Riemann Surfaces

Abstract: Idea of Riemann surface- Start with a surface (say like sphere or torus or cylinder that u can visualize in R^3, let X may be any of three .) ,suppose f : X to C is function , we want to define when f is holomorphic locally on X,but our object X is like a life (means i can touch it), so a tool which allow to do complex analysis on X (life) is known as Riemann surface .I will try to do it in two way. 1-by usual manifold in C and 2- by the graph of multivalued function in several complex variables.