470 lines
14 KiB
Plaintext
470 lines
14 KiB
Plaintext
import Mathlib.Analysis.Complex.TaylorSeries
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import Mathlib.Data.ENNReal.Basic
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noncomputable def primitive
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] :
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ℂ → (ℂ → E) → (ℂ → E) := by
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intro z₀
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intro f
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exact fun z ↦ (∫ (x : ℝ) in z₀.re..z.re, f ⟨x, z₀.im⟩) + Complex.I • ∫ (x : ℝ) in z₀.im..z.im, f ⟨z.re, x⟩
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theorem primitive_zeroAtBasepoint
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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(f : ℂ → E)
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(z₀ : ℂ) :
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(primitive z₀ f) z₀ = 0 := by
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unfold primitive
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simp
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theorem primitive_fderivAtBasepointZero
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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(f : ℂ → E)
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(hf : Continuous f) :
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HasDerivAt (primitive 0 f) (f 0) 0 := by
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unfold primitive
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simp
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apply hasDerivAt_iff_isLittleO.2
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simp
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rw [Asymptotics.isLittleO_iff]
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intro c hc
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have {z : ℂ} {e : E} : z • e = (∫ (_ : ℝ) in (0)..(z.re), e) + Complex.I • ∫ (_ : ℝ) in (0)..(z.im), e:= by
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simp
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rw [smul_comm]
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rw [← smul_assoc]
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simp
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have : z.re • e = (z.re : ℂ) • e := by exact rfl
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rw [this, ← add_smul]
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simp
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conv =>
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left
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intro x
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left
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arg 1
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arg 2
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rw [this]
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have {A B C D :E} : (A + B) - (C + D) = (A - C) + (B - D) := by
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abel
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have t₀ {r : ℝ} : IntervalIntegrable (fun x => f { re := x, im := 0 }) MeasureTheory.volume 0 r := by
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apply Continuous.intervalIntegrable
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apply Continuous.comp
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exact hf
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have : (fun x => ({ re := x, im := 0 } : ℂ)) = Complex.ofRealLI := by rfl
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rw [this]
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continuity
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have t₁ {r : ℝ} : IntervalIntegrable (fun _ => f 0) MeasureTheory.volume 0 r := by
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apply Continuous.intervalIntegrable
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apply Continuous.comp
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exact hf
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fun_prop
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have t₂ {a b : ℝ} : IntervalIntegrable (fun x_1 => f { re := a, im := x_1 }) MeasureTheory.volume 0 b := by
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apply Continuous.intervalIntegrable
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apply Continuous.comp hf
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have : (Complex.mk a) = (fun x => Complex.I • Complex.ofRealCLM x + { re := a, im := 0 }) := by
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funext x
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apply Complex.ext
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rw [Complex.add_re]
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simp
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simp
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rw [this]
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apply Continuous.add
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continuity
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fun_prop
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have t₃ {a : ℝ} : IntervalIntegrable (fun _ => f 0) MeasureTheory.volume 0 a := by
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apply Continuous.intervalIntegrable
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apply Continuous.comp
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exact hf
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fun_prop
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conv =>
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left
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intro x
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left
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arg 1
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rw [this]
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rw [← smul_sub]
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rw [← intervalIntegral.integral_sub t₀ t₁]
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rw [← intervalIntegral.integral_sub t₂ t₃]
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rw [Filter.eventually_iff_exists_mem]
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let s := f⁻¹' Metric.ball (f 0) (c / (4 : ℝ))
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have h₁s : IsOpen s := IsOpen.preimage hf Metric.isOpen_ball
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have h₂s : 0 ∈ s := by
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apply Set.mem_preimage.mpr
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apply Metric.mem_ball_self
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linarith
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obtain ⟨ε, h₁ε, h₂ε⟩ := Metric.isOpen_iff.1 h₁s 0 h₂s
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have h₃ε : ∀ y ∈ Metric.ball 0 ε, ‖(f y) - (f 0)‖ < (c / (4 : ℝ)) := by
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intro y hy
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apply mem_ball_iff_norm.mp (h₂ε hy)
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use Metric.ball 0 (ε / (4 : ℝ))
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constructor
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· apply Metric.ball_mem_nhds 0
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linarith
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· intro y hy
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have h₁y : |y.re| < ε / 4 := by
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calc |y.re|
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_ ≤ Complex.abs y := by apply Complex.abs_re_le_abs
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_ < ε / 4 := by
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let A := mem_ball_iff_norm.1 hy
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simp at A
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linarith
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have h₂y : |y.im| < ε / 4 := by
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calc |y.im|
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_ ≤ Complex.abs y := by apply Complex.abs_im_le_abs
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_ < ε / 4 := by
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let A := mem_ball_iff_norm.1 hy
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simp at A
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linarith
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have intervalComputation {x' y' : ℝ} (h : x' ∈ Ι 0 y') : |x'| ≤ |y'| := by
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let A := h.1
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let B := h.2
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rcases le_total 0 y' with hy | hy
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· simp [hy] at A
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simp [hy] at B
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rw [abs_of_nonneg hy]
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rw [abs_of_nonneg (le_of_lt A)]
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exact B
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· simp [hy] at A
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simp [hy] at B
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rw [abs_of_nonpos hy]
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rw [abs_of_nonpos]
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linarith [h.1]
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exact B
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have t₁ : ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ ≤ (c / (4 : ℝ)) * |y.re - 0| := by
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apply intervalIntegral.norm_integral_le_of_norm_le_const
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intro x hx
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have h₁x : |x| < ε / 4 := by
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calc |x|
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_ ≤ |y.re| := intervalComputation hx
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_ < ε / 4 := h₁y
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apply le_of_lt
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apply h₃ε { re := x, im := 0 }
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rw [mem_ball_iff_norm]
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simp
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have : { re := x, im := 0 } = (x : ℂ) := by rfl
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rw [this]
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rw [Complex.abs_ofReal]
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linarith
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have t₂ : ‖∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ ≤ (c / (4 : ℝ)) * |y.im - 0| := by
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apply intervalIntegral.norm_integral_le_of_norm_le_const
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intro x hx
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have h₁x : |x| < ε / 4 := by
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calc |x|
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_ ≤ |y.im| := intervalComputation hx
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_ < ε / 4 := h₂y
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apply le_of_lt
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apply h₃ε { re := y.re, im := x }
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simp
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calc Complex.abs { re := y.re, im := x }
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_ ≤ |y.re| + |x| := by
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apply Complex.abs_le_abs_re_add_abs_im { re := y.re, im := x }
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_ < ε := by
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linarith
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calc ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0) + Complex.I • ∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖
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_ ≤ ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ + ‖Complex.I • ∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ := by
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apply norm_add_le
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_ ≤ ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ + ‖∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ := by
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simp
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rw [norm_smul]
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simp
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_ ≤ (c / (4 : ℝ)) * |y.re - 0| + (c / (4 : ℝ)) * |y.im - 0| := by
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apply add_le_add
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exact t₁
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exact t₂
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_ ≤ (c / (4 : ℝ)) * (|y.re| + |y.im|) := by
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simp
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rw [mul_add]
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_ ≤ (c / (4 : ℝ)) * (4 * ‖y‖) := by
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have : |y.re| + |y.im| ≤ 4 * ‖y‖ := by
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calc |y.re| + |y.im|
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_ ≤ ‖y‖ + ‖y‖ := by
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apply add_le_add
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apply Complex.abs_re_le_abs
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apply Complex.abs_im_le_abs
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_ ≤ 4 * ‖y‖ := by
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rw [← two_mul]
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apply mul_le_mul
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linarith
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rfl
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exact norm_nonneg y
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linarith
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apply mul_le_mul
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rfl
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exact this
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apply add_nonneg
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apply abs_nonneg
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apply abs_nonneg
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linarith
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_ ≤ c * ‖y‖ := by
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linarith
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theorem primitive_translation
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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(f : ℂ → E)
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(z₀ t : ℂ) :
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primitive z₀ (f ∘ fun z ↦ (z - t)) = ((primitive (z₀ - t) f) ∘ fun z ↦ (z - t)) := by
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funext z
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unfold primitive
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simp
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let g : ℝ → E := fun x ↦ f ( {re := x, im := z₀.im - t.im} )
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have {x : ℝ} : f ({ re := x, im := z₀.im } - t) = g (1*x - t.re) := by
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congr 1
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apply Complex.ext <;> simp
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conv =>
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left
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left
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arg 1
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intro x
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rw [this]
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rw [intervalIntegral.integral_comp_mul_sub g one_ne_zero (t.re)]
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simp
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congr 1
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let g : ℝ → E := fun x ↦ f ( {re := z.re - t.re, im := x} )
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have {x : ℝ} : f ({ re := z.re, im := x} - t) = g (1*x - t.im) := by
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congr 1
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apply Complex.ext <;> simp
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conv =>
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left
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arg 1
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intro x
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rw [this]
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rw [intervalIntegral.integral_comp_mul_sub g one_ne_zero (t.im)]
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simp
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theorem primitive_hasDerivAtBasepoint
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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{f : ℂ → E}
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(hf : Continuous f)
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(z₀ : ℂ) :
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HasDerivAt (primitive z₀ f) (f z₀) z₀ := by
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let g := f ∘ fun z ↦ z + z₀
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have : Continuous g := by continuity
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let A := primitive_fderivAtBasepointZero g this
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simp at A
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let B := primitive_translation g z₀ z₀
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simp at B
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have : (g ∘ fun z ↦ (z - z₀)) = f := by
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funext z
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dsimp [g]
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simp
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rw [this] at B
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rw [B]
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have : f z₀ = (1 : ℂ) • (f z₀) := by
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exact (MulAction.one_smul (f z₀)).symm
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conv =>
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arg 2
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rw [this]
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apply HasDerivAt.scomp
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simp
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have : g 0 = f z₀ := by simp [g]
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rw [← this]
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exact A
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apply HasDerivAt.sub_const
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have : (fun (x : ℂ) ↦ x) = id := by
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funext x
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simp
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rw [this]
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exact hasDerivAt_id z₀
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lemma integrability₁
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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(f : ℂ → E)
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(hf : Differentiable ℂ f)
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(a₁ a₂ b : ℝ) :
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IntervalIntegrable (fun x => f { re := x, im := b }) MeasureTheory.volume a₁ a₂ := by
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apply Continuous.intervalIntegrable
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apply Continuous.comp
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exact Differentiable.continuous hf
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have : ((fun x => { re := x, im := b }) : ℝ → ℂ) = (fun x => Complex.ofRealCLM x + { re := 0, im := b }) := by
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funext x
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apply Complex.ext
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rw [Complex.add_re]
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simp
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rw [Complex.add_im]
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simp
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rw [this]
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continuity
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lemma integrability₂
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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(f : ℂ → E)
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(hf : Differentiable ℂ f)
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(a₁ a₂ b : ℝ) :
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IntervalIntegrable (fun x => f { re := b, im := x }) MeasureTheory.volume a₁ a₂ := by
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apply Continuous.intervalIntegrable
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apply Continuous.comp
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exact Differentiable.continuous hf
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have : (Complex.mk b) = (fun x => Complex.I • Complex.ofRealCLM x + { re := b, im := 0 }) := by
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funext x
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apply Complex.ext
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rw [Complex.add_re]
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simp
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simp
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rw [this]
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apply Continuous.add
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continuity
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fun_prop
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theorem primitive_additivity
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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(f : ℂ → E)
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(z₀ : ℂ)
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(R : ℝ)
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(hf : DifferentiableOn ℂ f (Metric.ball z₀ R))
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(z₁ : ℂ)
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(hz₁ : z₁ ∈ Metric.ball z₀ R) :
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∀ z ∈ Metric.ball z₁ (R - ‖z₁‖), (primitive z₀ f z) - (primitive z₁ f z) - (primitive z₀ f z₁) = 0 := by
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intro z _
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unfold primitive
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have : (∫ (x : ℝ) in z₀.re..z.re, f { re := x, im := z₀.im }) = (∫ (x : ℝ) in z₀.re..z₁.re, f { re := x, im := z₀.im }) + (∫ (x : ℝ) in z₁.re..z.re, f { re := x, im := z₀.im }) := by
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rw [intervalIntegral.integral_add_adjacent_intervals]
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sorry --apply integrability₁ f hf
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sorry --apply integrability₁ f hf
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rw [this]
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have : (∫ (x : ℝ) in z₀.im..z.im, f { re := z.re, im := x }) = (∫ (x : ℝ) in z₀.im..z₁.im, f { re := z.re, im := x }) + (∫ (x : ℝ) in z₁.im..z.im, f { re := z.re, im := x }) := by
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rw [intervalIntegral.integral_add_adjacent_intervals]
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sorry --apply integrability₂ f hf
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sorry --apply integrability₂ f hf
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rw [this]
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simp
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have {a b c d e f g h : E} : (a + b) + (c + d) - (e + f) - (g + h) = b + (a - g) - e - f + d - h + (c) := by
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abel
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rw [this]
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simp
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have H : DifferentiableOn ℂ f (Set.uIcc z₁.re z.re ×ℂ Set.uIcc z₀.im z₁.im) := by
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apply DifferentiableOn.mono hf
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intro x hx
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simp
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let A₀ : dist x.re z₀.re ≤ dist x.re z₁.re + dist z₁.re z₀.re := by apply dist_triangle
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sorry
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let A := Complex.integral_boundary_rect_eq_zero_of_differentiableOn f ⟨z₁.re, z₀.im⟩ ⟨z.re, z₁.im⟩ H
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have {x : ℝ} {w : ℂ} : ↑x + w.im * Complex.I = { re := x, im := w.im } := by
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apply Complex.ext
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· simp
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· simp
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simp_rw [this] at A
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have {x : ℝ} {w : ℂ} : w.re + x * Complex.I = { re := w.re, im := x } := by
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apply Complex.ext
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· simp
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· simp
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simp_rw [this] at A
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rw [← A]
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abel
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theorem primitive_additivity'
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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(f : ℂ → E)
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(hf : Differentiable ℂ f)
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(z₀ z₁ : ℂ) :
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primitive z₀ f = fun z ↦ (primitive z₁ f) z + (primitive z₀ f z₁) := by
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nth_rw 1 [← sub_zero (primitive z₀ f)]
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rw [← primitive_additivity f hf z₀ z₁]
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funext z
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simp
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abel
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theorem primitive_hasDerivAt
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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{f : ℂ → E}
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(hf : Differentiable ℂ f)
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(z₀ z : ℂ) :
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HasDerivAt (primitive z₀ f) (f z) z := by
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rw [primitive_additivity' f hf z₀ z]
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rw [← add_zero (f z)]
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apply HasDerivAt.add
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apply primitive_hasDerivAtBasepoint
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exact hf.continuous
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apply hasDerivAt_const
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theorem primitive_differentiable
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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{f : ℂ → E}
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(hf : Differentiable ℂ f)
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(z₀ : ℂ) :
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Differentiable ℂ (primitive z₀ f) := by
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intro z
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exact (primitive_hasDerivAt hf z₀ z).differentiableAt
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theorem primitive_hasFderivAt
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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{f : ℂ → E}
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(hf : Differentiable ℂ f)
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(z₀ : ℂ) :
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∀ z, HasFDerivAt (primitive z₀ f) ((ContinuousLinearMap.lsmul ℂ ℂ).flip (f z)) z := by
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intro z
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rw [hasFDerivAt_iff_hasDerivAt]
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simp
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exact primitive_hasDerivAt hf z₀ z
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theorem primitive_hasFderivAt'
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{f : ℂ → ℂ}
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(hf : Differentiable ℂ f)
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(z₀ : ℂ) :
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∀ z, HasFDerivAt (primitive z₀ f) (ContinuousLinearMap.lsmul ℂ ℂ (f z)) z := by
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intro z
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rw [hasFDerivAt_iff_hasDerivAt]
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simp
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exact primitive_hasDerivAt hf z₀ z
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theorem primitive_fderiv
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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{f : ℂ → E}
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(hf : Differentiable ℂ f)
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(z₀ : ℂ) :
|
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∀ z, (fderiv ℂ (primitive z₀ f) z) = (ContinuousLinearMap.lsmul ℂ ℂ).flip (f z) := by
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intro z
|
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apply HasFDerivAt.fderiv
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exact primitive_hasFderivAt hf z₀ z
|
||
|
||
|
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theorem primitive_fderiv'
|
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{f : ℂ → ℂ}
|
||
(hf : Differentiable ℂ f)
|
||
(z₀ : ℂ) :
|
||
∀ z, (fderiv ℂ (primitive z₀ f) z) = ContinuousLinearMap.lsmul ℂ ℂ (f z) := by
|
||
intro z
|
||
apply HasFDerivAt.fderiv
|
||
exact primitive_hasFderivAt' hf z₀ z
|