# DREs Over Complex Domains

## Abstract

In this chapter, we will introduce P.J. Davis’s result regarding the construction of DREs over complex domains (see [15]). From *Green’s theorem*, by using Schwarz functions, a double integral of an *analytic function* over a complex domain can be reduced to a *contour integral*. A subsequent application of the *radius theory* to the contour integral yields a DRE for the double integral. Since for any given *harmonic function* there always exists an analytic function the real part of which is the given harmonic function, a DRE for an analytic function leads to a DRE for the corresponding harmonic function. In Section 1, we will discuss the general method for constructing this type of DRE. Section 2 will give applications of the DRE in constructing quadrature formulas. In Section 3, we try to find (if possible) regions over which some given DREs and/or quadrature formulas of double integrals hold. Section 4 will discuss some additional topics on DREs over complex domains such as the construction of a Schwarz function regular in a slit region and applications of DREs in Fourier expansion.

## Keywords

Harmonic Function Quadrature Formula Fourier Expansion Complex Domain Bergman Kernel## Preview

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