antiderivative of complex function

By the of a complex function $f$ in a domain $D$ of $ℂ$, we 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

  $F(z):={\displaystyle {\int }_{{\gamma }_z}}f(t)𝑑t$(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 thepath ${\gamma }_z$ for each value of $z$, then (1) defines ananalytic function $F$ with domain $D$.  This function is anantiderivative of $f$ in $D$, i.e. (http://planetmath.org/Ie) at allpoints 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. (http://planetmath.org/Eg) 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 $U$.