In Arnold Sommerfeld's Lecture Vol.IV, it is mentioned that as per Riemann's theory of functions "If a two dimensional potential $v$ vanishes together with its normal derivative along a finite curve segment then $v$ vanishes identically in the whole plane".
I think this is somehow related to the uniqueness theorem.
Would you kindly suggest me how to prove this or from where I can get the proof/ relevant discussion on this statement?