The Inhomogeneous Cauchy-Riemann Equation and Runge’s Theorem

Raghavan Narasimhan and Yves Nievergelt
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Raghavan Narasimhan: University of Chicago, Department of Mathematics
Yves Nievergelt: Eastern Washington University, Department of Mathematics

Chapter Chapter 5 in Complex Analysis in One Variable, 2001, pp 97-114 from Springer

Abstract: Abstract Holomorphic functions are characterized by the equation ∂É/∂z = 0. In this chapter, we shall study the equation ∂É/∂̄z = g when g has compact support. We shall obtain an explicit solution which leads to a variant of the Cauchy integral formula. This variant can often be used instead of the usual Cauchy formula, and has the advantage of not involving winding numbers. We shall illustrate this principle with a variant of the argument principle and a proof of the Runge theorem.

Keywords: Compact Subset; Cauchy Integral Formula; Continuous Linear Form; Homology Form; Compact Connected Component (search for similar items in EconPapers)
Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4612-0175-5_5

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DOI: 10.1007/978-1-4612-0175-5_5

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