This research is motivated by the program of Reverse Mathematics. We investigate basic part of complex analysis within some weak subsystems of second order arithmetic, in order to determine what kind of set existence axioms are needed to prove theorems of basic analysis. We are especially concerned with Cauchy’s integral theorem. We show that a weak version of Cauchy’s integral theorem is proved in RCAo. Using this, we can prove that holomorphic functions are analytic in RCAo. On the other hand, (...) we show that a full version of Cauchy’s integral theorem cannot be proved in RCAo but is equivalent to weak König’s lemma over RCAo. (shrink)