Integral Calculus in the Complex Plane C

In Section I.5 we already encountered the problem of finding a primitive function for a given analytic function $f:D \to \mathcal{C}, D\subset \mathcal{C}$ open, i. e., an analytic function $F:D\to \mathcal{C}$ such that $F^\prime = f.$

In general, one may ask: Which functions $f : D \to \mathcal{C}, D \subset \mathcal{C}$ open, have a primitive? Recall that in the real case any continuous function $f : [a, b] \to \mathcal{R}, a < b$, has a primitive, namely, for example the integral

$$F(x): = \int^{x}_{a} f(t) dt.$$

Whether one uses the notion of a RIEMANN integral or the integral for regulated functions is irrelevant in this connection.