File: Mathlib/MeasureTheory/Integral/CircleIntegral.lean

full_name: circleMap
code: /-- The exponential map $θ ↦ c + R e^{θi}$. The range of this map is the circle in `ℂ` with center
`c` and radius `|R|`. -/
def circleMap (c : ℂ) (R : ℝ) : ℝ → ℂ 
goal: ℝ → ℂ

full_name: periodic_circleMap
code: /-- `circleMap` is `2π`-periodic. -/
theorem periodic_circleMap (c : ℂ) (R : ℝ) : Periodic (circleMap c R) (2 * π) 
goal: Function.Periodic (circleMap c R) (2 * π)

full_name: Set.Countable.preimage_circleMap
code: theorem Set.Countable.preimage_circleMap {s : Set ℂ} (hs : s.Countable) (c : ℂ) {R : ℝ}
    (hR : R ≠ 0) : (circleMap c R ⁻¹' s).Countable 
goal: (circleMap c R ⁻¹' s).Countable

full_name: circleMap_sub_center
code: @[simp]
theorem circleMap_sub_center (c : ℂ) (R : ℝ) (θ : ℝ) : circleMap c R θ - c = circleMap 0 R θ 
goal: circleMap c R θ - c = circleMap 0 R θ

full_name: circleMap_zero
code: theorem circleMap_zero (R θ : ℝ) : circleMap 0 R θ = R * exp (θ * I) 
goal: circleMap 0 R θ = ↑R * cexp (↑θ * Complex.I)

full_name: abs_circleMap_zero
code: @[simp]
theorem abs_circleMap_zero (R : ℝ) (θ : ℝ) : abs (circleMap 0 R θ) = |R| 
goal: Complex.abs (circleMap 0 R θ) = |R|

full_name: circleMap_mem_sphere'
code: theorem circleMap_mem_sphere' (c : ℂ) (R : ℝ) (θ : ℝ) : circleMap c R θ ∈ sphere c |R| 
goal: circleMap c R θ ∈ Metric.sphere c |R|

full_name: circleMap_mem_sphere
code: theorem circleMap_mem_sphere (c : ℂ) {R : ℝ} (hR : 0 ≤ R) (θ : ℝ) :
    circleMap c R θ ∈ sphere c R 
goal: circleMap c R θ ∈ Metric.sphere c R

full_name: circleMap_mem_closedBall
code: theorem circleMap_mem_closedBall (c : ℂ) {R : ℝ} (hR : 0 ≤ R) (θ : ℝ) :
    circleMap c R θ ∈ closedBall c R 
goal: circleMap c R θ ∈ Metric.closedBall c R

full_name: circleMap_not_mem_ball
code: theorem circleMap_not_mem_ball (c : ℂ) (R : ℝ) (θ : ℝ) : circleMap c R θ ∉ ball c R 
goal: circleMap c R θ ∉ Metric.ball c R

full_name: circleMap_ne_mem_ball
code: theorem circleMap_ne_mem_ball {c : ℂ} {R : ℝ} {w : ℂ} (hw : w ∈ ball c R) (θ : ℝ) :
    circleMap c R θ ≠ w 
goal: circleMap c R θ ≠ w

full_name: range_circleMap
code: /-- The range of `circleMap c R` is the circle with center `c` and radius `|R|`. -/
@[simp]
theorem range_circleMap (c : ℂ) (R : ℝ) : range (circleMap c R) = sphere c |R| 
goal: Set.range (circleMap c R) = Metric.sphere c |R|

full_name: image_circleMap_Ioc
code: /-- The image of `(0, 2π]` under `circleMap c R` is the circle with center `c` and radius `|R|`. -/
@[simp]
theorem image_circleMap_Ioc (c : ℂ) (R : ℝ) : circleMap c R '' Ioc 0 (2 * π) = sphere c |R| 
goal: circleMap c R '' Set.Ioc 0 (2 * π) = Metric.sphere c |R|

full_name: circleMap_eq_center_iff
code: @[simp]
theorem circleMap_eq_center_iff {c : ℂ} {R : ℝ} {θ : ℝ} : circleMap c R θ = c ↔ R = 0 
goal: circleMap c R θ = c ↔ R = 0

full_name: circleMap_zero_radius
code: @[simp]
theorem circleMap_zero_radius (c : ℂ) : circleMap c 0 = const ℝ c 
goal: circleMap c 0 = Function.const ℝ c

full_name: circleMap_ne_center
code: theorem circleMap_ne_center {c : ℂ} {R : ℝ} (hR : R ≠ 0) {θ : ℝ} : circleMap c R θ ≠ c 
goal: circleMap c R θ ≠ c

full_name: hasDerivAt_circleMap
code: theorem hasDerivAt_circleMap (c : ℂ) (R : ℝ) (θ : ℝ) :
    HasDerivAt (circleMap c R) (circleMap 0 R θ * I) θ 
goal: HasDerivAt (circleMap c R) (circleMap 0 R θ * Complex.I) θ

full_name: differentiable_circleMap
code: theorem differentiable_circleMap (c : ℂ) (R : ℝ) : Differentiable ℝ (circleMap c R) 
goal: Differentiable ℝ (circleMap c R)

full_name: continuous_circleMap
code: @[continuity, fun_prop]
theorem continuous_circleMap (c : ℂ) (R : ℝ) : Continuous (circleMap c R) 
goal: Continuous (circleMap c R)

full_name: measurable_circleMap
code: @[fun_prop, measurability]
theorem measurable_circleMap (c : ℂ) (R : ℝ) : Measurable (circleMap c R) 
goal: Measurable (circleMap c R)

full_name: deriv_circleMap
code: @[simp]
theorem deriv_circleMap (c : ℂ) (R : ℝ) (θ : ℝ) : deriv (circleMap c R) θ = circleMap 0 R θ * I 
goal: deriv (circleMap c R) θ = circleMap 0 R θ * Complex.I

full_name: deriv_circleMap_eq_zero_iff
code: theorem deriv_circleMap_eq_zero_iff {c : ℂ} {R : ℝ} {θ : ℝ} :
    deriv (circleMap c R) θ = 0 ↔ R = 0 
goal: deriv (circleMap c R) θ = 0 ↔ R = 0

full_name: deriv_circleMap_ne_zero
code: theorem deriv_circleMap_ne_zero {c : ℂ} {R : ℝ} {θ : ℝ} (hR : R ≠ 0) :
    deriv (circleMap c R) θ ≠ 0 
goal: deriv (circleMap c R) θ ≠ 0

full_name: lipschitzWith_circleMap
code: theorem lipschitzWith_circleMap (c : ℂ) (R : ℝ) : LipschitzWith (Real.nnabs R) (circleMap c R) 
goal: LipschitzWith (Real.nnabs R) (circleMap c R)

full_name: continuous_circleMap_inv
code: theorem continuous_circleMap_inv {R : ℝ} {z w : ℂ} (hw : w ∈ ball z R) :
    Continuous fun θ => (circleMap z R θ - w)⁻¹ 
goal: Continuous fun θ => (circleMap z R θ - w)⁻¹

full_name: CircleIntegrable
code: /-- We say that a function `f : ℂ → E` is integrable on the circle with center `c` and radius `R` if
the function `f ∘ circleMap c R` is integrable on `[0, 2π]`.

Note that the actual function used in the definition of `circleIntegral` is
`(deriv (circleMap c R) θ) • f (circleMap c R θ)`. Integrability of this function is equivalent
to integrability of `f ∘ circleMap c R` whenever `R ≠ 0`. -/
def CircleIntegrable (f : ℂ → E) (c : ℂ) (R : ℝ) : Prop 
goal: Prop

full_name: circleIntegrable_const
code: @[simp]
theorem circleIntegrable_const (a : E) (c : ℂ) (R : ℝ) : CircleIntegrable (fun _ => a) c R 
goal: CircleIntegrable (fun x => a) c R

full_name: CircleIntegrable.add
code: nonrec theorem add (hf : CircleIntegrable f c R) (hg : CircleIntegrable g c R) :
    CircleIntegrable (f + g) c R 
goal: CircleIntegrable (f + g) c R

full_name: CircleIntegrable.neg
code: nonrec theorem neg (hf : CircleIntegrable f c R) : CircleIntegrable (-f) c R 
goal: CircleIntegrable (-f) c R

full_name: CircleIntegrable.out
code: /-- The function we actually integrate over `[0, 2π]` in the definition of `circleIntegral` is
integrable. -/
theorem out [NormedSpace ℂ E] (hf : CircleIntegrable f c R) :
    IntervalIntegrable (fun θ : ℝ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0
      (2 * π) 
goal: IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) MeasureTheory.volume 0 (2 * π)

full_name: circleIntegrable_zero_radius
code: @[simp]
theorem circleIntegrable_zero_radius {f : ℂ → E} {c : ℂ} : CircleIntegrable f c 0 
goal: CircleIntegrable f c 0

full_name: circleIntegrable_iff
code: theorem circleIntegrable_iff [NormedSpace ℂ E] {f : ℂ → E} {c : ℂ} (R : ℝ) :
    CircleIntegrable f c R ↔ IntervalIntegrable (fun θ : ℝ =>
      deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π) 
goal: CircleIntegrable f c R ↔
  IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) MeasureTheory.volume 0 (2 * π)

full_name: ContinuousOn.circleIntegrable'
code: theorem ContinuousOn.circleIntegrable' {f : ℂ → E} {c : ℂ} {R : ℝ}
    (hf : ContinuousOn f (sphere c |R|)) : CircleIntegrable f c R 
goal: CircleIntegrable f c R

full_name: ContinuousOn.circleIntegrable
code: theorem ContinuousOn.circleIntegrable {f : ℂ → E} {c : ℂ} {R : ℝ} (hR : 0 ≤ R)
    (hf : ContinuousOn f (sphere c R)) : CircleIntegrable f c R 
goal: CircleIntegrable f c R

full_name: circleIntegrable_sub_zpow_iff
code: /-- The function `fun z ↦ (z - w) ^ n`, `n : ℤ`, is circle integrable on the circle with center `c`
and radius `|R|` if and only if `R = 0` or `0 ≤ n`, or `w` does not belong to this circle. -/
@[simp]
theorem circleIntegrable_sub_zpow_iff {c w : ℂ} {R : ℝ} {n : ℤ} :
    CircleIntegrable (fun z => (z - w) ^ n) c R ↔ R = 0 ∨ 0 ≤ n ∨ w ∉ sphere c |R| 
goal: CircleIntegrable (fun z => (z - w) ^ n) c R ↔ R = 0 ∨ 0 ≤ n ∨ w ∉ Metric.sphere c |R|

full_name: circleIntegrable_sub_inv_iff
code: @[simp]
theorem circleIntegrable_sub_inv_iff {c w : ℂ} {R : ℝ} :
    CircleIntegrable (fun z => (z - w)⁻¹) c R ↔ R = 0 ∨ w ∉ sphere c |R| 
goal: CircleIntegrable (fun z => (z - w)⁻¹) c R ↔ R = 0 ∨ w ∉ Metric.sphere c |R|

full_name: circleIntegral
code: /-- Definition for $\oint_{|z-c|=R} f(z)\,dz$. -/
def circleIntegral (f : ℂ → E) (c : ℂ) (R : ℝ) : E 
goal: E

full_name: «_aux_Mathlib_MeasureTheory_Integral_CircleIntegral___macroRules_term∮_InC(_,_),__1»
code: notation3 "∮ "(...)" in ""C("c", "R")"", "r:(scoped f => circleIntegral f c R) => r
goal: Lean.Macro

full_name: circleIntegral_def_Icc
code: theorem circleIntegral_def_Icc (f : ℂ → E) (c : ℂ) (R : ℝ) :
    (∮ z in C(c, R), f z) = ∫ θ in Icc 0 (2 * π),
    deriv (circleMap c R) θ • f (circleMap c R θ) 
goal: (∮ (z : ℂ) in C(c, R), f z) = ∫ (θ : ℝ) in Set.Icc 0 (2 * π), deriv (circleMap c R) θ • f (circleMap c R θ)

full_name: circleIntegral.integral_radius_zero
code: @[simp]
theorem integral_radius_zero (f : ℂ → E) (c : ℂ) : (∮ z in C(c, 0), f z) = 0 
goal: (∮ (z : ℂ) in C(c, 0), f z) = 0

full_name: circleIntegral.integral_congr
code: theorem integral_congr {f g : ℂ → E} {c : ℂ} {R : ℝ} (hR : 0 ≤ R) (h : EqOn f g (sphere c R)) :
    (∮ z in C(c, R), f z) = ∮ z in C(c, R), g z 
goal: (∮ (z : ℂ) in C(c, R), f z) = ∮ (z : ℂ) in C(c, R), g z

full_name: circleIntegral.integral_sub_inv_smul_sub_smul
code: theorem integral_sub_inv_smul_sub_smul (f : ℂ → E) (c w : ℂ) (R : ℝ) :
    (∮ z in C(c, R), (z - w)⁻¹ • (z - w) • f z) = ∮ z in C(c, R), f z 
goal: (∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • (z - w) • f z) = ∮ (z : ℂ) in C(c, R), f z

full_name: circleIntegral.integral_undef
code: theorem integral_undef {f : ℂ → E} {c : ℂ} {R : ℝ} (hf : ¬CircleIntegrable f c R) :
    (∮ z in C(c, R), f z) = 0 
goal: (∮ (z : ℂ) in C(c, R), f z) = 0

full_name: circleIntegral.integral_sub
code: theorem integral_sub {f g : ℂ → E} {c : ℂ} {R : ℝ} (hf : CircleIntegrable f c R)
    (hg : CircleIntegrable g c R) :
    (∮ z in C(c, R), f z - g z) = (∮ z in C(c, R), f z) - ∮ z in C(c, R), g z 
goal: (∮ (z : ℂ) in C(c, R), f z - g z) = (∮ (z : ℂ) in C(c, R), f z) - ∮ (z : ℂ) in C(c, R), g z

full_name: circleIntegral.norm_integral_le_of_norm_le_const'
code: theorem norm_integral_le_of_norm_le_const' {f : ℂ → E} {c : ℂ} {R C : ℝ}
    (hf : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ C) : ‖∮ z in C(c, R), f z‖ ≤ 2 * π * |R| * C 
goal: ‖∮ (z : ℂ) in C(c, R), f z‖ ≤ 2 * π * |R| * C

full_name: circleIntegral.norm_integral_le_of_norm_le_const
code: theorem norm_integral_le_of_norm_le_const {f : ℂ → E} {c : ℂ} {R C : ℝ} (hR : 0 ≤ R)
    (hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C) : ‖∮ z in C(c, R), f z‖ ≤ 2 * π * R * C 
goal: ‖∮ (z : ℂ) in C(c, R), f z‖ ≤ 2 * π * R * C

full_name: circleIntegral.norm_two_pi_i_inv_smul_integral_le_of_norm_le_const
code: theorem norm_two_pi_i_inv_smul_integral_le_of_norm_le_const {f : ℂ → E} {c : ℂ} {R C : ℝ}
    (hR : 0 ≤ R) (hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C) :
    ‖(2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), f z‖ ≤ R * C 
goal: ‖(2 * ↑π * Complex.I)⁻¹ • ∮ (z : ℂ) in C(c, R), f z‖ ≤ R * C

full_name: circleIntegral.norm_integral_lt_of_norm_le_const_of_lt
code: /-- If `f` is continuous on the circle `|z - c| = R`, `R > 0`, the `‖f z‖` is less than or equal to
`C : ℝ` on this circle, and this norm is strictly less than `C` at some point `z` of the circle,
then `‖∮ z in C(c, R), f z‖ < 2 * π * R * C`. -/
theorem norm_integral_lt_of_norm_le_const_of_lt {f : ℂ → E} {c : ℂ} {R C : ℝ} (hR : 0 < R)
    (hc : ContinuousOn f (sphere c R)) (hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C)
    (hlt : ∃ z ∈ sphere c R, ‖f z‖ < C) : ‖∮ z in C(c, R), f z‖ < 2 * π * R * C 
goal: ‖∮ (z : ℂ) in C(c, R), f z‖ < 2 * π * R * C

full_name: circleIntegral.integral_smul
code: @[simp]
theorem integral_smul {𝕜 : Type*} [RCLike 𝕜] [NormedSpace 𝕜 E] [SMulCommClass 𝕜 ℂ E] (a : 𝕜)
    (f : ℂ → E) (c : ℂ) (R : ℝ) : (∮ z in C(c, R), a • f z) = a • ∮ z in C(c, R), f z 
goal: (∮ (z : ℂ) in C(c, R), a • f z) = a • ∮ (z : ℂ) in C(c, R), f z

full_name: circleIntegral.integral_smul_const
code: @[simp]
theorem integral_smul_const (f : ℂ → ℂ) (a : E) (c : ℂ) (R : ℝ) :
    (∮ z in C(c, R), f z • a) = (∮ z in C(c, R), f z) • a 
goal: (∮ (z : ℂ) in C(c, R), f z • a) = (∮ (z : ℂ) in C(c, R), f z) • a

full_name: circleIntegral.integral_const_mul
code: @[simp]
theorem integral_const_mul (a : ℂ) (f : ℂ → ℂ) (c : ℂ) (R : ℝ) :
    (∮ z in C(c, R), a * f z) = a * ∮ z in C(c, R), f z 
goal: (∮ (z : ℂ) in C(c, R), a * f z) = a * ∮ (z : ℂ) in C(c, R), f z

full_name: circleIntegral.integral_sub_center_inv
code: @[simp]
theorem integral_sub_center_inv (c : ℂ) {R : ℝ} (hR : R ≠ 0) :
    (∮ z in C(c, R), (z - c)⁻¹) = 2 * π * I 
goal: (∮ (z : ℂ) in C(c, R), (z - c)⁻¹) = 2 * ↑π * Complex.I

full_name: circleIntegral.integral_eq_zero_of_hasDerivWithinAt'
code: /-- If `f' : ℂ → E` is a derivative of a complex differentiable function on the circle
`Metric.sphere c |R|`, then `∮ z in C(c, R), f' z = 0`. -/
theorem integral_eq_zero_of_hasDerivWithinAt' {f f' : ℂ → E} {c : ℂ} {R : ℝ}
    (h : ∀ z ∈ sphere c |R|, HasDerivWithinAt f (f' z) (sphere c |R|) z) :
    (∮ z in C(c, R), f' z) = 0 
goal: (∮ (z : ℂ) in C(c, R), f' z) = 0

full_name: circleIntegral.integral_eq_zero_of_hasDerivWithinAt
code: /-- If `f' : ℂ → E` is a derivative of a complex differentiable function on the circle
`Metric.sphere c R`, then `∮ z in C(c, R), f' z = 0`. -/
theorem integral_eq_zero_of_hasDerivWithinAt {f f' : ℂ → E} {c : ℂ} {R : ℝ} (hR : 0 ≤ R)
    (h : ∀ z ∈ sphere c R, HasDerivWithinAt f (f' z) (sphere c R) z) : (∮ z in C(c, R), f' z) = 0 
goal: (∮ (z : ℂ) in C(c, R), f' z) = 0

full_name: circleIntegral.integral_sub_zpow_of_undef
code: /-- If `n < 0` and `|w - c| = |R|`, then `(z - w) ^ n` is not circle integrable on the circle with
center `c` and radius `|R|`, so the integral `∮ z in C(c, R), (z - w) ^ n` is equal to zero. -/
theorem integral_sub_zpow_of_undef {n : ℤ} {c w : ℂ} {R : ℝ} (hn : n < 0)
    (hw : w ∈ sphere c |R|) : (∮ z in C(c, R), (z - w) ^ n) = 0 
goal: (∮ (z : ℂ) in C(c, R), (z - w) ^ n) = 0

full_name: circleIntegral.integral_sub_zpow_of_ne
code: /-- If `n ≠ -1` is an integer number, then the integral of `(z - w) ^ n` over the circle equals
zero. -/
theorem integral_sub_zpow_of_ne {n : ℤ} (hn : n ≠ -1) (c w : ℂ) (R : ℝ) :
    (∮ z in C(c, R), (z - w) ^ n) = 0 
goal: (∮ (z : ℂ) in C(c, R), (z - w) ^ n) = 0

full_name: cauchyPowerSeries
code: /-- The power series that is equal to
$\frac{1}{2πi}\sum_{n=0}^{\infty}
  \oint_{|z-c|=R} \left(\frac{w-c}{z - c}\right)^n \frac{1}{z-c}f(z)\,dz$ at
`w - c`. The coefficients of this power series depend only on `f ∘ circleMap c R`, and the power
series converges to `f w` if `f` is differentiable on the closed ball `Metric.closedBall c R` and
`w` belongs to the corresponding open ball. For any circle integrable function `f`, this power
series converges to the Cauchy integral for `f`. -/
def cauchyPowerSeries (f : ℂ → E) (c : ℂ) (R : ℝ) : FormalMultilinearSeries ℂ ℂ E 
goal: FormalMultilinearSeries ℂ ℂ E

full_name: cauchyPowerSeries_apply
code: theorem cauchyPowerSeries_apply (f : ℂ → E) (c : ℂ) (R : ℝ) (n : ℕ) (w : ℂ) :
    (cauchyPowerSeries f c R n fun _ => w) =
      (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (w / (z - c)) ^ n • (z - c)⁻¹ • f z 
goal: ((cauchyPowerSeries f c R n) fun x => w) =
  (2 * ↑π * Complex.I)⁻¹ • ∮ (z : ℂ) in C(c, R), (w / (z - c)) ^ n • (z - c)⁻¹ • f z

full_name: norm_cauchyPowerSeries_le
code: theorem norm_cauchyPowerSeries_le (f : ℂ → E) (c : ℂ) (R : ℝ) (n : ℕ) :
    ‖cauchyPowerSeries f c R n‖ ≤
      ((2 * π)⁻¹ * ∫ θ : ℝ in (0)..2 * π, ‖f (circleMap c R θ)‖) * |R|⁻¹ ^ n 
goal: ‖cauchyPowerSeries f c R n‖ ≤ ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0 ..2 * π, ‖f (circleMap c R θ)‖) * |R|⁻¹ ^ n

full_name: le_radius_cauchyPowerSeries
code: theorem le_radius_cauchyPowerSeries (f : ℂ → E) (c : ℂ) (R : ℝ≥0) :
    ↑R ≤ (cauchyPowerSeries f c R).radius 
goal: ↑R ≤ (cauchyPowerSeries f c ↑R).radius

full_name: hasSum_two_pi_I_cauchyPowerSeries_integral
code: /-- For any circle integrable function `f`, the power series `cauchyPowerSeries f c R` multiplied
by `2πI` converges to the integral `∮ z in C(c, R), (z - w)⁻¹ • f z` on the open disc
`Metric.ball c R`. -/
theorem hasSum_two_pi_I_cauchyPowerSeries_integral {f : ℂ → E} {c : ℂ} {R : ℝ} {w : ℂ}
    (hf : CircleIntegrable f c R) (hw : abs w < R) :
    HasSum (fun n : ℕ => ∮ z in C(c, R), (w / (z - c)) ^ n • (z - c)⁻¹ • f z)
      (∮ z in C(c, R), (z - (c + w))⁻¹ • f z) 
goal: HasSum (fun n => ∮ (z : ℂ) in C(c, R), (w / (z - c)) ^ n • (z - c)⁻¹ • f z)
  (∮ (z : ℂ) in C(c, R), (z - (c + w))⁻¹ • f z)

full_name: hasSum_cauchyPowerSeries_integral
code: /-- For any circle integrable function `f`, the power series `cauchyPowerSeries f c R`, `R > 0`,
converges to the Cauchy integral `(2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z` on the open
disc `Metric.ball c R`. -/
theorem hasSum_cauchyPowerSeries_integral {f : ℂ → E} {c : ℂ} {R : ℝ} {w : ℂ}
    (hf : CircleIntegrable f c R) (hw : abs w < R) :
    HasSum (fun n => cauchyPowerSeries f c R n fun _ => w)
      ((2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - (c + w))⁻¹ • f z) 
goal: HasSum (fun n => (cauchyPowerSeries f c R n) fun x => w)
  ((2 * ↑π * Complex.I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - (c + w))⁻¹ • f z)

full_name: sum_cauchyPowerSeries_eq_integral
code: /-- For any circle integrable function `f`, the power series `cauchyPowerSeries f c R`, `R > 0`,
converges to the Cauchy integral `(2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z` on the open
disc `Metric.ball c R`. -/
theorem sum_cauchyPowerSeries_eq_integral {f : ℂ → E} {c : ℂ} {R : ℝ} {w : ℂ}
    (hf : CircleIntegrable f c R) (hw : abs w < R) :
    (cauchyPowerSeries f c R).sum w = (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - (c + w))⁻¹ • f z 
goal: (cauchyPowerSeries f c R).sum w = (2 * ↑π * Complex.I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - (c + w))⁻¹ • f z

full_name: hasFPowerSeriesOn_cauchy_integral
code: /-- For any circle integrable function `f`, the power series `cauchyPowerSeries f c R`, `R > 0`,
converges to the Cauchy integral `(2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z` on the open
disc `Metric.ball c R`. -/
theorem hasFPowerSeriesOn_cauchy_integral {f : ℂ → E} {c : ℂ} {R : ℝ≥0}
    (hf : CircleIntegrable f c R) (hR : 0 < R) :
    HasFPowerSeriesOnBall (fun w => (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z)
      (cauchyPowerSeries f c R) c R 
goal: HasFPowerSeriesOnBall (fun w => (2 * ↑π * Complex.I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z)
  (cauchyPowerSeries f c ↑R) c ↑R

full_name: circleIntegral.integral_sub_inv_of_mem_ball
code: /-- Integral $\oint_{|z-c|=R} \frac{dz}{z-w} = 2πi$ whenever $|w-c| < R$. -/
theorem integral_sub_inv_of_mem_ball {c w : ℂ} {R : ℝ} (hw : w ∈ ball c R) :
    (∮ z in C(c, R), (z - w)⁻¹) = 2 * π * I 
goal: (∮ (z : ℂ) in C(c, R), (z - w)⁻¹) = 2 * ↑π * Complex.I