assume that there is some complex value and some such that for all in at which is defined. Then the new function: must be holomorphic on , with zeroes at the poles of f, and bounded by 1/ε. It can therefore be analytically continued (or continuously extended, or holomorphically extended) to all of V by Riemann's analytic continuation theorem. So the original function can be expressed in terms of : for all arguments z in V \ {z0}. Consider the two possible cases for If the limit is 0, then f has a pole at z0 . If the limit is not 0, then z0 is a removable singularity of f . Both possibilities contradict the assumption that the point z0 is an essential singularity of the function f . Hence the assumption is false and the theorem holds. History The history of this important theorem is described by Collingwood and Lohwater. It was published by Weierstrass in 1876 (in German) and by Sokhotski in 1868 in his Master thesis (in Russian). So it was called Sokhotski's theorem in the Russian literature and Weierstrass's theorem in the Western literature. The same theorem was published by Casorati in 1868, and by Briot and Bouquet in the first edition of their book (1859). However, Briot and Bouquet removed this theorem from the second edition (1875). References Section 31, Theorem 2 (pp. 124–125) of Theorems in complex analysis Articles containing proofs