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Testing analyticity on circles

From: American Journal of Mathematics
Volume 129, Number 3, June 2007
pp. 785-790 | 10.1353/ajm.2007.0015


Consider a continuous one parameter family of circles in a complex plane that contains two circles lying in the exterior of one another. We prove that if a continuous function on the union of the circles extends holomorphically into each circle, then the function is holomorphic in the interior of the union of the circles.