometric, and dilogarithm functions	277	(1)
Integration along curves	278	(4)
Complex line integrals	278	(1)
Complex derivatives of line integrals	279	(2)
Remainder of complex Taylor polynomials	281	(1)
H. A. Schwarz's reflection principle	281	(1)
Fundamental properties of holomorphic functions	282	(8)
The complex exponential function	282	(2)
Holomorphic functions	284	(1)
Bounds on the size of roots of polynomials	285	(2)
Principal branch of the complex square root	287	(1)
Complex square roots in celestial mechanics	288	(2)
Theorems of Weierstrass and Montel	290	(1)
Meromorphic functions	290	(7)
A complex Newton's method	291	(2)
Sequences of complex numbers	293	(4)
Covering Spaces and the Monodromy Theorem	297	(8)
Covering spaces and the lifting of curves	297	(2)
Examples of real or complex manifolds	297	(2)
Covering maps	299	(1)
The sheaf of germs of holomorphic functions	299	(1)
Covering spaces and integration along curves	300	(3)
The monodromy theorem and the homotopy form of Cauchy's theorem	303	(1)
Applications of the monodromy theorem	303	(2)
The Winding Number and the Residue Theorem	305	(8)
The winding number	305	(2)
The residue theorem	307	(3)
Applications of the residue theorem	310	(3)
Picard's Theorem	313	(2)
The Inhomogeneous Cauchy-Riemann Equation and Runge's Theorem	315	(16)
Partitions of unity	315	(1)
The equation ∂u/∂z = phis;	316	(7)
Complex differential forms	316	(5)
Rouche's theorem	321	(1)
Inhomogeneous Cauchy-Riemann equations	322	(1)
Runge's theorem	323	(8)
Applications of Runge's Theorem	331	(6)
The Mittag-Leffler theorem	331	(1)
The cohomology form of Cauchy's theorem	332	(1)
The theorem of Weierstrass	332	(3)
Ideals in H(Ω)	335	(2)
The Riemann Mapping Theorem and Simple Connectedness in the Plane	337	(6)
Analytic automorphisms of the disc and of the annulus	337	(3)
The Riemann mapping theorem	340	(2)
Simply connected plane domains	342	(1)
Functions of Several Complex Variables	343	(8)
Compact Riemann Surfaces	351	(10)
Definitions and basic theorems	351	(4)
The cohomology group H1(U, O)	355	(3)
Meromorphic functions on a compact Riemann surface	358	(3)
The Corona Theorem	361	(4)
The Poisson integral and the theorem of F. and M. Riesz	361	(4)
Subharmonic Functions and the Dirichlet Problem	365	(4)
Notes for the exercises	369	(4)
References for the exercises	373	(6)
Index	379