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DREs Over Complex Domains

  • Tian-Xiao He
Chapter

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 2001

Authors and Affiliations

  • Tian-Xiao He
    • 1
  1. 1.Department of Mathematics & Computer SienceIllinois Wesleyan UniversityBloomingtonUSA

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