Research proof index

For an anti-holomorphic function the anti-holomorphic Wirtinger derivative equals the complex Fréchet derivative of g at star u along the slot-I real coordinate direction, ∂̄_I (g ∘ star) = fderiv ℂ g (star u) (Pi.single I 1). Dual of dWirtingerCoord_eq_complex_fderiv_apply.

The two-term Wirtinger chain rule for dWirtingerAntiDir, the anti-holomorphic dual of dWirtingerDir_comp: ∂̄_v(g∘f) = (∂g/∂f)·∂̄_v f + (∂g/∂f̄)·∂̄_v f̄. Same outer ∂g/∂f, ∂g/∂f̄ coefficients, now each multiplying its anti-holomorphic inner derivative ∂̄_v f, ∂̄_v f̄; same proof as dWirtingerDir_comp.

The single-term coordinate chain rule for a holomorphic outer g, pointwise at u: ∂_I (g ∘ f) = deriv g (f u) · ∂_I f. From the two-term dWirtingerCoord_comp_apply: for holomorphic g the anti-holomorphic coefficient dWirtingerAntiDir g 1 (f u) vanishes and dWirtingerDir g 1 (f u) collapses to deriv g (f u), both off the ℂ-linearity `cline...

The two-term Wirtinger chain rule for dWirtingerDir, outer g : ℂ → ℂ and inner f : V → ℂ: ∂_v(g∘f) = (∂g/∂f)·∂_v f + (∂g/∂f̄)·∂_v f̄. realLinear_apply_eq_wirtinger splits the chain rule's outer ℝ-linear factor into the ∂g/∂f, ∂g/∂f̄ coefficients, each multiplying its inner directional derivative ∂_v f, ∂_v f̄.

Schwarz's theorem for the directional Wirtinger operators: on a C² field f the holomorphic and anti-holomorphic directional derivatives commute in any two directions, ∂_v ∂̄_w f = ∂̄_w ∂_v f. The commutation adds no analytic input. By the §G bridge each order expands into a real-linear combination of the second real Fréchet derivative `fderiv ℝ...

Conjugating the function swaps the operators up to an outer conjugation: ∂_v f̄ = conj (∂̄_v f).