DREs Over Complex Domains
In this chapter, we will introduce P.J. Davis’s result regarding the construction of DREs over complex domains (see ). 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.
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