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180bc03 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 | import OAI.NumberTheory.TwoPointCorrelations.HalaszPhaseCells
namespace OAI
/-! Exact primitive of the Archimedean phase, including the zero endpoint. -/
namespace TwoPointCorrelations
open MeasureTheory
lemma halasz_power_phase_eq_cpow (t x : ℝ) (hx : 0 < x) :
halaszPowerPhase t x = (x : ℂ) ^ ((-t : ℂ) * Complex.I) := by
rw [Complex.cpow_def_of_ne_zero (by exact_mod_cast hx.ne'),
← Complex.ofReal_log hx.le]
unfold halaszPowerPhase
congr 1
push_cast
ring
lemma halasz_power_phase_cpow_primitive (t x : ℝ) (hx : 0 ≤ x) :
(x : ℂ) ^ ((-t : ℂ) * Complex.I + 1) = (x : ℂ) * halaszPowerPhase t x := by
by_cases hx0 : x = 0
· subst x
simp only [Complex.ofReal_zero, zero_mul]
apply Complex.zero_cpow
intro h
have hr := congrArg Complex.re h
norm_num [Complex.mul_re] at hr
· have hxp : 0 < x := lt_of_le_of_ne hx (Ne.symm hx0)
rw [Complex.cpow_add _ _ (by exact_mod_cast hx0), Complex.cpow_one,
halasz_power_phase_eq_cpow t x hxp, mul_comm]
lemma halasz_power_phase_integral (t a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) :
(∫ x in a..b, halaszPowerPhase t x) =
((b : ℂ) * halaszPowerPhase t b - (a : ℂ) * halaszPowerPhase t a) /
(1 + (-t : ℂ) * Complex.I) := by
have he : (∫ x in a..b, halaszPowerPhase t x) =
∫ x : ℝ in a..b, (x : ℂ) ^ ((-t : ℂ) * Complex.I) := by
apply intervalIntegral.integral_congr_uIoo
intro x hx
exact halasz_power_phase_eq_cpow t x
(lt_of_le_of_lt (le_min ha hb) hx.1)
rw [he, integral_cpow (Or.inl (by norm_num [Complex.mul_re])),
halasz_power_phase_cpow_primitive t b hb,
halasz_power_phase_cpow_primitive t a ha, add_comm ((-t : ℂ) * Complex.I) 1]
lemma halasz_power_phase_intervalIntegrable (t a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) :
IntervalIntegrable (halaszPowerPhase t) volume a b := by
apply (intervalIntegral.intervalIntegrable_cpow' (r := (-t : ℂ) * Complex.I)
(by norm_num [Complex.mul_re])).congr_uIoo
intro x hx
exact (halasz_power_phase_eq_cpow t x
(lt_of_le_of_lt (le_min ha hb) hx.1)).symm
lemma halasz_power_phase_integral_norm (t a b : ℝ) :
‖∫ x in a..b, halaszPowerPhase t x‖ ≤ |b - a| := by
simpa using intervalIntegral.norm_integral_le_of_norm_le_const
(a := a) (b := b) (C := 1) (f := halaszPowerPhase t)
(fun x _ => (halasz_power_phase_norm t x).le)
end TwoPointCorrelations
end OAI
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