File: Mathlib/Analysis/Complex/AbelLimit.lean

full_name: Complex.stolzSet
code: /-- The Stolz set for a given `M`, roughly teardrop-shaped with the tip at 1 but tending to the
open unit disc as `M` tends to infinity. -/
def stolzSet (M : ℝ) : Set ℂ 
goal: Set ℂ

full_name: Complex.stolzCone
code: /-- The cone to the left of `1` with angle `2θ` such that `tan θ = s`. -/
def stolzCone (s : ℝ) : Set ℂ 
goal: Set ℂ

full_name: Complex.stolzSet_empty
code: theorem stolzSet_empty {M : ℝ} (hM : M ≤ 1) : stolzSet M = ∅ 
goal: Complex.stolzSet M = ∅

full_name: Complex.nhdsWithin_lt_le_nhdsWithin_stolzSet
code: theorem nhdsWithin_lt_le_nhdsWithin_stolzSet {M : ℝ} (hM : 1 < M) :
    (𝓝[<] 1).map ofReal' ≤ 𝓝[stolzSet M] 1 
goal: Filter.map Complex.ofReal' (𝓝[<] 1) ≤ 𝓝[Complex.stolzSet M] 1

full_name: _private.Mathlib.Analysis.Complex.AbelLimit.0.Complex.stolzCone_subset_stolzSet_aux'
code: private lemma stolzCone_subset_stolzSet_aux' (s : ℝ) :
    ∃ M ε, 0 < M ∧ 0 < ε ∧ ∀ x y, 0 < x → x < ε → |y| < s * x →
      sqrt (x ^ 2 + y ^ 2) < M * (1 - sqrt ((1 - x) ^ 2 + y ^ 2)) 
goal: ∃ M ε, 0 < M ∧ 0 < ε ∧ ∀ (x y : ℝ), 0 < x → x < ε → |y| < s * x → √(x ^ 2 + y ^ 2) < M * (1 - √((1 - x) ^ 2 + y ^ 2))

full_name: Complex.stolzCone_subset_stolzSet_aux
code: lemma stolzCone_subset_stolzSet_aux {s : ℝ} (hs : 0 < s) :
    ∃ M ε, 0 < M ∧ 0 < ε ∧ {z : ℂ | 1 - ε < z.re} ∩ stolzCone s ⊆ stolzSet M 
goal: ∃ M ε, 0 < M ∧ 0 < ε ∧ {z | 1 - ε < z.re} ∩ Complex.stolzCone s ⊆ Complex.stolzSet M

full_name: Complex.nhdsWithin_stolzCone_le_nhdsWithin_stolzSet
code: lemma nhdsWithin_stolzCone_le_nhdsWithin_stolzSet {s : ℝ} (hs : 0 < s) :
    ∃ M, 𝓝[stolzCone s] 1 ≤ 𝓝[stolzSet M] 1 
goal: ∃ M, 𝓝[Complex.stolzCone s] 1 ≤ 𝓝[Complex.stolzSet M] 1

full_name: Complex.abel_aux
code: /-- Auxiliary lemma for Abel's limit theorem. The difference between the sum `l` at 1 and the
power series's value at a point `z` away from 1 can be rewritten as `1 - z` times a power series
whose coefficients are tail sums of `l`. -/
lemma abel_aux (h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) {z : ℂ} (hz : ‖z‖ < 1) :
    Tendsto (fun n ↦ (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i)
      atTop (𝓝 (l - ∑' n, f n * z ^ n)) 
goal: Filter.Tendsto (fun n => (1 - z) * ∑ i ∈ Finset.range n, (l - ∑ j ∈ Finset.range (i + 1), f j) * z ^ i) Filter.atTop
  (𝓝 (l - ∑' (n : ℕ), f n * z ^ n))

full_name: Complex.tendsto_tsum_powerSeries_nhdsWithin_stolzSet
code: /-- **Abel's limit theorem**. Given a power series converging at 1, the corresponding function
is continuous at 1 when approaching 1 within a fixed Stolz set. -/
theorem tendsto_tsum_powerSeries_nhdsWithin_stolzSet
    (h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) {M : ℝ} :
    Tendsto (fun z ↦ ∑' n, f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l) 
goal: Filter.Tendsto (fun z => ∑' (n : ℕ), f n * z ^ n) (𝓝[Complex.stolzSet M] 1) (𝓝 l)

full_name: Complex.tendsto_tsum_powerSeries_nhdsWithin_stolzCone
code: /-- **Abel's limit theorem**. Given a power series converging at 1, the corresponding function
is continuous at 1 when approaching 1 within any fixed Stolz cone. -/
theorem tendsto_tsum_powerSeries_nhdsWithin_stolzCone
    (h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) {s : ℝ} (hs : 0 < s) :
    Tendsto (fun z ↦ ∑' n, f n * z ^ n) (𝓝[stolzCone s] 1) (𝓝 l) 
goal: Filter.Tendsto (fun z => ∑' (n : ℕ), f n * z ^ n) (𝓝[Complex.stolzCone s] 1) (𝓝 l)

full_name: Complex.tendsto_tsum_powerSeries_nhdsWithin_lt
code: theorem tendsto_tsum_powerSeries_nhdsWithin_lt
    (h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) :
    Tendsto (fun z ↦ ∑' n, f n * z ^ n) ((𝓝[<] 1).map ofReal') (𝓝 l) 
goal: Filter.Tendsto (fun z => ∑' (n : ℕ), f n * z ^ n) (Filter.map Complex.ofReal' (𝓝[<] 1)) (𝓝 l)

full_name: Real.tendsto_tsum_powerSeries_nhdsWithin_lt
code: /-- **Abel's limit theorem**. Given a real power series converging at 1, the corresponding function
is continuous at 1 when approaching 1 from the left. -/
theorem tendsto_tsum_powerSeries_nhdsWithin_lt
    (h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) :
    Tendsto (fun x ↦ ∑' n, f n * x ^ n) (𝓝[<] 1) (𝓝 l) 
goal: Filter.Tendsto (fun x => ∑' (n : ℕ), f n * x ^ n) (𝓝[<] 1) (𝓝 l)