File: Mathlib/MeasureTheory/Integral/CircleTransform.lean

full_name: Complex.circleTransform
code: /-- Given a function `f : ℂ → E`, `circleTransform R z w f` is the function mapping `θ` to
`(2 * ↑π * I)⁻¹ • deriv (circleMap z R) θ • ((circleMap z R θ) - w)⁻¹ • f (circleMap z R θ)`.

If `f` is differentiable and `w` is in the interior of the ball, then the integral from `0` to
`2 * π` of this gives the value `f(w)`. -/
def circleTransform (f : ℂ → E) (θ : ℝ) : E 
goal: E

full_name: Complex.circleTransformDeriv
code: /-- The derivative of `circleTransform` w.r.t `w`. -/
def circleTransformDeriv (f : ℂ → E) (θ : ℝ) : E 
goal: E

full_name: Complex.circleTransformDeriv_periodic
code: theorem circleTransformDeriv_periodic (f : ℂ → E) :
    Periodic (circleTransformDeriv R z w f) (2 * π) 
goal: Function.Periodic (Complex.circleTransformDeriv R z w f) (2 * π)

full_name: Complex.circleTransformDeriv_eq
code: theorem circleTransformDeriv_eq (f : ℂ → E) : circleTransformDeriv R z w f =
    fun θ => (circleMap z R θ - w)⁻¹ • circleTransform R z w f θ 
goal: Complex.circleTransformDeriv R z w f = fun θ => (circleMap z R θ - w)⁻¹ • Complex.circleTransform R z w f θ

full_name: Complex.integral_circleTransform
code: theorem integral_circleTransform (f : ℂ → E) :
    (∫ θ : ℝ in (0)..2 * π, circleTransform R z w f θ) =
      (2 * ↑π * I)⁻¹ • ∮ z in C(z, R), (z - w)⁻¹ • f z 
goal: ∫ (θ : ℝ) in 0 ..2 * π, Complex.circleTransform R z w f θ =
  (2 * ↑π * Complex.I)⁻¹ • ∮ (z : ℂ) in C(z, R), (z - w)⁻¹ • f z

full_name: Complex.continuous_circleTransform
code: theorem continuous_circleTransform {R : ℝ} (hR : 0 < R) {f : ℂ → E} {z w : ℂ}
    (hf : ContinuousOn f <| sphere z R) (hw : w ∈ ball z R) :
    Continuous (circleTransform R z w f) 
goal: Continuous (Complex.circleTransform R z w f)

full_name: Complex.continuous_circleTransformDeriv
code: theorem continuous_circleTransformDeriv {R : ℝ} (hR : 0 < R) {f : ℂ → E} {z w : ℂ}
    (hf : ContinuousOn f (sphere z R)) (hw : w ∈ ball z R) :
    Continuous (circleTransformDeriv R z w f) 
goal: Continuous (Complex.circleTransformDeriv R z w f)

full_name: Complex.circleTransformBoundingFunction
code: /-- A useful bound for circle integrals (with complex codomain)-/
def circleTransformBoundingFunction (R : ℝ) (z : ℂ) (w : ℂ × ℝ) : ℂ 
goal: ℂ

full_name: Complex.continuousOn_prod_circle_transform_function
code: theorem continuousOn_prod_circle_transform_function {R r : ℝ} (hr : r < R) {z : ℂ} :
    ContinuousOn (fun w : ℂ × ℝ => (circleMap z R w.snd - w.fst)⁻¹ ^ 2)
      (closedBall z r ×ˢ univ) 
goal: ContinuousOn (fun w => (circleMap z R w.2 - w.1)⁻¹ ^ 2) (Metric.closedBall z r ×ˢ Set.univ)

full_name: Complex.continuousOn_abs_circleTransformBoundingFunction
code: theorem continuousOn_abs_circleTransformBoundingFunction {R r : ℝ} (hr : r < R) (z : ℂ) :
    ContinuousOn (abs ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ) 
goal: ContinuousOn (⇑Complex.abs ∘ Complex.circleTransformBoundingFunction R z) (Metric.closedBall z r ×ˢ Set.univ)

full_name: Complex.abs_circleTransformBoundingFunction_le
code: theorem abs_circleTransformBoundingFunction_le {R r : ℝ} (hr : r < R) (hr' : 0 ≤ r) (z : ℂ) :
    ∃ x : closedBall z r ×ˢ [[0, 2 * π]], ∀ y : closedBall z r ×ˢ [[0, 2 * π]],
    abs (circleTransformBoundingFunction R z y) ≤ abs (circleTransformBoundingFunction R z x) 
goal: ∃ x,
  ∀ (y : ↑(Metric.closedBall z r ×ˢ [[0, 2 * π]])),
    Complex.abs (Complex.circleTransformBoundingFunction R z ↑y) ≤
      Complex.abs (Complex.circleTransformBoundingFunction R z ↑x)

full_name: Complex.circleTransformDeriv_bound
code: /-- The derivative of a `circleTransform` is locally bounded. -/
theorem circleTransformDeriv_bound {R : ℝ} (hR : 0 < R) {z x : ℂ} {f : ℂ → ℂ} (hx : x ∈ ball z R)
    (hf : ContinuousOn f (sphere z R)) : ∃ B ε : ℝ, 0 < ε ∧
      ball x ε ⊆ ball z R ∧ ∀ (t : ℝ), ∀ y ∈ ball x ε, ‖circleTransformDeriv R z y f t‖ ≤ B 
goal: ∃ B ε,
  0 < ε ∧
    Metric.ball x ε ⊆ Metric.ball z R ∧ ∀ (t : ℝ), ∀ y ∈ Metric.ball x ε, ‖Complex.circleTransformDeriv R z y f t‖ ≤ B