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In this paper we prove a result on the Pompeiu problem. If the Schwarz function 4> of the boundary of a simply-connected domain fi C R2 extends meromorphically into a certain portion E of ii with a pole at some point zn. 6 E , then ii has the Pompeiu property unless <1> is a Mobius transformation, in which case Q is a disk.doi:10.2307/2154945 fatcat:7ctliereonas7du6voz44lwpw4