Characterization of solutions to the generalized Cauchy-Riemann system |
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Authors: | Peter A McCoy |
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Institution: | Mathematics Department, United States Naval Academy, Annapolis, Maryland 21402 U.S.A. |
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Abstract: | For a generalized biaxially symmetric potential U on a semi-disk D+, a harmonic conjugate V is defined by the generalized Cauchy-Riemann system. There is an associated boundary value theory for the Dirichlet problem. The converse to the Dirichlet problem is considered by determining the boundary functions to which U and V converge. The unique limits are hyperfunctions on the ?D+. In fact, the space of hyperfunctions is isomorphic to the spaces of generalized biaxially symmetric potentials and their harmonic conjugates. A representation theorem is given for U and V terms of convolutions of certain Poisson kernels with continuous functions that satisfy a growth condition on the ?D+. |
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