In this thesis we study the alternative method of constructing Riemann Surfaces of functions of one complex variable by means of analytic continuation. This approach was initiated both by Karl Weirstrass and by Bernhard Riemann. Starting with a function element (D, f) we first construct the triple (R, r , F ) associated to this (D, f) and call it the complete analytic function in the sense of Weirstrass. Here R is a dimensional complex manifold and r , F are holomorphic functions on R. We then discuss the problem of extending the triple (R, r , F) to (R, r , F). In such a way that R, is a one dimensional compact complex manifold p and F are the holomorphic extensions of the maps r and F respectively.

It turned out that this extension is possible if and only if the initial function element (D, f) satisfies a non trivial polynomial relation P(z, f(z)) = 0 on D.