A Course on Complex Analysis in One Variable Book

	Preface and Acknowledgments
1	First Concepts	1
1.1	Fundamentals of the complex field	1
1.2	Holomorphic functions	3
1.3	Some important examples	5
1.4	The Cauchy-Riemann equations	10
1.5	Some elementary differential equations	14
1.6	Conformality	16
1.7	Power series	18
2	Integration Along a Contour	21
2.1	Curves and their trajectories	21
2.2	Change of Parameter and a Fundamental Inequality	24
2.3	Some important examples of contour integration	27
2.4	The Cauchy theorem in simply connected domains	29
2.5	Some immediate consequences of Cauchy's theorem for a simply connected domain	39
3	The Main Consequences of Cauchy's theorem	43
3.1	The Cauchy theorem in multiply connected domains and the pre-residue theorem	43
3.2	The Cauchy integral formula and its consequences	45
3.3	Analyticity, Taylor's theorem and the identity theorem	53
3.4	The area formula and some consequences	61
3.5	Application to spaces of square integrable holomorphic functions	64
3.6	Spaces of holomorphic functions and Montel's theorem	67
3.7	The maximum modulus theorem and Schwarz' lemma	70
4	Singularities	75
4.1	Classification of isolated singularities, the theorems of Riemann and Casorati-Weierstrass	75
4.2	The principle of the argument	80
4.3	Rouche's theorem and its consequences	86
4.4	The study of a transcendental equation	91
4.5	Laurent expansion	94
4.6	The calculation of residues at an isolated singularity, the residue theorem	99
4.7	Application to the calculation of real integrals	103
4.8	A more general removable singularities theorem and the Schwarz reflection principle	108
5	Conformal Mappings	113
5.1	Linear fractional transformations, equivalence of the unit disk and the upper half plane	113
5.2	Automorphism groups of the disk, upper half plane and entire plane	114
5.3	Annuli	120
5.4	The Riemann mapping theorem for planar domains	123
6	Applications of Complex Analysis to Lie Theory	131
6.1	Applications of the identity theorem: Complete reducibility of representations according to Hermann Weyl and the functional equation for the exponential map of a real Lie group	131
6.2	Application of residues: The surjectivity of the exponential map for U(p,q)	134
6.3	Application of Liouville's theorem and the maximum modulus theorem: The Zariski density of cofinite volume subgroups of complex Lie groups	138
6.4	Applications of the identity theorem to differential topology and Lie groups	140
	Bibliography	143
	Index	145