Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-16
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The rectifiable Riemann–Stieltjes definition on an explicit polygonal contour with corners

Example

Let γ follow the three segments 0→1→1+i→i, and let f(z)=z. Then the componentwise Riemann–Stieltjes definition gives ∫γz dz=−12, the same value as the piecewise-C1 parametric formula. The corners require no matching derivatives.

Facts & Assumptions

Given: The polygonal contour and affine integrand in the Example.

[L1]

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

[L2]

On piecewise-C1 contours it agrees with the parametric complex integral (For piecewise-C1 contours the Riemann–Stieltjes integral agrees with the parametric complex integral and the published real line integrals).

[L3]

Integrals add under concatenation (Complex line integrals change sign under reversal and add under concatenation).

Verification

technique · direct
1.1L1algebra

On each affine segment, the four Stieltjes components in [L1] reduce to ordinary integrals against constant coordinate derivatives. Recombination gives ∫01γj(t)γj′(t)dt on that segment.

2.1step 1.1L3algebra

Each segment integral is (z12−z02)/2. Adding the three endpoint increments by [L3] telescopes to (i2−02)/2=−1/2.

3.1step 1.1step 2.1L2∎

Formula [L2] gives the same three parametric integrals. The one-sided derivatives at the two corners need not agree.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources