import Mathlib.Analysis.Analytic.Constructions import Mathlib.Analysis.Analytic.IsolatedZeros import Mathlib.Analysis.Complex.Basic theorem AnalyticOn.order_eq_nat_iff {f : ℂ → ℂ} {U : Set ℂ} {z₀ : ℂ} (hf : AnalyticOn ℂ f U) (hz₀ : z₀ ∈ U) (n : ℕ) : (hf z₀ hz₀).order = ↑n ↔ ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ g z₀ ≠ 0 ∧ ∀ z, f z = (z - z₀) ^ n • g z := by constructor -- Direction → intro hn obtain ⟨gloc, h₁gloc, h₂gloc, h₃gloc⟩ := (AnalyticAt.order_eq_nat_iff (hf z₀ hz₀) n).1 hn -- Define a candidate function; this is (f z) / (z - z₀) ^ n with the -- removable singularity removed let g : ℂ → ℂ := fun z ↦ if z = z₀ then gloc z₀ else (f z) / (z - z₀) ^ n -- Describe g near z₀ have g_near_z₀ : ∀ᶠ (z : ℂ) in nhds z₀, g z = gloc z := by rw [eventually_nhds_iff] obtain ⟨t, h₁t, h₂t, h₃t⟩ := eventually_nhds_iff.1 h₃gloc use t constructor · intro y h₁y by_cases h₂y : y = z₀ · dsimp [g]; simp [h₂y] · dsimp [g]; simp [h₂y] rw [div_eq_iff_mul_eq, eq_comm, mul_comm] exact h₁t y h₁y norm_num rw [sub_eq_zero] tauto · constructor · assumption · assumption -- Describe g near points z₁ that are different from z₀ have g_near_z₁ {z₁ : ℂ} : z₁ ≠ z₀ → ∀ᶠ (z : ℂ) in nhds z₁, g z = f z / (z - z₀) ^ n := by intro hz₁ rw [eventually_nhds_iff] use {z₀}ᶜ constructor · intro y hy simp at hy simp [g, hy] · exact ⟨isOpen_compl_singleton, hz₁⟩ -- Use g and show that it has all required properties use g constructor · -- AnalyticOn ℂ g U intro z h₁z by_cases h₂z : z = z₀ · rw [h₂z] apply AnalyticAt.congr h₁gloc exact Filter.EventuallyEq.symm g_near_z₀ · simp_rw [eq_comm] at g_near_z₁ apply AnalyticAt.congr _ (g_near_z₁ h₂z) apply AnalyticAt.div exact hf z h₁z apply AnalyticAt.pow apply AnalyticAt.sub apply analyticAt_id apply analyticAt_const simp rw [sub_eq_zero] tauto · constructor · simp [g]; tauto · intro z by_cases h₂z : z = z₀ · rw [h₂z, g_near_z₀.self_of_nhds] exact h₃gloc.self_of_nhds · rw [(g_near_z₁ h₂z).self_of_nhds] simp [h₂z] rw [div_eq_mul_inv, mul_comm, mul_assoc, inv_mul_cancel] simp; norm_num rw [sub_eq_zero] tauto -- direction ← intro h obtain ⟨g, h₁g, h₂g, h₃g⟩ := h rw [AnalyticAt.order_eq_nat_iff] use g exact ⟨h₁g z₀ hz₀, ⟨h₂g, Filter.eventually_of_forall h₃g⟩⟩ theorem AnalyticAt.order_mul {f₁ f₂ : ℂ → ℂ} {z₀ : ℂ} (hf₁ : AnalyticAt ℂ f₁ z₀) (hf₂ : AnalyticAt ℂ f₂ z₀) : (AnalyticAt.mul hf₁ hf₂).order = hf₁.order + hf₂.order := by by_cases h₂f₁ : hf₁.order = ⊤ · simp [h₂f₁] rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff] rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff] at h₂f₁ obtain ⟨t, h₁t, h₂t, h₃t⟩ := h₂f₁ use t constructor · intro y hy rw [h₁t y hy] ring · exact ⟨h₂t, h₃t⟩ · by_cases h₂f₂ : hf₂.order = ⊤ · simp [h₂f₂] rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff] rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff] at h₂f₂ obtain ⟨t, h₁t, h₂t, h₃t⟩ := h₂f₂ use t constructor · intro y hy rw [h₁t y hy] ring · exact ⟨h₂t, h₃t⟩ · obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (AnalyticAt.order_eq_nat_iff hf₁ ↑hf₁.order.toNat).1 (eq_comm.1 (ENat.coe_toNat h₂f₁)) obtain ⟨g₂, h₁g₂, h₂g₂, h₃g₂⟩ := (AnalyticAt.order_eq_nat_iff hf₂ ↑hf₂.order.toNat).1 (eq_comm.1 (ENat.coe_toNat h₂f₂)) rw [← ENat.coe_toNat h₂f₁, ← ENat.coe_toNat h₂f₂, ← ENat.coe_add] rw [AnalyticAt.order_eq_nat_iff (AnalyticAt.mul hf₁ hf₂) ↑(hf₁.order.toNat + hf₂.order.toNat)] use g₁ * g₂ constructor · exact AnalyticAt.mul h₁g₁ h₁g₂ · constructor · simp; tauto · obtain ⟨t₁, h₁t₁, h₂t₁, h₃t₁⟩ := eventually_nhds_iff.1 h₃g₁ obtain ⟨t₂, h₁t₂, h₂t₂, h₃t₂⟩ := eventually_nhds_iff.1 h₃g₂ rw [eventually_nhds_iff] use t₁ ∩ t₂ constructor · intro y hy rw [h₁t₁ y hy.1, h₁t₂ y hy.2] simp; ring · constructor · exact IsOpen.inter h₂t₁ h₂t₂ · exact Set.mem_inter h₃t₁ h₃t₂ theorem AnalyticOn.eliminateZeros {f : ℂ → ℂ} {U : Set ℂ} {A : Finset U} (hf : AnalyticOn ℂ f U) (n : ℂ → ℕ) : (∀ a ∈ A, (hf a.1 a.2).order = n a) → ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ a ∈ A, g a ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (n a)) • g z := by apply Finset.induction (α := U) (p := fun A ↦ (∀ a ∈ A, (hf a.1 a.2).order = n a) → ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ a ∈ A, g a ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (n a)) • g z) -- case empty simp use f simp exact hf -- case insert intro b₀ B hb iHyp intro hBinsert obtain ⟨g₀, h₁g₀, h₂g₀, h₃g₀⟩ := iHyp (fun a ha ↦ hBinsert a (Finset.mem_insert_of_mem ha)) have : (h₁g₀ b₀ b₀.2).order = n b₀ := by rw [← hBinsert b₀ (Finset.mem_insert_self b₀ B)] let φ := fun z ↦ (∏ a ∈ B, (z - a.1) ^ n a.1) have : f = fun z ↦ φ z * g₀ z := by funext z rw [h₃g₀ z] rfl simp_rw [this] have h₁φ : AnalyticAt ℂ φ b₀ := by dsimp [φ] apply Finset.analyticAt_prod intro b _ apply AnalyticAt.pow apply AnalyticAt.sub apply analyticAt_id ℂ exact analyticAt_const have h₂φ : h₁φ.order = (0 : ℕ) := by rw [AnalyticAt.order_eq_nat_iff h₁φ 0] use φ constructor · assumption · constructor · dsimp [φ] push_neg rw [Finset.prod_ne_zero_iff] intro a ha simp have : ¬ (b₀.1 - a.1 = 0) := by by_contra C rw [sub_eq_zero] at C rw [SetCoe.ext C] at hb tauto tauto · simp rw [AnalyticAt.order_mul h₁φ (h₁g₀ b₀ b₀.2)] rw [h₂φ] simp obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (AnalyticOn.order_eq_nat_iff h₁g₀ b₀.2 (n b₀)).1 this use g₁ constructor · exact h₁g₁ · constructor · intro a h₁a by_cases h₂a : a = b₀ · rwa [h₂a] · let A' := Finset.mem_of_mem_insert_of_ne h₁a h₂a let B' := h₃g₁ a let C' := h₂g₀ a A' rw [B'] at C' exact right_ne_zero_of_smul C' · intro z let A' := h₃g₀ z rw [h₃g₁ z] at A' rw [A'] rw [← smul_assoc] congr simp rw [Finset.prod_insert] ring exact hb