diff --git a/Nevanlinna/complexHarmonic.lean b/Nevanlinna/complexHarmonic.lean index 8c19276..7cfa999 100644 --- a/Nevanlinna/complexHarmonic.lean +++ b/Nevanlinna/complexHarmonic.lean @@ -12,19 +12,13 @@ import Mathlib.Analysis.RCLike.Basic import Mathlib.Topology.Algebra.InfiniteSum.Module import Mathlib.Topology.Instances.RealVectorSpace import Nevanlinna.cauchyRiemann +import Nevanlinna.laplace import Nevanlinna.partialDeriv - -noncomputable def Complex.laplace : (ℂ → ℂ) → (ℂ → ℂ) := by - intro f - let fx := partialDeriv ℝ 1 f - let fxx := partialDeriv ℝ 1 fx - let fy := partialDeriv ℝ Complex.I f - let fyy := partialDeriv ℝ Complex.I fy - exact fxx + fyy +variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] -def Harmonic (f : ℂ → ℂ) : Prop := +def Harmonic (f : ℂ → F) : Prop := (ContDiff ℝ 2 f) ∧ (∀ z, Complex.laplace f z = 0) @@ -86,3 +80,10 @@ theorem holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) : exact fI_is_real_differentiable -- Differentiable ℝ (Real.partialDeriv 1 f) exact fI_is_real_differentiable + + +theorem re_of_holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) : + Harmonic (Complex.reCLM ∘ f) := by + + + sorry diff --git a/Nevanlinna/laplace.lean b/Nevanlinna/laplace.lean new file mode 100644 index 0000000..49d35bf --- /dev/null +++ b/Nevanlinna/laplace.lean @@ -0,0 +1,25 @@ +import Mathlib.Data.Fin.Tuple.Basic +import Mathlib.Analysis.Complex.Basic +import Mathlib.Analysis.Complex.TaylorSeries +import Mathlib.Analysis.Calculus.LineDeriv.Basic +import Mathlib.Analysis.Calculus.ContDiff.Defs +import Mathlib.Analysis.Calculus.FDeriv.Basic +import Mathlib.Analysis.Calculus.FDeriv.Symmetric +import Mathlib.Data.Complex.Module +import Mathlib.Data.Complex.Order +import Mathlib.Data.Complex.Exponential +import Mathlib.Analysis.RCLike.Basic +import Mathlib.Topology.Algebra.InfiniteSum.Module +import Mathlib.Topology.Instances.RealVectorSpace +import Nevanlinna.cauchyRiemann +import Nevanlinna.partialDeriv + +variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] + +noncomputable def Complex.laplace : (ℂ → F) → (ℂ → F) := by + intro f + let fx := partialDeriv ℝ 1 f + let fxx := partialDeriv ℝ 1 fx + let fy := partialDeriv ℝ Complex.I f + let fyy := partialDeriv ℝ Complex.I fy + exact fxx + fyy