Complex Analysis 20 | Antiderivatives

Complex antiderivatives (also called primitives) let integrals in the complex plane be computed purely from endpoint values: if a holomorphic function f has a primitive F on a domain U, then the contour integral of f along any curve from a to b depends only on where the curve starts and ends, not on the path taken. Concretely, for a curve γ in U with γ(a)=z_start and γ(b)=z_end, the integral ∫_γ f(z) dz equals F(z_end)−F(z_start). This path-independence is the complex analogue of the real fundamental theorem of calculus, but it comes with the added strength that complex derivatives are more restrictive, making the existence of primitives a powerful structural condition.

The result is proved by combining the chain rule with the real fundamental theorem of calculus. Writing the complex line integral over a parametrized curve γ(t), the integrand becomes F′(γ(t))·γ′(t). Since F′ equals f, the integral reduces to an expression of the form ∫ d/dt[F(γ(t))] dt, which evaluates to F(γ(b))−F(γ(a)). A key corollary follows immediately: if the curve is closed (start and end points coincide) and f has a primitive on U, then the contour integral must be zero. In other words, primitives force “no net accumulation” around closed loops.

Examples sharpen the distinction between functions that look similar but behave differently around singularities. On U=ℂ\{0}, the function f(z)=1/z^2 has an antiderivative F(z)=−1/z. Because a primitive exists on the punctured plane, every closed contour integral of 1/z^2 is zero—even for loops that wind around the origin. By contrast, the function 1/z does not admit an antiderivative on ℂ\{0}. The earlier computation of the integral around a circle enclosing the origin gives ∮ 1/z dz = 2πi, which cannot be reconciled with the corollary that closed integrals would have to vanish if a primitive existed.

The logarithm clarifies the “almost” relationship. Although log(z) has derivative 1/z, it only works as a primitive on a smaller domain where the logarithm can be defined consistently—specifically, the complex plane with a branch cut along the negative real axis (excluding that axis, including 0). On such a slit domain, curves cannot enclose the origin in the same way, so the obstruction that produced the 2πi winding effect disappears. The takeaway is that isolated singularities matter: for 1/z, enclosing the singularity changes the integral by 2πi, while for 1/z^2 the integral remains zero. This special role of 1/z is highlighted as a central theme for later work on complex integration.