antiderivative of complex function

By the antiderivative of a complex function $f$ in a domain $D$ of $\mathbb{C}$ , we mean every complex function $F$ which in $D$ satisfies the condition $$\frac{d}{dz}F(z) = f(z).$$

• If $f$ is a continuous complex function in a domain $D$ and if the integral

  (1)

  
  where the path ${\gamma_z}$ begins at a fixed point $z_0$ of $D$ and ends at the point $z$ of $D$ , is independent of the path $\gamma_z$ for each value of $z$ , then (1) defines an analytic function $F$ with domain $D$ . This function is an antiderivative of $f$ in $D$ , i.e. in all of $D$ , the condition $$\frac{d}{dz}\int_{\gamma_z}f(t)\,dt = f(z)$$ is true.
• If $f$ is an analytic function in a simply connected open domain $U$ , then $f$ has an antiderivative in $U$ , e.g. the function $F$ defined by (1) where the path $\gamma_z$ is within $U$ . If $\gamma$ lies within $U$ and connects the points $z_0$ and $z_1$ , then $$\int_{\gamma}f(z)\,dz = F(z_1)-F(z_0),$$ where $F$ is an arbitrary antiderivative of $f$ in $U$ .