File: Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean

full_name: Complex.hasSum_cos'
code: theorem Complex.hasSum_cos' (z : ℂ) :
    HasSum (fun n : ℕ => (z * Complex.I) ^ (2 * n) / ↑(2 * n)!) (Complex.cos z) 
goal: HasSum (fun n => (z * Complex.I) ^ (2 * n) / ↑(2 * n)!) (Complex.cos z)

full_name: Complex.hasSum_sin'
code: theorem Complex.hasSum_sin' (z : ℂ) :
    HasSum (fun n : ℕ => (z * Complex.I) ^ (2 * n + 1) / ↑(2 * n + 1)! / Complex.I)
      (Complex.sin z) 
goal: HasSum (fun n => (z * Complex.I) ^ (2 * n + 1) / ↑(2 * n + 1)! / Complex.I) (Complex.sin z)

full_name: Complex.hasSum_cos
code: /-- The power series expansion of `Complex.cos`. -/
theorem Complex.hasSum_cos (z : ℂ) :
    HasSum (fun n : ℕ => (-1) ^ n * z ^ (2 * n) / ↑(2 * n)!) (Complex.cos z) 
goal: HasSum (fun n => (-1) ^ n * z ^ (2 * n) / ↑(2 * n)!) (Complex.cos z)

full_name: Complex.hasSum_sin
code: /-- The power series expansion of `Complex.sin`. -/
theorem Complex.hasSum_sin (z : ℂ) :
    HasSum (fun n : ℕ => (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)!) (Complex.sin z) 
goal: HasSum (fun n => (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)!) (Complex.sin z)

full_name: Complex.cos_eq_tsum'
code: theorem Complex.cos_eq_tsum' (z : ℂ) :
    Complex.cos z = ∑' n : ℕ, (z * Complex.I) ^ (2 * n) / ↑(2 * n)! 
goal: Complex.cos z = ∑' (n : ℕ), (z * Complex.I) ^ (2 * n) / ↑(2 * n)!

full_name: Complex.sin_eq_tsum'
code: theorem Complex.sin_eq_tsum' (z : ℂ) :
    Complex.sin z = ∑' n : ℕ, (z * Complex.I) ^ (2 * n + 1) / ↑(2 * n + 1)! / Complex.I 
goal: Complex.sin z = ∑' (n : ℕ), (z * Complex.I) ^ (2 * n + 1) / ↑(2 * n + 1)! / Complex.I

full_name: Complex.cos_eq_tsum
code: theorem Complex.cos_eq_tsum (z : ℂ) :
    Complex.cos z = ∑' n : ℕ, (-1) ^ n * z ^ (2 * n) / ↑(2 * n)! 
goal: Complex.cos z = ∑' (n : ℕ), (-1) ^ n * z ^ (2 * n) / ↑(2 * n)!

full_name: Complex.sin_eq_tsum
code: theorem Complex.sin_eq_tsum (z : ℂ) :
    Complex.sin z = ∑' n : ℕ, (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)! 
goal: Complex.sin z = ∑' (n : ℕ), (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)!

full_name: Real.hasSum_cos
code: /-- The power series expansion of `Real.cos`. -/
theorem Real.hasSum_cos (r : ℝ) :
    HasSum (fun n : ℕ => (-1) ^ n * r ^ (2 * n) / ↑(2 * n)!) (Real.cos r) 
goal: HasSum (fun n => (-1) ^ n * r ^ (2 * n) / ↑(2 * n)!) (Real.cos r)

full_name: Real.hasSum_sin
code: /-- The power series expansion of `Real.sin`. -/
theorem Real.hasSum_sin (r : ℝ) :
    HasSum (fun n : ℕ => (-1) ^ n * r ^ (2 * n + 1) / ↑(2 * n + 1)!) (Real.sin r) 
goal: HasSum (fun n => (-1) ^ n * r ^ (2 * n + 1) / ↑(2 * n + 1)!) (Real.sin r)

full_name: Real.cos_eq_tsum
code: theorem Real.cos_eq_tsum (r : ℝ) : Real.cos r = ∑' n : ℕ, (-1) ^ n * r ^ (2 * n) / ↑(2 * n)! 
goal: Real.cos r = ∑' (n : ℕ), (-1) ^ n * r ^ (2 * n) / ↑(2 * n)!

full_name: Real.sin_eq_tsum
code: theorem Real.sin_eq_tsum (r : ℝ) :
    Real.sin r = ∑' n : ℕ, (-1) ^ n * r ^ (2 * n + 1) / ↑(2 * n + 1)! 
goal: Real.sin r = ∑' (n : ℕ), (-1) ^ n * r ^ (2 * n + 1) / ↑(2 * n + 1)!

full_name: Complex.hasSum_cosh
code: /-- The power series expansion of `Complex.cosh`. -/
lemma hasSum_cosh (z : ℂ) : HasSum (fun n ↦ z ^ (2 * n) / ↑(2 * n)!) (cosh z) 
goal: HasSum (fun n => z ^ (2 * n) / ↑(2 * n)!) (Complex.cosh z)

full_name: Complex.hasSum_sinh
code: /-- The power series expansion of `Complex.sinh`. -/
lemma hasSum_sinh (z : ℂ) : HasSum (fun n ↦ z ^ (2 * n + 1) / ↑(2 * n + 1)!) (sinh z) 
goal: HasSum (fun n => z ^ (2 * n + 1) / ↑(2 * n + 1)!) (Complex.sinh z)

full_name: Complex.cosh_eq_tsum
code: lemma cosh_eq_tsum (z : ℂ) : cosh z = ∑' n, z ^ (2 * n) / ↑(2 * n)! 
goal: Complex.cosh z = ∑' (n : ℕ), z ^ (2 * n) / ↑(2 * n)!

full_name: Complex.sinh_eq_tsum
code: lemma sinh_eq_tsum (z : ℂ) : sinh z = ∑' n, z ^ (2 * n + 1) / ↑(2 * n + 1)! 
goal: Complex.sinh z = ∑' (n : ℕ), z ^ (2 * n + 1) / ↑(2 * n + 1)!

full_name: Real.hasSum_cosh
code: /-- The power series expansion of `Real.cosh`. -/
lemma hasSum_cosh (r : ℝ) : HasSum (fun n  ↦ r ^ (2 * n) / ↑(2 * n)!) (cosh r) 
goal: HasSum (fun n => r ^ (2 * n) / ↑(2 * n)!) (Real.cosh r)

full_name: Real.hasSum_sinh
code: /-- The power series expansion of `Real.sinh`. -/
lemma hasSum_sinh (r : ℝ) : HasSum (fun n ↦ r ^ (2 * n + 1) / ↑(2 * n + 1)!) (sinh r) 
goal: HasSum (fun n => r ^ (2 * n + 1) / ↑(2 * n + 1)!) (Real.sinh r)

full_name: Real.cosh_eq_tsum
code: lemma cosh_eq_tsum (r : ℝ) : cosh r = ∑' n, r ^ (2 * n) / ↑(2 * n)! 
goal: Real.cosh r = ∑' (n : ℕ), r ^ (2 * n) / ↑(2 * n)!

full_name: Real.sinh_eq_tsum
code: lemma sinh_eq_tsum (r : ℝ) : sinh r = ∑' n, r ^ (2 * n + 1) / ↑(2 * n + 1)! 
goal: Real.sinh r = ∑' (n : ℕ), r ^ (2 * n + 1) / ↑(2 * n + 1)!

full_name: Real.cosh_le_exp_half_sq
code: lemma cosh_le_exp_half_sq (x : ℝ) : cosh x ≤ exp (x ^ 2 / 2) 
goal: Real.cosh x ≤ Real.exp (x ^ 2 / 2)