Abstract

This is joint work with Elias M. Stein (Princeton University).

The classical Cauchy theorem and Cauchy integral formula for analytic functions of one complex variable give rise to a plethora of applications to Physics and Engineering, and as such are essential components of the Complex Analysis Toolbox.

Two crucial features of the integration kernel of the Cauchy integral (Cauchy kernel, for short) are its ``analyticity’’ (the Cauchy kernel is an analytic function of the output variable) and its ``universality’’ (the Cauchy integral is meaningful for almost any contour shape). One drawback of the Cauchy kernel is that it lacks a good transformation law under conformal maps (with a few exceptions).

This brings up two questions: Are there other integration kernels that retain the main features of the Cauchy kernel but also have good transformation laws under conformal maps? And: is there an analog of the Cauchy kernel for analytic functions of two (or more) complex variables that retains the aforementioned crucial features?

In this talk I will give a survey of what is known of these matters, with an eye towards enriching the Complex Analysis Toolbox as we know it, and towards building a ``Multivariable Complex Analysis Toolbox’’.