Synopsis
I Complex Analysis in One Variable.- 1 Elementary Theory of Holomorphic Functions.- 2 Covering Spaces and the Monodromy Theorem.- 3 The Winding Number and the Residue Theorem.- 4 Picard's Theorem.- 5 Inhomogeneous Cauchy-Riemann Equation and Runge's Theorem.- 6 Applications of Runge's Theorem.- 7 Riemann Mapping Theorem and Simple Connectedness in the Plane.- 8 Functions of Several Complex Variables.- 9 Compact Riemann Surfaces.- 10 The Corona Theorem.- 11 Subharmonic Functions and the Dirichlet Problem.- II Exercises.- 0 Review of Complex Numbers.- 1 Elementary Theory of Holomorphic Functions.- 2 Covering Spaces and the Monodromy Theorem.- 3 The Winding Number and the Residue Theorem.- 4 Picard's Theorem.- 5 The Inhomogeneous Cauchy-Riemann Equation and Runge's Theorem.- 6 Applications of Runge's Theorem.- 7 The Riemann Mapping Theorem and Simple Connectedness in the Plane.- 8 Functions of Several Complex Variables.- 9 Compact Riemann Surfaces.- 10 The Corona Theorem.- 11 Subharmonic Functions and the Dirichlet Problem.- Notes for the exercises.- References for the exercises.
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