Introduction to the Cauchy-Riemann operator

A set of notes based on lectures I gave in the Analysis Seminar at Portland State University during the 2020-2021 academic year. The exposition begins with the usual Cauchy-Riemann equations, pointing out how they connect the notions of real and complex differentiability. After observing that these equations can be written in complex form as \(\overline{\partial}f=0\), where \[ \overline{\partial}=\frac{1}{2}\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y} \right), \] we proceed to the non-homogeneous Cauchy-Riemann equations, which assume the complex form \(\overline{\partial}f=u\). We show how this equation can be used to give non-classical proofs of some classical theorems on the existence of analytic functions with certain prescribed properties. We use these results to examine the ideal structure of the ring of analytic functions on a plane domain.

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