PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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Complex line integrals are linear in the integrand

Statement

For continuous $f, g$ on the trace of a rectifiable contour $γ$ and $α, β \in C$ , $\int_{γ} (α f + β g) d z = α \int_{γ} f d z + β \int_{γ} g d z .$

Facts & Assumptions

Given: A rectifiable contour, continuous $f, g$ , and complex scalars $α, β$ .

[L1]

The complex line integral is the stated combination of four real Riemann–Stieltjes integrals (The complex line integral over a rectifiable path as a componentwise Riemann–Stieltjes integral).

[L2]

Riemann–Stieltjes integrals are linear in the integrand and integrator whenever the displayed integrals exist (Linearity and interval additivity of the Riemann–Stieltjes integral).

Proof

technique · direct

1.1

Expand the real and imaginary parts of $α f + β g$ and apply [L2] to each component integral in [L1].

L1L2algebra

2.1

Recombining the real and imaginary identities gives the displayed complex linearity. Zero scalars and singleton paths are included.

step 1.1L1∎

Depends on

Used by

The unit-circle integral of exp(z)/z is 2 pi i by uniform termwise integration Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 59 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

L. Ahlfors, Complex Analysis, 3rd ed., Ch. 4 (standard reference, not scraped)