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The Inhomogeneous Cauchy-Riemann Equation and Runge’s Theorem

  • Raghavan Narasimhan

Abstract

Holomorphic functions are characterized by the equation \( \partial f/\partial \bar z = 0 \). In this chapter, we shall study the equation \( \partial f/\partial \bar z = g \). 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 Compact Hausdorff Space Argument Principle Cauchy Integral Formula Continuous Linear Form 
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Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • Raghavan Narasimhan
    • 1
  1. 1.Department of MathematicsThe University of ChicagoChicagoUSA

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