File: Mathlib/Analysis/Complex/TaylorSeries.lean

full_name: Complex.hasSum_taylorSeries_on_ball
code: /-- A function that is complex differentiable on the open ball of radius `r` around `c`
is given by evaluating its Taylor series at `c` on this open ball. -/
lemma hasSum_taylorSeries_on_ball :
    HasSum (fun n : ℕ ↦ (n ! : ℂ)⁻¹ • (z - c) ^ n • iteratedDeriv n f c) (f z) 
goal: HasSum (fun n => (↑n.factorial)⁻¹ • (z - c) ^ n • iteratedDeriv n f c) (f z)

full_name: Complex.taylorSeries_eq_on_ball
code: /-- A function that is complex differentiable on the open ball of radius `r` around `c`
is given by evaluating its Taylor series at `c` on this open ball. -/
lemma taylorSeries_eq_on_ball :
    ∑' n : ℕ, (n ! : ℂ)⁻¹ • (z - c) ^ n • iteratedDeriv n f c = f z 
goal: ∑' (n : ℕ), (↑n.factorial)⁻¹ • (z - c) ^ n • iteratedDeriv n f c = f z

full_name: Complex.taylorSeries_eq_on_ball'
code: /-- A function that is complex differentiable on the open ball of radius `r` around `c`
is given by evaluating its Taylor series at `c` on this open ball. -/
lemma taylorSeries_eq_on_ball' {f : ℂ → ℂ} (hf : DifferentiableOn ℂ f (Metric.ball c r)) :
    ∑' n : ℕ, (n ! : ℂ)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = f z 
goal: ∑' (n : ℕ), (↑n.factorial)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = f z

full_name: Complex.hasSum_taylorSeries_on_emetric_ball
code: /-- A function that is complex differentiable on the open ball of radius `r ≤ ∞` around `c`
is given by evaluating its Taylor series at `c` on this open ball. -/
lemma hasSum_taylorSeries_on_emetric_ball :
    HasSum (fun n : ℕ ↦ (n ! : ℂ)⁻¹ • (z - c) ^ n • iteratedDeriv n f c) (f z) 
goal: HasSum (fun n => (↑n.factorial)⁻¹ • (z - c) ^ n • iteratedDeriv n f c) (f z)

full_name: Complex.taylorSeries_eq_on_emetric_ball
code: /-- A function that is complex differentiable on the open ball of radius `r ≤ ∞` around `c`
is given by evaluating its Taylor series at `c` on this open ball. -/
lemma taylorSeries_eq_on_emetric_ball :
    ∑' n : ℕ, (n ! : ℂ)⁻¹ • (z - c) ^ n • iteratedDeriv n f c = f z 
goal: ∑' (n : ℕ), (↑n.factorial)⁻¹ • (z - c) ^ n • iteratedDeriv n f c = f z

full_name: Complex.taylorSeries_eq_on_emetric_ball'
code: /-- A function that is complex differentiable on the open ball of radius `r ≤ ∞` around `c`
is given by evaluating its Taylor series at `c` on this open ball. -/
lemma taylorSeries_eq_on_emetric_ball' {f : ℂ → ℂ} (hf : DifferentiableOn ℂ f (EMetric.ball c r)) :
    ∑' n : ℕ, (n ! : ℂ)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = f z 
goal: ∑' (n : ℕ), (↑n.factorial)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = f z

full_name: Complex.hasSum_taylorSeries_of_entire
code: /-- A function that is complex differentiable on the complex plane is given by evaluating
its Taylor series at any point `c`. -/
lemma hasSum_taylorSeries_of_entire :
    HasSum (fun n : ℕ ↦ (n ! : ℂ)⁻¹ • (z - c) ^ n • iteratedDeriv n f c) (f z) 
goal: HasSum (fun n => (↑n.factorial)⁻¹ • (z - c) ^ n • iteratedDeriv n f c) (f z)

full_name: Complex.taylorSeries_eq_of_entire
code: /-- A function that is complex differentiable on the complex plane is given by evaluating
its Taylor series at any point `c`. -/
lemma taylorSeries_eq_of_entire :
    ∑' n : ℕ, (n ! : ℂ)⁻¹ • (z - c) ^ n • iteratedDeriv n f c = f z 
goal: ∑' (n : ℕ), (↑n.factorial)⁻¹ • (z - c) ^ n • iteratedDeriv n f c = f z

full_name: Complex.taylorSeries_eq_of_entire'
code: /-- A function that is complex differentiable on the complex plane is given by evaluating
its Taylor series at any point `c`. -/
lemma taylorSeries_eq_of_entire' {f : ℂ → ℂ} (hf : Differentiable ℂ f) :
    ∑' n : ℕ, (n ! : ℂ)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = f z 
goal: ∑' (n : ℕ), (↑n.factorial)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = f z