Atom 1: compact-support planar Stokes for dbar #
Unit: planar-stokes-atoms (docs/design/planar-stokes.md §6). Proves the honest 2-D Stokes
theorem this unit exists to supply: a compactly-supported C¹ function's dbar integrates to zero
over the whole plane (integral_wirtingerDbar_eq_zero), by picking a rectangle strictly
containing the support and invoking Complex.integral_boundary_rect_of_differentiableOn_real
(mathlib's rectangle divergence theorem, specialized to ℂ) — the boundary terms vanish since g
is ≡ 0 there, leaving exactly the wirtingerDbar-area identity. The holomorphic-multiplier
corollary (integral_wirtingerDbar_mul_eq_zero_of_differentiableOn, Atom 1b) is the "no pole in
this chart" case residue-theorem needs for every PoU piece that misses every pole.
Atom 1 (compact-support planar Stokes for dbar): the dbar of a compactly-supported C¹
function integrates to zero over the whole plane.
Atom 1b (immediate corollary, the "no pole in this chart" case residue-theorem needs for
every PoU piece that does not touch a pole): if f is holomorphic throughout U, the
dbar-weighted integral against f also vanishes.