Update specialFunctions_Integrals.lean

This commit is contained in:
Stefan Kebekus 2024-08-13 09:46:30 +02:00
parent 0cb1914b18
commit bc8fed96b0
1 changed files with 47 additions and 1 deletions

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@ -5,6 +5,8 @@ import Mathlib.MeasureTheory.Integral.CircleIntegral
open scoped Interval Topology
open Real Filter MeasureTheory intervalIntegral
-- The following theorem was suggested by Gareth Ma on Zulip
theorem logInt
{t : }
(ht : 0 < t) :
@ -63,6 +65,50 @@ theorem logInt
lemma int₁ :
∫ x in (0)..(2 * π), log ‖circleMap 0 1 x - 1‖ = 0 := by
dsimp [circleMap]
have {x : } : Complex.normSq (circleMap 0 1 x - 1) = 2 - 2 * cos x := by
calc Complex.normSq (circleMap 0 1 x - 1)
_ = (cos x - 1) * (cos x - 1) + sin x * sin x := by
dsimp [circleMap]
rw [Complex.normSq_apply]
simp
_ = sin x ^ 2 + cos x ^ 2 + 1 - 2 * cos x := by
ring
_ = 2 - 2 * cos x := by
rw [sin_sq_add_cos_sq]
norm_num
have {x : } : 2 - 2 * cos x = 4 * sin (x / 2) ^ 2 := by
calc 2 - 2 * cos x
_ = 2 - 2 * cos (2 * (x / 2)) := by
rw [← mul_div_assoc]
congr; norm_num
_ = 4 - 4 * Real.cos (x / 2) ^ 2 := by
rw [cos_two_mul]
norm_num
_ = 4 * sin (x / 2) ^ 2 := by
nth_rw 1 [← mul_one 4]
nth_rw 1 [← sin_sq_add_cos_sq (x / 2)]
rw [mul_add]
abel
dsimp [Complex.abs]
sorry
sorry
sorry