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Lecture notes on several complex variables - math.tamu.edu

Math 650, fall 2007 Texas A&M UniversityLecture notes on several complexvariablesHarold P. BoasDraft of December 7, 2007 Contents1 Power series .. Integral representations .. Partial differential equations .. Geometry .. 32 Power Domain of convergence .. Characterization of domains of convergence .. Natural boundaries .. Summary .. 93 Real convexity .. Convexity with respect to a class of functions .. Polynomial convexity .. Linear and rational convexity .. Holomorphic convexity .. Pseudoconvexity .. The Levi problem .. The Levi form .. Applications of the problem.

Thus every convergence domain is a union of polydiscs centered at the origin. By expanding the Cauchy kernel in a power series, one finds from the iterated Cauchy formula (just as in the one-variable case) that a function holomorphic in a polydisc, or

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