Complex primitive

If \(F\) is a primitive of \(f\) on \(D\) and \(C\) is a contour in \(D\) from \(z_0\) to \(z_1\), then \(\int_C f\,dz=F(z_1)-F(z_0)\), whatever the shape of the path. It follows from the chain rule: \(\frac{d}{dt}F(\gamma(t))=f(\gamma(t))\gamma'(t)\), integrated in \(t\). Primitives are computed with the same rules as in real calculus: \(z^n\to\frac{z^{n+1}}{n+1}\), \(e^z\), \(\sin z\to-\cos z\), etc.

With a primitive, closed integrals are zero, because the endpoints coincide. So \(\oint_{|z|=1}z^3\,dz=0\) and also \(\oint_{|z|=1}\frac{dz}{z^2}=0\), since \(-1/z\) is a primitive on \(\mathbb{C}\setminus\{0\}\). Conversely, if \(f\) is continuous and all its closed integrals in \(D\) are zero, then \(F(z)=\int_{z_0}^{z}f\) is a primitive.

By the Cauchy-Goursat theorem, every function holomorphic on a simply connected open set has a primitive there. The borderline case is \(1/z\) on \(\mathbb{C}\setminus\{0\}\), which is not simply connected and has no primitive: its integral over \(|z|=1\) is \(2\pi i\). The logarithm serves as a primitive only on regions that do not surround the origin, and the value of \(\int dz/z\) depends on the path.

A typical mistake is forgetting that the primitive must exist on the whole set the path travels: having a formula for \(F\) on a piece is not enough. Another is using \(F(z_1)-F(z_0)\) when \(f\) is not even holomorphic, like \(\bar z\), which has no primitive.