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LeanPool.JacobianDiffgeo.PlanarStokes.CompactSupport

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 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 ) — 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.

theorem RS.integral_wirtingerDbar_eq_zero {g : } {U : Set } (hU : IsOpen U) (hg : ContDiffOn 1 g U) (hcs : HasCompactSupport g) (hsub : tsupport gU) :
(w : ), wirtingerDbar g w = 0

Atom 1 (compact-support planar Stokes for dbar): the dbar of a compactly-supported function integrates to zero over the whole plane.

theorem RS.integral_wirtingerDbar_mul_eq_zero_of_differentiableOn {g f : } {U : Set } (hU : IsOpen U) (hg : ContDiffOn 1 g U) (hcs : HasCompactSupport g) (hsub : tsupport gU) (hf : DifferentiableOn f U) :
(w : ), wirtingerDbar g w * f w = 0

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.