Complex Line Integrals

Core Idea

The integral of f(z) along a path γ from a to b is ∫_γ f(z) dz = ∫_a^b f(γ(t)) γ'(t) dt. For a general continuous f, this integral depends on the path. But for holomorphic f on simply connected domains, the integral is path-independent and depends only on the endpoints, a fact made rigorous by Cauchy's theorem.

How It's Best Learned

Compute ∫_γ z dz along two different paths from 0 to 1+i; verify that they give the same result. Then try f(z) = 1/z along paths that wind around the origin; observe the path-dependence.

Common Misconceptions

Assuming all line integrals are path-independent; this is true only for holomorphic functions on simply connected domains. Forgetting to parametrize the contour when computing the integral.

Explainer

From your work with scalar line integrals in real analysis, you know how to integrate a function along a curve in the plane by parametrizing the curve and reducing to a single-variable integral. The complex line integral does something analogous but richer: it integrates a complex-valued function f(z) along a directed path γ in the complex plane. The definition ∫_γ f(z) dz = ∫_a^b f(γ(t)) γ'(t) dt is the direct translation — parametrize the path, substitute, and integrate. The factor γ'(t) accounts for both the speed and direction of travel along the path, just as arc-length differentials do in real line integrals.

The key new ingredient is holomorphicity. For real line integrals, path-independence required the integrand to be a gradient (a conservative field). For complex line integrals, holomorphicity plays the analogous role — but it is a much stronger condition. A holomorphic function satisfies the Cauchy-Riemann equations, which impose tight coupling between its real and imaginary parts. Because of this coupling, holomorphic functions on simply connected domains are automatically path-independent: ∫_γ f(z) dz depends only on the endpoints of γ, not on the specific path taken.

Path-dependence arises precisely when holomorphicity fails. The canonical example is f(z) = 1/z, which has a singularity at z = 0. If you integrate 1/z along a small circle centered at the origin, you get 2πi regardless of the radius; if you integrate along a path that doesn't wind around the origin, you get 0. The winding number of the path around the singularity determines the answer. This is the first glimpse of why complex analysis connects integration to topology — the value of an integral can depend on how the path encircles singularities, not just where it starts and ends.

Computing a complex line integral in practice means choosing a parametrization γ(t) for a ≤ t ≤ b, then evaluating ∫_a^b f(γ(t)) γ'(t) dt as an ordinary real integral of a complex-valued function (integrate real and imaginary parts separately). For the upper half of the unit circle from 1 to −1, set γ(t) = e^it for 0 ≤ t ≤ π; then γ'(t) = ie^it dt, and you substitute f(γ(t)) accordingly. This parametrization technique is the computational foundation for everything in contour integration and Cauchy's theorem that follows.