Purdue University Graduate School
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Algebraic Formulas for Kernel Functions on Representative Two-Connected Domains

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posted on 2022-12-06, 20:51 authored by Raymond Leonard Polak IIIRaymond Leonard Polak III

We write down explicit algebraic formulas for the Szeg\H{o}, Garabedian and Bergman kernels for specific two-connected planar domains. We use these results to derive integral representations for a biholomorphic invariant relating the Bergman and Szeg\H{o} kernels. We use the formulas to study the asymptotic behavior of these kernels as a family of two-connected domains approaches the unit disc. We derive an explicit formula for the Green's function for the Laplacian for special values on two-connected domains. Every two-connected domain is biholomorphic to a unique two-connected domain of the type we consider. This allows one to write down formulas for the kernel functions on a general two-connected domain.


NSF DMS 1764167


Degree Type

  • Doctor of Philosophy


  • Mathematics

Campus location

  • West Lafayette

Advisor/Supervisor/Committee Chair

Steven R. Bell

Additional Committee Member 2

Laszlo Lempert

Additional Committee Member 3

Johnny Brown

Additional Committee Member 4

Sai-Kee Yeung