The edgewise Riemann integral around a complex triangle for an integrable pullback

For an ordered complex triangle and a function on its boundary trace, if the pullback along each affine edge multiplied by that edge's constant velocity is Riemann integrable, define the edgewise triangle integral to be the sum of those three complex Riemann integrals.

Precisely, let the directed edges ${\mathrm{\ell }}_{ab}$, ${\mathrm{\ell }}_{bc}$, and ${\mathrm{\ell }}_{ca}$ be those of Filled complex triangles, their oriented three-edge boundary contours, diameter, and perimeter, and let $f$ be defined on their combined trace. When the functions

$t⟼f({\mathrm{\ell }}_{ab}(t))(b-a),t⟼f({\mathrm{\ell }}_{bc}(t))(c-b),t⟼f({\mathrm{\ell }}_{ca}(t))(a-c)$

are Riemann integrable on $[0,1]$ as maps into $\mathbb{C}={\mathbb{R}}^2$ ($\mathbb{C}$ is the real coordinate plane, with coordinate arithmetic, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral), put

${\int }_{\mathrm{\partial }\mathrm{\Delta }[a,b,c]}^{\mathrm{e}\mathrm{d}\mathrm{g}\mathrm{e}}f(z)dz:={\int }_0^{1}f({\mathrm{\ell }}_{ab}(t))(b-a)dt+{\int }_0^{1}f({\mathrm{\ell }}_{bc}(t))(c-b)dt+{\int }_0^{1}f({\mathrm{\ell }}_{ca}(t))(a-c)dt\mathrm{.}$

The integrability hypothesis makes every term a uniquely defined complex number, so the displayed sum is well-defined. If $f$ is continuous on the trace, For piecewise-C1 contours the Riemann–Stieltjes integral agrees with the parametric complex integral and the published real line integrals identifies this edgewise value with the published contour integral ${\int }_{\mathrm{\partial }\mathrm{\Delta }[a,b,c]}f(z)dz$. Constant edges cause no ambiguity because their velocity is zero.