Unique continuation on a line for harmonic functions

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Abstract
In this paper, we discuss local unique continuation for a harmonic function on lines. By using complex extension, we prove a conditional stability estimation for a harmonic function on a line. Our unique continuation is an intermediate property between the classical unique continuation for a harmonic function and the analytic continuation for a holomorphic function. As an application, we show conditional stability up to the boundary in a Cauchy problem of the Laplace equation.