Complex Analysis 20 | Antiderivatives [dark version]

Complex antiderivatives (primitives) let contour integrals be computed purely from endpoints: if a holomorphic function f has a complex antiderivative F on a domain U, then for any curve γ in U from a to b,

∫_γ f(z) dz = F(b) − F(a).

That single endpoint rule is the central payoff, because it makes the integral independent of the path taken between the same start and end points. The restriction is strong—complex differentiation is more rigid than real differentiation—but the reward is equally clear: once an antiderivative exists, contour integration becomes a straightforward subtraction.

The result follows from the complex chain rule and the fundamental theorem of calculus applied to the parameterization of γ. Writing γ(t) for t in an interval [a,b], the integrand f(γ(t)) dz transforms into F′(γ(t))·γ′(t) dt, and the integral collapses to F(γ(b)) − F(γ(a)). Because the argument relies on real Riemann integration after parameterization, the usual calculus machinery carries over cleanly into the complex setting.

A direct corollary sharpens the message: if γ is a closed curve (start point equals end point) and f has an antiderivative on U, then the contour integral must be zero. This “closed integral equals 0” criterion becomes a practical test for whether an antiderivative can exist.

The examples highlight why complex functions behave differently around singularities. On U = ℂ \ {0}, the function f(z) = 1/z^2 has an antiderivative F(z) = −1/z. Since F exists on the punctured plane, every closed contour integral of 1/z^2 must vanish—so looping around the origin changes nothing.

But the similar-looking function f(z) = 1/z does not share that property on the same domain. A known computation gives ∮ 1/z dz = 2πi for a circle around the origin, which contradicts the corollary’s prediction that closed integrals would be 0 if an antiderivative existed. Therefore, 1/z has no antiderivative on ℂ \ {0}.

The logarithm resolves the apparent paradox. Although log(z) has derivative 1/z, it cannot be defined on the entire punctured plane without ambiguity: the domain must be cut to remove a branch issue. Using a slit domain (the complex plane with the negative real axis excluded, including 0) allows a principal branch of the logarithm, and on that smaller domain the logarithm becomes a valid antiderivative of 1/z. In that setting, contours that would “wrap around” the excluded region are no longer allowed, so closed-curve integrals return to the expected zero behavior.

Overall, the key lesson is that isolated singularities matter: for 1/z, enclosing the singularity produces a nonzero 2πi, while for 1/z^2 it does not. That difference foreshadows why the existence and domain of primitives—especially near singularities—are decisive in complex analysis.