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namespace OAI
open intervalIntegral Filter
open scoped Topology
noncomputable section
namespace CylinderCovering.RationalIntegrals
theorem ceiling_integral (α c₀ c₂ : ℝ) :
(∫ y in (-1 / 4 : ℝ)..(1 / 4 : ℝ), α * (c₀ + c₂ * y ^ 2) * y ^ 2) =
α * (c₀ / 96 + c₂ / 2560) := by
have hf : (fun y : ℝ => α * (c₀ + c₂ * y ^ 2) * y ^ 2) =
(fun y : ℝ => (α * c₀) * y ^ 2 + (α * c₂) * y ^ 4) := by
funext y
ring
rw [hf, integral_add]
· simp only [integral_const_mul, integral_pow]
norm_num
ring
· exact (continuous_const.mul (continuous_id.pow 2)).intervalIntegrable _ _
· exact (continuous_const.mul (continuous_id.pow 4)).intervalIntegrable _ _
theorem limiting_ceiling_integral :
(∫ y in (-1 / 4 : ℝ)..(1 / 4 : ℝ), ((1 / 16 : ℝ) + y ^ 2) * y ^ 2) =
(1 / 960 : ℝ) := by
have h := ceiling_integral 1 (1 / 16) 1
norm_num at h ⊢
exact h
theorem limiting_energy_angular_integral :
(∫ z in (-1 / 2 : ℝ)..(1 / 2 : ℝ),
((1 / 16 : ℝ) + z ^ 2 / 4) ^ 2 + (1 / 2 : ℝ) * (z / 2) ^ 2) =
(17 / 960 : ℝ) := by
have hf : (fun z : ℝ => ((1 / 16 : ℝ) + z ^ 2 / 4) ^ 2 +
(1 / 2 : ℝ) * (z / 2) ^ 2) =
(fun z : ℝ => (1 / 256 : ℝ) + (5 / 32 : ℝ) * z ^ 2 +
(1 / 16 : ℝ) * z ^ 4) := by
funext z
ring
rw [hf, integral_add, integral_add]
· simp only [integral_const, integral_const_mul, integral_pow]
norm_num
· exact continuous_const.intervalIntegrable _ _
· exact (continuous_const.mul (continuous_id.pow 2)).intervalIntegrable _ _
· exact (continuous_const.add (continuous_const.mul (continuous_id.pow 2))).intervalIntegrable _ _
· exact (continuous_const.mul (continuous_id.pow 4)).intervalIntegrable _ _
theorem limiting_energy_integral :
(1 / 2 : ℝ) * (∫ r in (0 : ℝ)..(1 / 2 : ℝ), r) *
(∫ z in (-1 / 2 : ℝ)..(1 / 2 : ℝ),
((1 / 16 : ℝ) + z ^ 2 / 4) ^ 2 + (1 / 2 : ℝ) * (z / 2) ^ 2) =
(17 / 15360 : ℝ) := by
rw [limiting_energy_angular_integral, integral_id]
norm_num
theorem limiting_coefficient_gap :
(17 / 15360 : ℝ) - (1 / 960 : ℝ) = (1 / 15360 : ℝ) ∧
0 < (17 / 15360 : ℝ) - (1 / 960 : ℝ) := by
norm_num
theorem rational_ceiling_coefficient_formula (d : ℝ) :
(1 / (1 + 2 * d)) *
(((1 / 2 : ℝ) * (1 / 2 + 2 * d) / (16 * (1 / 2 + d) ^ 2)) / 96 +
(1 / (4 * (1 / 2 + d) ^ 2)) / 2560) =
(2 + 5 * d) / (1920 * (1 + 2 * d) ^ 3) := by
by_cases hd : 1 + 2 * d = 0
· have he : d = -(1 / 2 : ℝ) := by linarith
subst d
norm_num
· have h : (1 / 2 : ℝ) + d ≠ 0 := by intro he; apply hd; linarith
field_simp
ring
theorem rational_ceiling_integral_tendsto :
Tendsto (fun d : ℝ =>
∫ y in (-1 / 4 : ℝ)..(1 / 4 : ℝ),
(1 / (1 + 2 * d)) *
(((1 / 2 : ℝ) * (1 / 2 + 2 * d) / (16 * (1 / 2 + d) ^ 2)) +
(1 / (4 * (1 / 2 + d) ^ 2)) * y ^ 2) * y ^ 2)
(𝓝 0) (𝓝 (1 / 960 : ℝ)) := by
simp_rw [ceiling_integral, rational_ceiling_coefficient_formula]
have h : ContinuousAt (fun d : ℝ =>
(2 + 5 * d) / (1920 * (1 + 2 * d) ^ 3)) 0 := by
fun_prop (disch := norm_num)
convert h.tendsto using 1
norm_num
theorem rational_ceiling_integral_tendsto_right :
Tendsto (fun d : ℝ =>
∫ y in (-1 / 4 : ℝ)..(1 / 4 : ℝ),
(1 / (1 + 2 * d)) *
(((1 / 2 : ℝ) * (1 / 2 + 2 * d) / (16 * (1 / 2 + d) ^ 2)) +
(1 / (4 * (1 / 2 + d) ^ 2)) * y ^ 2) * y ^ 2)
(𝓝[Set.Ioi 0] 0) (𝓝 (1 / 960 : ℝ)) :=
rational_ceiling_integral_tendsto.mono_left nhdsWithin_le_nhds
theorem uncentered_ceiling_integral (c : ℝ) :
(∫ u in (-c)..c, (c ^ 2 * u ^ 2 / 2 + u ^ 4 / 2)) = (8 / 15 : ℝ) * c ^ 5 := by
have he : (fun u : ℝ => c ^ 2 * u ^ 2 / 2 + u ^ 4 / 2) =
(fun u : ℝ => (c ^ 2 / 2) * u ^ 2 + (1 / 2 : ℝ) * u ^ 4) := by
funext u
ring
rw [he, intervalIntegral.integral_add]
· rw [intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul,
integral_pow, integral_pow]
norm_num
ring
· exact (continuous_const.mul (continuous_id.pow 2)).intervalIntegrable _ _
· exact (continuous_const.mul (continuous_id.pow 4)).intervalIntegrable _ _
theorem uncentered_energy_integral (c : ℝ) :
(1 / 4 : ℝ) * (∫ b in (-c)..c, ((c ^ 2 + b ^ 2) ^ 2 / 4 + b ^ 2)) =
c ^ 3 / 6 + 7 * c ^ 5 / 30 := by
have he : (fun b : ℝ => (c ^ 2 + b ^ 2) ^ 2 / 4 + b ^ 2) =
(fun b : ℝ => c ^ 4 / 4 + (c ^ 2 / 2 + 1) * b ^ 2 + (1 / 4 : ℝ) * b ^ 4) := by
funext b
ring
rw [he, intervalIntegral.integral_add, intervalIntegral.integral_add]
· rw [intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul,
integral_pow, integral_pow, intervalIntegral.integral_const]
norm_num
ring
· exact continuous_const.intervalIntegrable _ _
· exact (continuous_const.mul (continuous_id.pow 2)).intervalIntegrable _ _
· exact (continuous_const.add (continuous_const.mul (continuous_id.pow 2))).intervalIntegrable _ _
· exact (continuous_const.mul (continuous_id.pow 4)).intervalIntegrable _ _
theorem uncentered_limiting_gap {c : ℝ} (hc : c ^ 2 = (1 / 2 : ℝ)) :
(1 / 4 : ℝ) * (∫ b in (-c)..c, ((c ^ 2 + b ^ 2) ^ 2 / 4 + b ^ 2)) -
(∫ u in (-c)..c, (c ^ 2 * u ^ 2 / 2 + u ^ 4 / 2)) = c ^ 3 / 60 := by
rw [uncentered_energy_integral, uncentered_ceiling_integral]
have h5 : c ^ 5 = c ^ 3 / 2 := by
calc c ^ 5 = c ^ 3 * c ^ 2 := by ring
_ = c ^ 3 / 2 := by rw [hc]; ring
rw [h5]
ring
end CylinderCovering.RationalIntegrals
end
end OAI
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