laplacianWithin_eq_iteratedFDerivWithin_complexPlane

English

Special case of Laplacian formula for complex plane with standard real inner product structure.

Русский

Особый случай формулы лаплациана для комплексной плоскости с обычной вещественной структурой внутреннего произведения.

LaTeX

$$Δ f = f^''(z)_{zz̄} + f^''(z)_{z z} в комплексной форме$$

Lean4

/-- Special case of the standard formula for functions on `ℂ`, with the standard real inner product
structure.
-/
theorem laplacianWithin_eq_iteratedFDerivWithin_complexPlane {e : ℂ} {s : Set ℂ} (f : ℂ → F) (hs : UniqueDiffOn ℝ s)
    (he : e ∈ s) :
    (Δ[s]f) e = iteratedFDerivWithin ℝ 2 f s e ![1, 1] + iteratedFDerivWithin ℝ 2 f s e ![Complex.I, Complex.I] := by
  simp [laplacianWithin_eq_iteratedFDerivWithin_orthonormalBasis f hs he Complex.orthonormalBasisOneI]

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