This paper deals with a generalization of the Vekua-Riemann function. The linear differential equationwith holomorphic coefficients and holomorphic right side is considered. The Riemann function of this differential equation is defined recursively by some Goursat conditions. As a special case the well known Vekua-Riemann function is obtained. With the help of the generalized Riemann function, every holomorphic solution possesses an integral representation, in which the Goursat conditions enter. This method can also be applied to ordinary differential equations. A fundamental system can be computed by solving a single initial value problem.