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.

H. Begehr and A. Kumar, “Boundary Value Problems for the Inhomogeneous Polyanalytic Equation I,” Analysis: International Mathematical Journal of Analysis and its Application, Vol. 25, No. 1, 2005, pp. 55-71.

D. Jinyuan and W. Yufeng, “On Boundary Value Problems of Polyanalytic Functions on the Real Axis,” Complex Variables, Vol. 48, No. 6, 2003, pp. 527-542.
doi:10.1080/0278107031000103412

B. F. Fatulaev, “The Main Haseman Type Boundary Value Problem for Metaanalytic Function in the Case of Circular Domains,” Mathematical Modelling and Analysis, Vol. 6, No. 1, 2001, pp. 68-76.

A. S. Mshimba, “A Mixed Boundary Value Problem for Polyanalytic Function of Order n in the Sobolev Space Wn, p(D),” Complex Variables, Vol. 47, No. 12, 2002, pp. 278-1077.

W. Yufeng and D. Jinyuan, “Hilbert Boundary Value Problems of Polyanalytic Functions on the Unit Circumference,” Complex Variables and Elliptic Equations, Vol. 51, No. 8-11, 2006, pp. 923-943.
doi:10.1080/17476930600667692

L. Xing, “A Class of Periodic Riemann Boundary Value Inverse Problems,” Proceedings of the Second Asian Mathematical Conference, Nakhon Ratchasima, 1995, pp. 397-400.

M. H. Wang, “Inverse Riemann Boundary Value Problems for Generalized Analytic Functions,” Journal of Ningxia University of Natural Resources and Life Sciences Education, Vol. 27, No. 1, 2006, pp. 18-24.

X. Q. Wen and M. Z. Li, “A Class of Inverse Riemann Boundary Value Problems for Generalized Holomorphic Functions,” Journal of Mathematical, Vol. 24, No. 4, 2004, pp. 457-464.

L. X. Cao, P.-R. Li and P. Sun, “The Hilbert Boundary Value Problem With Parametric Unknown Function on Upper Half-Plane,” Mathematics in Practice and Theory, Vol. 42, No. 2, 2012, pp. 189-194.