Integracija na kompaktnim Riemannovim plohama

In this paper we present theory of Riemann surfaces. In first chapter we introduce definition of Riemann surfaces and functions on them. Then we give basic examples of Riemann surfaces such as Riemann sphere, complex torus and smooth affine/projective plane curves. Also, we determine meromorphic functions on mentioned examples of Riemann surfaces. For Riemann sphere we showed that every meromorphic function on it is rational function. On complex torus example of meromorphic function is ratio of translated theta functions, while on smooth affine/projective plane curves example of meromorphic function is ratio of polynomials - homogeneous of the same degree in projective case. Furthermore, we study holomorphic maps between Riemann surfaces. We introduce concept of degree of holomorphic map between compact Riemann surfaces which we use to get information about geometry of Riemann surface through algebraic argument. For example, if map is of degree 1, than Riemann surfaces are isomorphic. Since Riemann surfaces are locally isomorphic with open subsets of \(\mathbb{C}\), question whether classical theorems of complex analysis are true in theory of Riemann surfaces comes naturally. The answer is affirmative; some theorems we can be transfered directly, some come in simpler form, while there are theorems inherent only to theory of Riemann surfaces since they are concerning compact Riemann surfaces. In second chapter we study integration on Riemann surfaces. The most important construction in this chapter is construction of differential forms on Riemann surface which is done by the theory of sheaves. After that, we define integration of differential 1 and 2 forms and we easily show that integration is well defined. The main theorem of this chapter is Residue theorem which is of fundamental importance for theory of compact Riemann surfaces and for the proof of its connection with algebraic geometry.