Primitive of Function on Connected Domain

Theorem
Let $f: D \to \C$ be a continuous complex function, where $D$ is a connected domain.

Then the following three conditions are equivalent:


 * $(1): \quad$ $f$ has a primitive.


 * $(2): \quad$ For any two contours $C_1, C_2$ in $D$ with identical start points $z_1 \in D$ and endpoints $z_2 \in D$, we have:


 * $\displaystyle \int_{C_1} f \left({z}\right) \ \mathrm dz = \int_{C_2} f \left({z}\right) \ \mathrm dz$


 * $(3): \quad$ For all closed contours $C$ in $D$, we have:


 * $\displaystyle \oint_{C} f \left({z}\right) \ \mathrm dz = 0$

If the conditions hold, we can choose any $z_0 \in D$ and define a primitive $F: D \to \C$ of $f$ by:


 * $\displaystyle F \left({w}\right) = \int_{C_w} f \left({z}\right) \ \mathrm dz$

where $C_w$ is any contour in $D$ with start point $z_0$ and endpoint $w$.

$(1)$ implies $(2)$
If $F$ is a primitive of $f$, we have:

$(2)$ implies $(3)$
Let $C$ be a closed contour in $D$ with endpoints $z_0$.

Let the constant function $\gamma: \left[{0\,.\,.\,1}\right] \to D$ with $\gamma \left({t}\right) = z_0$ be the parameterization of a contour $C_0$.

Then:

$(3)$ implies $(1)$
This follows from Zero Staircase Integral Condition for Primitive.

Construction of a Primitive
If the conditions hold, we choose $z_0 \in D$ and define a function $F: D \to \C$ of $f$ by:


 * $\displaystyle F \left({w}\right) = \int_{C_w} f \left({z}\right) \rd z$

where $C_w$ is any contour in $D$ with start point $z_0$ and endpoint $w$.

From Connected Domain is Connected by Staircase Contours, it follows that we can choose $C_w$ to be a staircase contour.

If $C_w'$ is another contour in $D$ with the same endpoints as $C_w$, then:


 * $\displaystyle \int_{C_w'} f \left({z}\right) \rd z = \int_{C_w} f \left({z}\right) \rd z$

by condition $(2)$, so $F$ is well-defined.

From Zero Staircase Integral Condition for Primitive, it follows that $F$ is a primitive of $f$.