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Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

Trigonometric functions as sums of infinite series #

In this file we express trigonometric functions in terms of their series expansion.

Main results #

cos and sin for ℝ and ℂ #

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theorem Complex.hasSum_cos' (z : ℂ) :
HasSum (fun (n : ℕ) => (z * Complex.I) ^ (2 * n) / ↑(2 * n).factorial) (Complex.cos z)
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theorem Complex.hasSum_sin' (z : ℂ) :
HasSum (fun (n : ℕ) => (z * Complex.I) ^ (2 * n + 1) / ↑(2 * n + 1).factorial / Complex.I) (Complex.sin z)
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theorem Complex.hasSum_cos (z : ℂ) :
HasSum (fun (n : ℕ) => (-1) ^ n * z ^ (2 * n) / ↑(2 * n).factorial) (Complex.cos z)

The power series expansion of Complex.cos.

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theorem Complex.hasSum_sin (z : ℂ) :
HasSum (fun (n : ℕ) => (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1).factorial) (Complex.sin z)

The power series expansion of Complex.sin.

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theorem Complex.cos_eq_tsum' (z : ℂ) :
Complex.cos z = ∑' (n : ℕ), (z * Complex.I) ^ (2 * n) / ↑(2 * n).factorial
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theorem Complex.sin_eq_tsum' (z : ℂ) :
Complex.sin z = ∑' (n : ℕ), (z * Complex.I) ^ (2 * n + 1) / ↑(2 * n + 1).factorial / Complex.I
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theorem Complex.cos_eq_tsum (z : ℂ) :
Complex.cos z = ∑' (n : ℕ), (-1) ^ n * z ^ (2 * n) / ↑(2 * n).factorial
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theorem Complex.sin_eq_tsum (z : ℂ) :
Complex.sin z = ∑' (n : ℕ), (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1).factorial
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theorem Real.hasSum_cos (r : ℝ) :
HasSum (fun (n : ℕ) => (-1) ^ n * r ^ (2 * n) / ↑(2 * n).factorial) (Real.cos r)

The power series expansion of Real.cos.

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theorem Real.hasSum_sin (r : ℝ) :
HasSum (fun (n : ℕ) => (-1) ^ n * r ^ (2 * n + 1) / ↑(2 * n + 1).factorial) (Real.sin r)

The power series expansion of Real.sin.

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theorem Real.cos_eq_tsum (r : ℝ) :
Real.cos r = ∑' (n : ℕ), (-1) ^ n * r ^ (2 * n) / ↑(2 * n).factorial
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theorem Real.sin_eq_tsum (r : ℝ) :
Real.sin r = ∑' (n : ℕ), (-1) ^ n * r ^ (2 * n + 1) / ↑(2 * n + 1).factorial

cosh and sinh for ℝ and ℂ #

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theorem Complex.hasSum_cosh (z : ℂ) :
HasSum (fun (n : ℕ) => z ^ (2 * n) / ↑(2 * n).factorial) (Complex.cosh z)

The power series expansion of Complex.cosh.

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theorem Complex.hasSum_sinh (z : ℂ) :
HasSum (fun (n : ℕ) => z ^ (2 * n + 1) / ↑(2 * n + 1).factorial) (Complex.sinh z)

The power series expansion of Complex.sinh.

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theorem Complex.cosh_eq_tsum (z : ℂ) :
Complex.cosh z = ∑' (n : ℕ), z ^ (2 * n) / ↑(2 * n).factorial
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theorem Complex.sinh_eq_tsum (z : ℂ) :
Complex.sinh z = ∑' (n : ℕ), z ^ (2 * n + 1) / ↑(2 * n + 1).factorial
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theorem Real.hasSum_cosh (r : ℝ) :
HasSum (fun (n : ℕ) => r ^ (2 * n) / ↑(2 * n).factorial) (Real.cosh r)

The power series expansion of Real.cosh.

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theorem Real.hasSum_sinh (r : ℝ) :
HasSum (fun (n : ℕ) => r ^ (2 * n + 1) / ↑(2 * n + 1).factorial) (Real.sinh r)

The power series expansion of Real.sinh.

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theorem Real.cosh_eq_tsum (r : ℝ) :
Real.cosh r = ∑' (n : ℕ), r ^ (2 * n) / ↑(2 * n).factorial
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theorem Real.sinh_eq_tsum (r : ℝ) :
Real.sinh r = ∑' (n : ℕ), r ^ (2 * n + 1) / ↑(2 * n + 1).factorial
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theorem Real.cosh_le_exp_half_sq (x : ℝ) :
Real.cosh x ≤ Real.exp (x ^ 2 / 2)