Abstract
We consider a Hilbert boundary value problem with an unknown parametric function on arbitrary infinite straight line passing through the origin. We propose to transform the Hilbert boundary value problem to Riemann boundary value problem, and address it by defining symmetric extension for holomorphic functions about an arbitrary straight line passing through the origin. Finally, we develop the general solution and the solvable conditions for the Hilbert boundary value problem.
Highlights
Various kinds of boundary value problems (BVPs) for analytic functions or polyanalytic functions have been widely investigated [1,2,3,4,5,6,7,8]
We consider a Hilbert boundary value problem with an unknown parametric function on arbitrary infinite straight line passing through the origin
We propose to transform the Hilbert boundary value problem to Riemann boundary value problem, and address it by defining symmetric extension for holomorphic functions about an arbitrary straight line passing through the origin
Summary
Various kinds of boundary value problems (BVPs) for analytic functions or polyanalytic functions have been widely investigated [1,2,3,4,5,6,7,8]. Inverse Riemann BVPs for generalized holomorphic functions or bianalytic functions have been investigated [9,10,11,12]. We first define the symmetric extension of holomorphic function about an infinite straight line passing through the origin, and discuss its several important properties. We propose a Hilbert BVP with an unknown parametric function on arbitrary half-plane with its boundary passing through the origin. We transform the Hilbert BVP into a Riemann BVP on the infinite straight line using the defined symmetric extension. We discuss the solvable conditions and the solution for the Hilbert BVP
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