Synopsis
1 Elementary Theory of Holomorphic Functions.- 1 Some basic properties of ?-differentiable and holomorphic functions.- 2 Integration along curves.- 3 Fundamental properties of holomorphic functions.- 4 The theorems of Weierstrass and Montel.- 5 Meromorphic functions.- 6 The Looman-Menchoff theorem.- Notes on Chapter 1.- References : Chapter 1.- 2 Covering Spaces and the Monodromy Theorem.- 1 Covering spaces and the lifting of curves.- 2 The sheaf of germs of holomorphic functions.- 3 Covering spaces and integration along curves.- 4 The monodromy theorem and the homotopy form of Cauchy's theorem.- 5 Applications of the monodromy theorem.- Notes on Chapter 2.- References : Chapter 2.- 3 The Winding Number and the Residue Theorem.- 1 The winding number.- 2 The residue theorem.- 3 Applications of the residue theorem.- Notes on Chapter 3.- References : Chapter 3.- 4 Picard's Theorem.- Notes on Chapter 4.- References : Chapter 4.- 5 The Inhomogeneous Cauchy-Riemann Equation and Runge's Theorem.- 1 Partitions of unity.- 2 The equation % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq % GHciITcaWG1baabaGaeyOaIyRabmOEayaaraaaaiabg2da9iabew9a % Mbaa!3DAD!$$ \[\frac{{\partial u}} {{\partial \bar z}} = \phi \]$$.- 3 Runge's theorem.- 4 The homology form of Cauchy's theorem.- Notes on Chapter 5.- References : Chapter 5.- 6 Applications of Runge's Theorem.- 1 The Mittag-Leffler theorem.- 2 The cohomology form of Cauchy's theorem.- 3 The theorem of Weierstrass.- 4 Ideals in ? (?).- Notes on Chapter 6.- References : Chapter 6.- 7 The Riemann Mapping Theorem and Simple Connectedness in the Plane.- 1 Analytic automorphisms of the disc and of the annulus.- 2 The Riemann mapping theorem.- 3 Simply connected plane domains.- Notes on Chapter 7.- References : Chapter 7.- 8 Functions of Several Complex Variables.- Notes on Chapter 8.- References : Chapter 8.- 9 Compact Riemann Surfaces.- 1 Definitions and basic theorems.- 2 Meromorphic functions.- 3 The cohomology group H1(𝖀𝒪).- 4 A theorem from functional analysis.- 5 The finiteness theorem.- 6 Meromorphic functions on a compact Riemann surface.- Notes on Chapter 9.- References : Chapter 9.- 10 The Corona Theorem.- 1 The Poisson Integral and the theorem of F and M Riesz.- 2 The corona theorem.- Notes on Chapter 10.- References: Chapter 10.- 11 Subharmonic Functions and the Dirichlet Problem.- 1 Semi-continuous functions.- 2 Harmonic functions and Harnack's principle.- 3 Convex functions.- 4 Subharmonic functions : Definition and basic properties.- 5 Subharmonic functions : Further properties and application to convexity theorems.- 6 Harmonic and subharmonic functions on Riemann surfaces.- 7 The Dirichlet problem.- 8 The Radó-Cartan theorem.- Notes on Chapter 11.- References : Chapter 11.
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